all about matrices

7/29/2019 All About Matrices

1/14

Matrix (mathematics)From Wikipedia, the free encyclopedia

"Matrix theory" redirects here. For the physics topic, see Matrix string theory.

Specific elements of a matrix are often denoted by a variable with twosubscripts. For instance, a2,1 represents the

element at the second row and first column of a matrix A.

Inmathematics, a matrix (plural matrices) is arectangulararrayofnumbers,symbols, orexpressions,

arranged inrows andcolumns.[1][2]

The individual items in a matrix are called its elements orentries. An

example of a matrix with 2 rows and 3 columns is

Matrices of the same size can beaddedor subtracted element by element. But the rule formatrix

multiplicationis that two matrices can be multiplied only when the number of columns in the first equals the

number of rows in the second. A major application of matrices is to represen tlinear transformations, that is,

generalizations oflinear functionssuch as f(x) = 4x. For example, therotationofvectorsin

threedimensionalspace is a linear transformation. IfR is arotation matrixand v is acolumn vector(a

matrix with only one column) describing thepositionof a point in space, the product Rv is a column vector

describing the position of that point after a rotation. The product of two matrices is a matrix that represents

thecompositionof two linear transformations. Another application of matrices is in the solution of a system

of linear equations. If the matrix issquare, it is possible to deduce some of its properties by computing

itsdeterminant. For example, a square matrix has aninverseif and only ifits determinant is

notzero.Eigenvalues and eigenvectorsprovide insight into thegeometryof linear transformations.

Applications of matrices are found in most scientific fields. In every branch ofphysics,

includingclassical mechanics,optics,electromagnetism,quantum mechanics, andquantum

electrodynamics, they are used to study physical phenomena, such as the motion ofrigid bodies.

Incomputer graphics, they are used to project a 3-dimensional image onto a 2-dimensional screen.

Inprobability theoryandstatistics,stochastic matricesare used to describe sets of probabilities; for

instance, they are used within thePageRankalgorithm that ranks the pages in a Google
http://en.wikipedia.org/wiki/Matrix_string_theoryhttp://en.wikipedia.org/wiki/Matrix_string_theoryhttp://en.wikipedia.org/wiki/Matrix_string_theoryhttp://en.wikipedia.org/wiki/Index_notationhttp://en.wikipedia.org/wiki/Index_notationhttp://en.wikipedia.org/wiki/Index_notationhttp://en.wikipedia.org/wiki/Mathematicshttp://en.wikipedia.org/wiki/Mathematicshttp://en.wikipedia.org/wiki/Mathematicshttp://en.wikipedia.org/wiki/Rectanglehttp://en.wikipedia.org/wiki/Rectanglehttp://en.wiktionary.org/wiki/arrayhttp://en.wiktionary.org/wiki/arrayhttp://en.wiktionary.org/wiki/arrayhttp://en.wikipedia.org/wiki/Numberhttp://en.wikipedia.org/wiki/Numberhttp://en.wikipedia.org/wiki/Numberhttp://en.wikipedia.org/wiki/Symbol_(formal)http://en.wikipedia.org/wiki/Symbol_(formal)http://en.wikipedia.org/wiki/Symbol_(formal)http://en.wikipedia.org/wiki/Expression_(mathematics)http://en.wikipedia.org/wiki/Expression_(mathematics)http://en.wikipedia.org/wiki/Expression_(mathematics)http://en.wiktionary.org/wiki/rowhttp://en.wiktionary.org/wiki/rowhttp://en.wiktionary.org/wiki/rowhttp://en.wiktionary.org/wiki/columnhttp://en.wiktionary.org/wiki/columnhttp://en.wiktionary.org/wiki/columnhttp://en.wikipedia.org/wiki/Matrix_(mathematics)#cite_note-1http://en.wikipedia.org/wiki/Matrix_(mathematics)#cite_note-1http://en.wikipedia.org/wiki/Matrix_(mathematics)#cite_note-1http://en.wikipedia.org/wiki/Matrix_additionhttp://en.wikipedia.org/wiki/Matrix_additionhttp://en.wikipedia.org/wiki/Matrix_additionhttp://en.wikipedia.org/wiki/Matrix_multiplicationhttp://en.wikipedia.org/wiki/Matrix_multiplicationhttp://en.wikipedia.org/wiki/Matrix_multiplicationhttp://en.wikipedia.org/wiki/Matrix_multiplicationhttp://en.wikipedia.org/wiki/Linear_transformationhttp://en.wikipedia.org/wiki/Linear_transformationhttp://en.wikipedia.org/wiki/Linear_transformationhttp://en.wikipedia.org/wiki/Linear_functionshttp://en.wikipedia.org/wiki/Linear_functionshttp://en.wikipedia.org/wiki/Linear_functionshttp://en.wikipedia.org/wiki/Rotation_(mathematics)http://en.wikipedia.org/wiki/Rotation_(mathematics)http://en.wikipedia.org/wiki/Rotation_(mathematics)http://en.wikipedia.org/wiki/Euclidean_vectorhttp://en.wikipedia.org/wiki/Euclidean_vectorhttp://en.wikipedia.org/wiki/Euclidean_vectorhttp://en.wikipedia.org/wiki/Dimensionhttp://en.wikipedia.org/wiki/Dimensionhttp://en.wikipedia.org/wiki/Dimensionhttp://en.wikipedia.org/wiki/Rotation_matrixhttp://en.wikipedia.org/wiki/Rotation_matrixhttp://en.wikipedia.org/wiki/Rotation_matrixhttp://en.wikipedia.org/wiki/Column_vectorhttp://en.wikipedia.org/wiki/Column_vectorhttp://en.wikipedia.org/wiki/Column_vectorhttp://en.wikipedia.org/wiki/Position_(vector)http://en.wikipedia.org/wiki/Position_(vector)http://en.wikipedia.org/wiki/Position_(vector)http://en.wikipedia.org/wiki/Function_compositionhttp://en.wikipedia.org/wiki/Function_compositionhttp://en.wikipedia.org/wiki/Function_compositionhttp://en.wikipedia.org/wiki/System_of_linear_equationshttp://en.wikipedia.org/wiki/System_of_linear_equationshttp://en.wikipedia.org/wiki/System_of_linear_equationshttp://en.wikipedia.org/wiki/System_of_linear_equationshttp://en.wikipedia.org/wiki/Matrix_(mathematics)#Square_matriceshttp://en.wikipedia.org/wiki/Matrix_(mathematics)#Square_matriceshttp://en.wikipedia.org/wiki/Matrix_(mathematics)#Square_matriceshttp://en.wikipedia.org/wiki/Determinanthttp://en.wikipedia.org/wiki/Determinanthttp://en.wikipedia.org/wiki/Determinanthttp://en.wikipedia.org/wiki/Inverse_matrixhttp://en.wikipedia.org/wiki/Inverse_matrixhttp://en.wikipedia.org/wiki/If_and_only_ifhttp://en.wikipedia.org/wiki/If_and_only_ifhttp://en.wikipedia.org/wiki/If_and_only_ifhttp://en.wikipedia.org/wiki/Zerohttp://en.wikipedia.org/wiki/Zerohttp://en.wikipedia.org/wiki/Zerohttp://en.wikipedia.org/wiki/Eigenvalues_and_eigenvectorshttp://en.wikipedia.org/wiki/Eigenvalues_and_eigenvectorshttp://en.wikipedia.org/wiki/Eigenvalues_and_eigenvectorshttp://en.wikipedia.org/wiki/Geometryhttp://en.wikipedia.org/wiki/Geometryhttp://en.wikipedia.org/wiki/Geometryhttp://en.wikipedia.org/wiki/Physicshttp://en.wikipedia.org/wiki/Physicshttp://en.wikipedia.org/wiki/Physicshttp://en.wikipedia.org/wiki/Classical_mechanicshttp://en.wikipedia.org/wiki/Classical_mechanicshttp://en.wikipedia.org/wiki/Classical_mechanicshttp://en.wikipedia.org/wiki/Opticshttp://en.wikipedia.org/wiki/Opticshttp://en.wikipedia.org/wiki/Opticshttp://en.wikipedia.org/wiki/Electromagnetismhttp://en.wikipedia.org/wiki/Electromagnetismhttp://en.wikipedia.org/wiki/Quantum_mechanicshttp://en.wikipedia.org/wiki/Quantum_mechanicshttp://en.wikipedia.org/wiki/Quantum_mechanicshttp://en.wikipedia.org/wiki/Quantum_electrodynamicshttp://en.wikipedia.org/wiki/Quantum_electrodynamicshttp://en.wikipedia.org/wiki/Quantum_electrodynamicshttp://en.wikipedia.org/wiki/Quantum_electrodynamicshttp://en.wikipedia.org/wiki/Rigid_bodyhttp://en.wikipedia.org/wiki/Rigid_bodyhttp://en.wikipedia.org/wiki/Rigid_bodyhttp://en.wikipedia.org/wiki/Computer_graphicshttp://en.wikipedia.org/wiki/Computer_graphicshttp://en.wikipedia.org/wiki/Computer_graphicshttp://en.wikipedia.org/wiki/Probability_theoryhttp://en.wikipedia.org/wiki/Probability_theoryhttp://en.wikipedia.org/wiki/Probability_theoryhttp://en.wikipedia.org/wiki/Statisticshttp://en.wikipedia.org/wiki/Statisticshttp://en.wikipedia.org/wiki/Statisticshttp://en.wikipedia.org/wiki/Stochastic_matrixhttp://en.wikipedia.org/wiki/Stochastic_matrixhttp://en.wikipedia.org/wiki/Stochastic_matrixhttp://en.wikipedia.org/wiki/PageRankhttp://en.wikipedia.org/wiki/PageRankhttp://en.wikipedia.org/wiki/PageRankhttp://en.wikipedia.org/wiki/File:Matrix.svghttp://en.wikipedia.org/wiki/File:Matrix.svghttp://en.wikipedia.org/wiki/PageRankhttp://en.wikipedia.org/wiki/Stochastic_matrixhttp://en.wikipedia.org/wiki/Statisticshttp://en.wikipedia.org/wiki/Probability_theoryhttp://en.wikipedia.org/wiki/Computer_graphicshttp://en.wikipedia.org/wiki/Rigid_bodyhttp://en.wikipedia.org/wiki/Quantum_electrodynamicshttp://en.wikipedia.org/wiki/Quantum_electrodynamicshttp://en.wikipedia.org/wiki/Quantum_mechanicshttp://en.wikipedia.org/wiki/Electromagnetismhttp://en.wikipedia.org/wiki/Opticshttp://en.wikipedia.org/wiki/Classical_mechanicshttp://en.wikipedia.org/wiki/Physicshttp://en.wikipedia.org/wiki/Geometryhttp://en.wikipedia.org/wiki/Eigenvalues_and_eigenvectorshttp://en.wikipedia.org/wiki/Zerohttp://en.wikipedia.org/wiki/If_and_only_ifhttp://en.wikipedia.org/wiki/Inverse_matrixhttp://en.wikipedia.org/wiki/Determinanthttp://en.wikipedia.org/wiki/Matrix_(mathematics)#Square_matriceshttp://en.wikipedia.org/wiki/System_of_linear_equationshttp://en.wikipedia.org/wiki/System_of_linear_equationshttp://en.wikipedia.org/wiki/Function_compositionhttp://en.wikipedia.org/wiki/Position_(vector)http://en.wikipedia.org/wiki/Column_vectorhttp://en.wikipedia.org/wiki/Rotation_matrixhttp://en.wikipedia.org/wiki/Dimensionhttp://en.wikipedia.org/wiki/Euclidean_vectorhttp://en.wikipedia.org/wiki/Rotation_(mathematics)http://en.wikipedia.org/wiki/Linear_functionshttp://en.wikipedia.org/wiki/Linear_transformationhttp://en.wikipedia.org/wiki/Matrix_multiplicationhttp://en.wikipedia.org/wiki/Matrix_multiplicationhttp://en.wikipedia.org/wiki/Matrix_additionhttp://en.wikipedia.org/wiki/Matrix_(mathematics)#cite_note-1http://en.wikipedia.org/wiki/Matrix_(mathematics)#cite_note-1http://en.wiktionary.org/wiki/columnhttp://en.wiktionary.org/wiki/rowhttp://en.wikipedia.org/wiki/Expression_(mathematics)http://en.wikipedia.org/wiki/Symbol_(formal)http://en.wikipedia.org/wiki/Numberhttp://en.wiktionary.org/wiki/arrayhttp://en.wikipedia.org/wiki/Rectanglehttp://en.wikipedia.org/wiki/Mathematicshttp://en.wikipedia.org/wiki/Index_notationhttp://en.wikipedia.org/wiki/Matrix_string_theory


2/14

search.[3]

Matrix calculusgeneralizes classicalanalyticalnotions such

asderivativesandexponentialsto higher dimensions.

A major branch ofnumerical analysisis devoted to the development of efficient algorithms for matrix

computations, a subject that is centuries old and is today an expanding area of research .Matrixdecomposition methodssimplify computations, both theoretically and practically. Algorithms that are

tailored to particular matrix structures, such assparse matricesandnear-diagonal matrices, expedite

computations infinite element methodand other computations. Infinite matrices occur in planetary

theory and in atomic theory. A simple example of an infinite matrix is the matrix representing

thederivativeoperator, which acts on theTaylor seriesof a function.

Definition[edit source|editbeta]

A matrixis a rectangular array ofnumbersor other mathematical objects, for which operations such

asadditionandmultiplicationare defined.[4]

Most commonly, a matrix over afieldFis a rectangulararray of scalars from F.

[5][6]Most of this article focuses on realand complex matrices, i.e., matrices

whose elements arereal numbersorcomplex numbers, respectively. More general types of entries

are discussedbelow. For instance, this is a real matrix:

The numbers, symbols or expressions in the matrix are called its entries or its elements. The

horizontal and vertical lines in a matrix are called rows and columns, respectively.

Size[edit source|editbeta]

The size of a matrix is defined by the number of rows and columns that it contains. A matrix

with m rows and n columns is called an m n matrix orm-by-n matrix, while m and n are called

itsdimensions. For example, the matrix A above is a 3 2 matrix.

Matrices which have a single row are calledrow vectors, and those which have a single column

are calledcolumn vectors. A matrix which has the same number of rows and columns is called

asquare matrix. A matrix with an infinite number of rows or columns (or both) is called aninfinite

matrix. In some contexts such as computer algebra programs it is useful to consider a matrix with

no rows or no columns, called anempty matrix.

Name Size Example Description

Row

vector1 n A matrix with one row, sometimes used to represent a vector
http://en.wikipedia.org/wiki/Matrix_(mathematics)#cite_note-3http://en.wikipedia.org/wiki/Matrix_(mathematics)#cite_note-3http://en.wikipedia.org/wiki/Matrix_calculushttp://en.wikipedia.org/wiki/Matrix_calculushttp://en.wikipedia.org/wiki/Matrix_calculushttp://en.wikipedia.org/wiki/Mathematical_analysishttp://en.wikipedia.org/wiki/Mathematical_analysishttp://en.wikipedia.org/wiki/Derivativehttp://en.wikipedia.org/wiki/Derivativehttp://en.wikipedia.org/wiki/Derivativehttp://en.wikipedia.org/wiki/Exponentialshttp://en.wikipedia.org/wiki/Exponentialshttp://en.wikipedia.org/wiki/Exponentialshttp://en.wikipedia.org/wiki/Numerical_analysishttp://en.wikipedia.org/wiki/Numerical_analysishttp://en.wikipedia.org/wiki/Numerical_analysishttp://en.wikipedia.org/wiki/Matrix_(mathematics)#Matrix_decomposition_methodshttp://en.wikipedia.org/wiki/Matrix_(mathematics)#Matrix_decomposition_methodshttp://en.wikipedia.org/wiki/Matrix_(mathematics)#Matrix_decomposition_methodshttp://en.wikipedia.org/wiki/Matrix_(mathematics)#Matrix_decomposition_methodshttp://en.wikipedia.org/wiki/Sparse_matrixhttp://en.wikipedia.org/wiki/Sparse_matrixhttp://en.wikipedia.org/wiki/Sparse_matrixhttp://en.wikipedia.org/wiki/Diagonal_matrixhttp://en.wikipedia.org/wiki/Diagonal_matrixhttp://en.wikipedia.org/wiki/Diagonal_matrixhttp://en.wikipedia.org/wiki/Finite_element_methodhttp://en.wikipedia.org/wiki/Finite_element_methodhttp://en.wikipedia.org/wiki/Finite_element_methodhttp://en.wikipedia.org/wiki/Derivative_(calculus)http://en.wikipedia.org/wiki/Derivative_(calculus)http://en.wikipedia.org/wiki/Derivative_(calculus)http://en.wikipedia.org/wiki/Taylor_serieshttp://en.wikipedia.org/wiki/Taylor_serieshttp://en.wikipedia.org/wiki/Taylor_serieshttp://en.wikipedia.org/w/index.php?title=Matrix_(mathematics)&action=edit&section=1http://en.wikipedia.org/w/index.php?title=Matrix_(mathematics)&action=edit&section=1http://en.wikipedia.org/w/index.php?title=Matrix_(mathematics)&action=edit&section=1http://en.wikipedia.org/w/index.php?title=Matrix_(mathematics)&veaction=edit&section=1http://en.wikipedia.org/w/index.php?title=Matrix_(mathematics)&veaction=edit&section=1http://en.wikipedia.org/w/index.php?title=Matrix_(mathematics)&veaction=edit&section=1http://en.wikipedia.org/w/index.php?title=Matrix_(mathematics)&veaction=edit&section=1http://en.wikipedia.org/wiki/Numberhttp://en.wikipedia.org/wiki/Numberhttp://en.wikipedia.org/wiki/Numberhttp://en.wikipedia.org/wiki/Matrix_(mathematics)#Basic_operationshttp://en.wikipedia.org/wiki/Matrix_(mathematics)#Basic_operationshttp://en.wikipedia.org/wiki/Matrix_(mathematics)#Basic_operationshttp://en.wikipedia.org/wiki/Matrix_(mathematics)#Matrix_multiplicationhttp://en.wikipedia.org/wiki/Matrix_(mathematics)#Matrix_multiplicationhttp://en.wikipedia.org/wiki/Matrix_(mathematics)#Matrix_multiplicationhttp://en.wikipedia.org/wiki/Matrix_(mathematics)#cite_note-4http://en.wikipedia.org/wiki/Matrix_(mathematics)#cite_note-4http://en.wikipedia.org/wiki/Matrix_(mathematics)#cite_note-4http://en.wikipedia.org/wiki/Field_(mathematics)http://en.wikipedia.org/wiki/Field_(mathematics)http://en.wikipedia.org/wiki/Field_(mathematics)http://en.wikipedia.org/wiki/Matrix_(mathematics)#cite_note-5http://en.wikipedia.org/wiki/Matrix_(mathematics)#cite_note-5http://en.wikipedia.org/wiki/Matrix_(mathematics)#cite_note-5http://en.wikipedia.org/wiki/Real_numbershttp://en.wikipedia.org/wiki/Real_numbershttp://en.wikipedia.org/wiki/Real_numbershttp://en.wikipedia.org/wiki/Complex_numbershttp://en.wikipedia.org/wiki/Complex_numbershttp://en.wikipedia.org/wiki/Complex_numbershttp://en.wikipedia.org/wiki/Matrix_(mathematics)#More_general_entrieshttp://en.wikipedia.org/wiki/Matrix_(mathematics)#More_general_entrieshttp://en.wikipedia.org/wiki/Matrix_(mathematics)#More_general_entrieshttp://en.wikipedia.org/w/index.php?title=Matrix_(mathematics)&action=edit&section=2http://en.wikipedia.org/w/index.php?title=Matrix_(mathematics)&action=edit&section=2http://en.wikipedia.org/w/index.php?title=Matrix_(mathematics)&action=edit&section=2http://en.wikipedia.org/w/index.php?title=Matrix_(mathematics)&veaction=edit&section=2http://en.wikipedia.org/w/index.php?title=Matrix_(mathematics)&veaction=edit&section=2http://en.wikipedia.org/w/index.php?title=Matrix_(mathematics)&veaction=edit&section=2http://en.wikipedia.org/w/index.php?title=Matrix_(mathematics)&veaction=edit&section=2http://en.wikipedia.org/wiki/Row_vectorhttp://en.wikipedia.org/wiki/Row_vectorhttp://en.wikipedia.org/wiki/Row_vectorhttp://en.wikipedia.org/wiki/Column_vectorshttp://en.wikipedia.org/wiki/Column_vectorshttp://en.wikipedia.org/wiki/Column_vectorshttp://en.wikipedia.org/wiki/Square_matrixhttp://en.wikipedia.org/wiki/Square_matrixhttp://en.wikipedia.org/wiki/Matrix_(mathematics)#Infinite_matriceshttp://en.wikipedia.org/wiki/Matrix_(mathematics)#Infinite_matriceshttp://en.wikipedia.org/wiki/Matrix_(mathematics)#Infinite_matriceshttp://en.wikipedia.org/wiki/Matrix_(mathematics)#Infinite_matriceshttp://en.wikipedia.org/wiki/Matrix_(mathematics)#Empty_matriceshttp://en.wikipedia.org/wiki/Matrix_(mathematics)#Empty_matriceshttp://en.wikipedia.org/wiki/Matrix_(mathematics)#Empty_matriceshttp://en.wikipedia.org/wiki/Row_vectorhttp://en.wikipedia.org/wiki/Row_vectorhttp://en.wikipedia.org/wiki/Row_vectorhttp://en.wikipedia.org/wiki/Row_vectorhttp://en.wikipedia.org/wiki/Row_vectorhttp://en.wikipedia.org/wiki/Matrix_(mathematics)#Empty_matriceshttp://en.wikipedia.org/wiki/Matrix_(mathematics)#Infinite_matriceshttp://en.wikipedia.org/wiki/Matrix_(mathematics)#Infinite_matriceshttp://en.wikipedia.org/wiki/Square_matrixhttp://en.wikipedia.org/wiki/Column_vectorshttp://en.wikipedia.org/wiki/Row_vectorhttp://en.wikipedia.org/w/index.php?title=Matrix_(mathematics)&veaction=edit&section=2http://en.wikipedia.org/w/index.php?title=Matrix_(mathematics)&action=edit&section=2http://en.wikipedia.org/wiki/Matrix_(mathematics)#More_general_entrieshttp://en.wikipedia.org/wiki/Complex_numbershttp://en.wikipedia.org/wiki/Real_numbershttp://en.wikipedia.org/wiki/Matrix_(mathematics)#cite_note-5http://en.wikipedia.org/wiki/Matrix_(mathematics)#cite_note-5http://en.wikipedia.org/wiki/Field_(mathematics)http://en.wikipedia.org/wiki/Matrix_(mathematics)#cite_note-4http://en.wikipedia.org/wiki/Matrix_(mathematics)#Matrix_multiplicationhttp://en.wikipedia.org/wiki/Matrix_(mathematics)#Basic_operationshttp://en.wikipedia.org/wiki/Numberhttp://en.wikipedia.org/w/index.php?title=Matrix_(mathematics)&veaction=edit&section=1http://en.wikipedia.org/w/index.php?title=Matrix_(mathematics)&action=edit&section=1http://en.wikipedia.org/wiki/Taylor_serieshttp://en.wikipedia.org/wiki/Derivative_(calculus)http://en.wikipedia.org/wiki/Finite_element_methodhttp://en.wikipedia.org/wiki/Diagonal_matrixhttp://en.wikipedia.org/wiki/Sparse_matrixhttp://en.wikipedia.org/wiki/Matrix_(mathematics)#Matrix_decomposition_methodshttp://en.wikipedia.org/wiki/Matrix_(mathematics)#Matrix_decomposition_methodshttp://en.wikipedia.org/wiki/Numerical_analysishttp://en.wikipedia.org/wiki/Exponentialshttp://en.wikipedia.org/wiki/Derivativehttp://en.wikipedia.org/wiki/Mathematical_analysishttp://en.wikipedia.org/wiki/Matrix_calculushttp://en.wikipedia.org/wiki/Matrix_(mathematics)#cite_note-3


3/14

Column

vectorn 1 A matrix with one column, sometimes used to represent a vector

Square

matrixn n

A matrix with the same number of rows and columns,

sometimes used to represent alinear transformationfrom a

vector space to itself, such asreflection,rotation, orshearing.

Notation[edit source|editbeta]

Matrices are commonly written in box brackets:

An alternative notation uses largeparenthesesinstead ofbox brackets:

The specifics of symbolic matrix notation varies widely, with some prevailing trends.

Matrices are usually symbolized usingupper-caseletters (such as A in the examples

above), while the correspondinglower-caseletters, with two subscript indices (e.g., a11,

ora1,1), represent the entries. In addition to using upper-case letters to symbolize

matrices, many authors use a specialtypographical style, commonly boldface upright

(non-italic), to further distinguish matrices from other mathematical objects. An

alternative notation involves the use of a double-underline with the variable name, with or

without boldface style, (e.g., ).

The entry in the i-th row andj-th column of a matrix A is sometimes referred to as the i,j,

(i,j), or (i,j)th

entry of the matrix, and most commonly denoted as ai,j, oraij. Alternative

notations for that entry areA[i,j] orAi,j. For example, the (1,3) entry of the following

matrix A is 5 (also denoted a13, a1,3,A[1,3] orA1,3):

Sometimes, the entries of a matrix can be defined by a formula such as ai,j= f(i,j).For example, each of the entries of the following matrix A is determined by aij= ij.
http://en.wikipedia.org/wiki/Column_vectorhttp://en.wikipedia.org/wiki/Column_vectorhttp://en.wikipedia.org/wiki/Column_vectorhttp://en.wikipedia.org/wiki/Square_matrixhttp://en.wikipedia.org/wiki/Square_matrixhttp://en.wikipedia.org/wiki/Matrix_(mathematics)#Linear_transformationshttp://en.wikipedia.org/wiki/Matrix_(mathematics)#Linear_transformationshttp://en.wikipedia.org/wiki/Matrix_(mathematics)#Linear_transformationshttp://en.wikipedia.org/wiki/Reflection_(mathematics)http://en.wikipedia.org/wiki/Reflection_(mathematics)http://en.wikipedia.org/wiki/Rotation_(mathematics)http://en.wikipedia.org/wiki/Rotation_(mathematics)http://en.wikipedia.org/wiki/Rotation_(mathematics)http://en.wikipedia.org/wiki/Shear_mappinghttp://en.wikipedia.org/wiki/Shear_mappinghttp://en.wikipedia.org/wiki/Shear_mappinghttp://en.wikipedia.org/w/index.php?title=Matrix_(mathematics)&action=edit&section=3http://en.wikipedia.org/w/index.php?title=Matrix_(mathematics)&action=edit&section=3http://en.wikipedia.org/w/index.php?title=Matrix_(mathematics)&action=edit&section=3http://en.wikipedia.org/w/index.php?title=Matrix_(mathematics)&veaction=edit&section=3http://en.wikipedia.org/w/index.php?title=Matrix_(mathematics)&veaction=edit&section=3http://en.wikipedia.org/w/index.php?title=Matrix_(mathematics)&veaction=edit&section=3http://en.wikipedia.org/w/index.php?title=Matrix_(mathematics)&veaction=edit&section=3http://en.wikipedia.org/wiki/Parenshttp://en.wikipedia.org/wiki/Parenshttp://en.wikipedia.org/wiki/Parenshttp://en.wikipedia.org/wiki/Box_brackethttp://en.wikipedia.org/wiki/Box_brackethttp://en.wikipedia.org/wiki/Box_brackethttp://en.wikipedia.org/wiki/Upper-casehttp://en.wikipedia.org/wiki/Upper-casehttp://en.wikipedia.org/wiki/Upper-casehttp://en.wikipedia.org/wiki/Lower-casehttp://en.wikipedia.org/wiki/Lower-casehttp://en.wikipedia.org/wiki/Lower-casehttp://en.wikipedia.org/wiki/Emphasis_(typography)http://en.wikipedia.org/wiki/Emphasis_(typography)http://en.wikipedia.org/wiki/Emphasis_(typography)http://en.wikipedia.org/wiki/Emphasis_(typography)http://en.wikipedia.org/wiki/Lower-casehttp://en.wikipedia.org/wiki/Upper-casehttp://en.wikipedia.org/wiki/Box_brackethttp://en.wikipedia.org/wiki/Parenshttp://en.wikipedia.org/w/index.php?title=Matrix_(mathematics)&veaction=edit&section=3http://en.wikipedia.org/w/index.php?title=Matrix_(mathematics)&action=edit&section=3http://en.wikipedia.org/wiki/Shear_mappinghttp://en.wikipedia.org/wiki/Rotation_(mathematics)http://en.wikipedia.org/wiki/Reflection_(mathematics)http://en.wikipedia.org/wiki/Matrix_(mathematics)#Linear_transformationshttp://en.wikipedia.org/wiki/Square_matrixhttp://en.wikipedia.org/wiki/Square_matrixhttp://en.wikipedia.org/wiki/Column_vectorhttp://en.wikipedia.org/wiki/Column_vector


4/14

In this case, the matrix itself is sometimes defined by that formula, within square

brackets or double parenthesis. For example, the matrix above is defined as A =

[i-j], orA = ((i-j)). If matrix size ism n, the above-mentioned formula f(i,j) is

valid for any i= 1, ..., m and anyj= 1, ..., n. This can be either specified

separately, or using m n as a subscript. For instance, the matrix A above is 4

3 and can be defined as A = [ij] (i= 1, ..., 4;j= 1, 2, 3), orA = [ij]43.

Some programming languages utilize doubly-subscripted arrays (or arrays of

arrays) to represent an m--n matrix. Some programming languages start the

numbering of array indexes at zero, in which case the entries of an m-by-

n matrix are indexed by 0 im 1 and 0 jn 1.[7]

This article follows the

more common convention in mathematical writing where enumeration starts

from 1.

Thesetof all m-by-n matrices is denoted (m, n).

Basic operations[edit source|editbeta]

There are a number of basic operations that can be applied to modify matrices, called matrix

addition, scalar multiplication, transposition, matrix multiplication, row operations, and submatrix.[9]

Addition, scalar multiplication and transposition[edit source|editbeta]

Main articles:Matrix addition,Scalar multiplication, andTranspose

Operation Definition Example

Addition

The sumA+B of two m-by-

n matrices A and B is

calculated entrywise:

(A + B)i,j= Ai,j+ Bi,j,

where 1 im and

1 jn.

Scalar

multipli

cation

The scalar multiplication

cA of a matrix A and a

numberc(also called

ascalarin the parlance

ofabstract algebra) is

given by multiplying

every entry ofA by c:

(cA)i,j= c Ai,j.
http://en.wikipedia.org/wiki/Matrix_(mathematics)#cite_note-7http://en.wikipedia.org/wiki/Matrix_(mathematics)#cite_note-7http://en.wikipedia.org/wiki/Matrix_(mathematics)#cite_note-7http://en.wikipedia.org/wiki/Set_(mathematics)http://en.wikipedia.org/wiki/Set_(mathematics)http://en.wikipedia.org/wiki/Set_(mathematics)http://en.wikipedia.org/w/index.php?title=Matrix_(mathematics)&action=edit&section=4http://en.wikipedia.org/w/index.php?title=Matrix_(mathematics)&action=edit&section=4http://en.wikipedia.org/w/index.php?title=Matrix_(mathematics)&action=edit&section=4http://en.wikipedia.org/w/index.php?title=Matrix_(mathematics)&veaction=edit&section=4http://en.wikipedia.org/w/index.php?title=Matrix_(mathematics)&veaction=edit&section=4http://en.wikipedia.org/w/index.php?title=Matrix_(mathematics)&veaction=edit&section=4http://en.wikipedia.org/w/index.php?title=Matrix_(mathematics)&veaction=edit&section=4http://en.wikipedia.org/wiki/Matrix_(mathematics)#cite_note-9http://en.wikipedia.org/wiki/Matrix_(mathematics)#cite_note-9http://en.wikipedia.org/wiki/Matrix_(mathematics)#cite_note-9http://en.wikipedia.org/w/index.php?title=Matrix_(mathematics)&action=edit&section=5http://en.wikipedia.org/w/index.php?title=Matrix_(mathematics)&action=edit&section=5http://en.wikipedia.org/w/index.php?title=Matrix_(mathematics)&action=edit&section=5http://en.wikipedia.org/w/index.php?title=Matrix_(mathematics)&veaction=edit&section=5http://en.wikipedia.org/w/index.php?title=Matrix_(mathematics)&veaction=edit&section=5http://en.wikipedia.org/w/index.php?title=Matrix_(mathematics)&veaction=edit&section=5http://en.wikipedia.org/w/index.php?title=Matrix_(mathematics)&veaction=edit&section=5http://en.wikipedia.org/wiki/Matrix_additionhttp://en.wikipedia.org/wiki/Matrix_additionhttp://en.wikipedia.org/wiki/Matrix_additionhttp://en.wikipedia.org/wiki/Scalar_multiplicationhttp://en.wikipedia.org/wiki/Scalar_multiplicationhttp://en.wikipedia.org/wiki/Scalar_multiplicationhttp://en.wikipedia.org/wiki/Transposehttp://en.wikipedia.org/wiki/Transposehttp://en.wikipedia.org/wiki/Transposehttp://en.wikipedia.org/wiki/Matrix_additionhttp://en.wikipedia.org/wiki/Matrix_additionhttp://en.wikipedia.org/wiki/Scalar_multiplicationhttp://en.wikipedia.org/wiki/Scalar_multiplicationhttp://en.wikipedia.org/wiki/Scalar_multiplicationhttp://en.wikipedia.org/wiki/Scalar_multiplicationhttp://en.wikipedia.org/wiki/Scalar_(mathematics)http://en.wikipedia.org/wiki/Scalar_(mathematics)http://en.wikipedia.org/wiki/Scalar_(mathematics)http://en.wikipedia.org/wiki/Abstract_algebrahttp://en.wikipedia.org/wiki/Abstract_algebrahttp://en.wikipedia.org/wiki/Abstract_algebrahttp://en.wikipedia.org/wiki/Abstract_algebrahttp://en.wikipedia.org/wiki/Scalar_(mathematics)http://en.wikipedia.org/wiki/Scalar_multiplicationhttp://en.wikipedia.org/wiki/Scalar_multiplicationhttp://en.wikipedia.org/wiki/Scalar_multiplicationhttp://en.wikipedia.org/wiki/Matrix_additionhttp://en.wikipedia.org/wiki/Transposehttp://en.wikipedia.org/wiki/Scalar_multiplicationhttp://en.wikipedia.org/wiki/Matrix_additionhttp://en.wikipedia.org/w/index.php?title=Matrix_(mathematics)&veaction=edit&section=5http://en.wikipedia.org/w/index.php?title=Matrix_(mathematics)&action=edit&section=5http://en.wikipedia.org/wiki/Matrix_(mathematics)#cite_note-9http://en.wikipedia.org/w/index.php?title=Matrix_(mathematics)&veaction=edit&section=4http://en.wikipedia.org/w/index.php?title=Matrix_(mathematics)&action=edit&section=4http://en.wikipedia.org/wiki/Set_(mathematics)http://en.wikipedia.org/wiki/Matrix_(mathematics)#cite_note-7


5/14

Transpose

The transpose of

an m-by-

n matrix A is the n-

by-m matrix AT

(also

denoted A

tr

or

t

A)formed by turning

rows into columns

and vice versa:

(AT)i,j= Aj,i.

Familiar properties of numbers extend to these operations of matrices: for example, addition

iscommutative, i.e., the matrix sum does not depend on the order of the

summands:A + B = B + A.[10]

The transpose is compatible with addition and scalar multiplication, as

expressed by (cA)T

= c(AT) and (A + B)

T= A

T+ B

T. Finally, (A

T)T

= A.

Matrix multiplication

Schematic depiction of the matrix product AB of two matrices A and B.

Multiplication of two matrices is defined only if the number of columns of the left matrix is the same as

the number of rows of the right matrix. IfA is an m-by-n matrix and B is an n-by-p matrix, then

theirmatrix productAB is the m-by-p matrix whose entries are given bydot productof the

corresponding row ofA and the corresponding column ofB:

,

where 1 imand 1 jp.[11]

For example, the underlined entry 2340 in the product is calculated

as (2 1000) + (3 100) + (4 10) = 2340:

Matrix multiplication satisfies the rules (AB)C = A(BC) (associativity), and (A+B)C = AC+BC as well

as C(A+B) = CA+CB (left and rightdistributivity), whenever the size of the matrices is such that thevarious products are defined.

[12]The product AB may be defined without BA being defined, namely
http://en.wikipedia.org/wiki/Transposehttp://en.wikipedia.org/wiki/Transposehttp://en.wikipedia.org/wiki/Commutativehttp://en.wikipedia.org/wiki/Commutativehttp://en.wikipedia.org/wiki/Commutativehttp://en.wikipedia.org/wiki/Matrix_(mathematics)#cite_note-10http://en.wikipedia.org/wiki/Matrix_(mathematics)#cite_note-10http://en.wikipedia.org/wiki/Matrix_(mathematics)#cite_note-10http://en.wikipedia.org/wiki/Dot_producthttp://en.wikipedia.org/wiki/Dot_producthttp://en.wikipedia.org/wiki/Dot_producthttp://en.wikipedia.org/wiki/Matrix_(mathematics)#cite_note-11http://en.wikipedia.org/wiki/Matrix_(mathematics)#cite_note-11http://en.wikipedia.org/wiki/Matrix_(mathematics)#cite_note-11http://en.wikipedia.org/wiki/Associativityhttp://en.wikipedia.org/wiki/Associativityhttp://en.wikipedia.org/wiki/Associativityhttp://en.wikipedia.org/wiki/Distributivityhttp://en.wikipedia.org/wiki/Distributivityhttp://en.wikipedia.org/wiki/Distributivityhttp://en.wikipedia.org/wiki/Matrix_(mathematics)#cite_note-12http://en.wikipedia.org/wiki/Matrix_(mathematics)#cite_note-12http://en.wikipedia.org/wiki/Matrix_(mathematics)#cite_note-12http://en.wikipedia.org/wiki/File:Matrix_multiplication_diagram_2.svghttp://en.wikipedia.org/wiki/File:Matrix_multiplication_diagram_2.svghttp://en.wikipedia.org/wiki/File:Matrix_multiplication_diagram_2.svghttp://en.wikipedia.org/wiki/File:Matrix_multiplication_diagram_2.svghttp://en.wikipedia.org/wiki/Matrix_(mathematics)#cite_note-12http://en.wikipedia.org/wiki/Distributivityhttp://en.wikipedia.org/wiki/Associativityhttp://en.wikipedia.org/wiki/Matrix_(mathematics)#cite_note-11http://en.wikipedia.org/wiki/Dot_producthttp://en.wikipedia.org/wiki/Matrix_(mathematics)#cite_note-10http://en.wikipedia.org/wiki/Commutativehttp://en.wikipedia.org/wiki/Transpose


6/14

ifA and B are m-by-n and n-by-kmatrices, respectively, and mk. Even if both products are defined,

they need not be equal, i.e., generally one has

ABBA,

i.e., matrix multiplication is notcommutative, in marked contrast to (rational, real, or complex)

numbers whose product is independent of the order of the factors. An example of two matrices not

commuting with each other is:

whereas

Besides the ordinary matrix multiplication just described, there exist other less frequently used

operations on matrices that can be considered forms of multiplication, such as theHadamard

productand theKronecker product.[13]

They arise in solving matrix equations such as theSylvester

equation.

Row operations

There are three types of row operations:

1. row addition, that is adding a row to another.

2. row multiplication, that is multiplying all entries of a row by a constant;

3. row switching, that is interchanging two rows of a matrix;

These operations are used in a number of ways, including solvinglinear equationsand findingmatrix

inverses.

Submatrix

A submatrix of a matrix is obtained by deleting any collection of rows and/or columns. For example,

for the following 3-by-4 matrix, we can construct a 2-by-3 submatrix by removing row 3 and column 2:

Theminorsand cofactors of a matrix are found by computing thedeterminantof certain

submatrices.

Linear equations

Matrices can be used to compactly write and work with multiple linear equations, i.e., systems of

linear equations. For example, ifA is an m-by-n matrix, x designates a column vector (i.e., n1-

matrix) ofn variablesx1,x2, ...,xn, and b is an m1-column vector, then the matrix equation

Ax = b
http://en.wikipedia.org/wiki/Commutativityhttp://en.wikipedia.org/wiki/Commutativityhttp://en.wikipedia.org/wiki/Commutativityhttp://en.wikipedia.org/wiki/Hadamard_product_(matrices)http://en.wikipedia.org/wiki/Hadamard_product_(matrices)http://en.wikipedia.org/wiki/Hadamard_product_(matrices)http://en.wikipedia.org/wiki/Hadamard_product_(matrices)http://en.wikipedia.org/wiki/Kronecker_producthttp://en.wikipedia.org/wiki/Kronecker_producthttp://en.wikipedia.org/wiki/Matrix_(mathematics)#cite_note-13http://en.wikipedia.org/wiki/Matrix_(mathematics)#cite_note-13http://en.wikipedia.org/wiki/Matrix_(mathematics)#cite_note-13http://en.wikipedia.org/wiki/Sylvester_equationhttp://en.wikipedia.org/wiki/Sylvester_equationhttp://en.wikipedia.org/wiki/Sylvester_equationhttp://en.wikipedia.org/wiki/Sylvester_equationhttp://en.wikipedia.org/wiki/Linear_equationhttp://en.wikipedia.org/wiki/Linear_equationhttp://en.wikipedia.org/wiki/Linear_equationhttp://en.wikipedia.org/wiki/Matrix_inversehttp://en.wikipedia.org/wiki/Matrix_inversehttp://en.wikipedia.org/wiki/Matrix_inversehttp://en.wikipedia.org/wiki/Matrix_inversehttp://en.wikipedia.org/wiki/Minor_(linear_algebra)http://en.wikipedia.org/wiki/Minor_(linear_algebra)http://en.wikipedia.org/wiki/Minor_(linear_algebra)http://en.wikipedia.org/wiki/Determinanthttp://en.wikipedia.org/wiki/Determinanthttp://en.wikipedia.org/wiki/Determinanthttp://en.wikipedia.org/wiki/Determinanthttp://en.wikipedia.org/wiki/Minor_(linear_algebra)http://en.wikipedia.org/wiki/Matrix_inversehttp://en.wikipedia.org/wiki/Matrix_inversehttp://en.wikipedia.org/wiki/Linear_equationhttp://en.wikipedia.org/wiki/Sylvester_equationhttp://en.wikipedia.org/wiki/Sylvester_equationhttp://en.wikipedia.org/wiki/Matrix_(mathematics)#cite_note-13http://en.wikipedia.org/wiki/Kronecker_producthttp://en.wikipedia.org/wiki/Hadamard_product_(matrices)http://en.wikipedia.org/wiki/Hadamard_product_(matrices)http://en.wikipedia.org/wiki/Commutativity


7/14

is equivalent to the system of linear equations

A1,1x1 +A1,2x2 + ... +A1,nxn = b1

...

Am,1

x1

+Am,2

x2

+ ... +Am,n

xn

= bm

.

Linear transformations[edit source|editbeta]

Main articles:Linear transformationandTransformation matrix

The vectors represented by a 2-by-2 matrix correspond to the sides of a unit square transformed into a parallelogram.

Matrices and matrix multiplication reveal their essential features when related to linear

transformations, also known as linear maps.A real m-by-n matrixA gives rise to a linear

transformation RnR

mmapping each vectorxin R

nto the (matrix) productAx, which is a vector

in Rm. Conversely, each linear transformation f: R

nR

marises from a unique m-by-n matrixA :

explicitly, the (i, j)-entry ofA is the ith

coordinate of f(ej), whereej= (0,...,0,1,0,...,0) is theunit

vectorwith 1 in the jth

position and 0 elsewhere. The matrix A is said to represent the linear map f,

and A is called thetransformation matrixoff.

For example, the 22 matrix

can be viewed as the transform of theunit squareinto aparallelogramwith vertices at (0,

0), (a, b), (a + c, b + d), and (c, d). The parallelogram pictured at the right is obtained by

multiplying A with each of the column vectors and in turn. These vectors

define the vertices of the unit square.
http://en.wikipedia.org/w/index.php?title=Matrix_(mathematics)&action=edit&section=10http://en.wikipedia.org/w/index.php?title=Matrix_(mathematics)&action=edit&section=10http://en.wikipedia.org/w/index.php?title=Matrix_(mathematics)&action=edit&section=10http://en.wikipedia.org/w/index.php?title=Matrix_(mathematics)&veaction=edit&section=10http://en.wikipedia.org/w/index.php?title=Matrix_(mathematics)&veaction=edit&section=10http://en.wikipedia.org/w/index.php?title=Matrix_(mathematics)&veaction=edit&section=10http://en.wikipedia.org/w/index.php?title=Matrix_(mathematics)&veaction=edit&section=10http://en.wikipedia.org/wiki/Linear_transformationhttp://en.wikipedia.org/wiki/Linear_transformationhttp://en.wikipedia.org/wiki/Linear_transformationhttp://en.wikipedia.org/wiki/Transformation_matrixhttp://en.wikipedia.org/wiki/Transformation_matrixhttp://en.wikipedia.org/wiki/Transformation_matrixhttp://en.wikipedia.org/wiki/Unit_vectorhttp://en.wikipedia.org/wiki/Unit_vectorhttp://en.wikipedia.org/wiki/Unit_vectorhttp://en.wikipedia.org/wiki/Unit_vectorhttp://en.wikipedia.org/wiki/Unit_squarehttp://en.wikipedia.org/wiki/Unit_squarehttp://en.wikipedia.org/wiki/Unit_squarehttp://en.wikipedia.org/wiki/Parallelogramhttp://en.wikipedia.org/wiki/Parallelogramhttp://en.wikipedia.org/wiki/Parallelogramhttp://en.wikipedia.org/wiki/File:Area_parallellogram_as_determinant.svghttp://en.wikipedia.org/wiki/File:Area_parallellogram_as_determinant.svghttp://en.wikipedia.org/wiki/File:Area_parallellogram_as_determinant.svghttp://en.wikipedia.org/wiki/File:Area_parallellogram_as_determinant.svghttp://en.wikipedia.org/wiki/File:Area_parallellogram_as_determinant.svghttp://en.wikipedia.org/wiki/File:Area_parallellogram_as_determinant.svghttp://en.wikipedia.org/wiki/File:Area_parallellogram_as_determinant.svghttp://en.wikipedia.org/wiki/File:Area_parallellogram_as_determinant.svghttp://en.wikipedia.org/wiki/File:Area_parallellogram_as_determinant.svghttp://en.wikipedia.org/wiki/File:Area_parallellogram_as_determinant.svghttp://en.wikipedia.org/wiki/Parallelogramhttp://en.wikipedia.org/wiki/Unit_squarehttp://en.wikipedia.org/wiki/Unit_vectorhttp://en.wikipedia.org/wiki/Unit_vectorhttp://en.wikipedia.org/wiki/Transformation_matrixhttp://en.wikipedia.org/wiki/Linear_transformationhttp://en.wikipedia.org/w/index.php?title=Matrix_(mathematics)&veaction=edit&section=10http://en.wikipedia.org/w/index.php?title=Matrix_(mathematics)&action=edit&section=10


8/14

The following table shows a number of2-by-2 matriceswith the associated linear maps ofR2.

The blue original is mapped to the green grid and shapes. The origin (0,0) is marked with a black

point.

Horizontal

shearwith m=1.25.Horizontal flip

Squeezemappingwith

r=3/2

Scalingby a

factor of 3/2Rotationby /6

R= 30

Under the1-to-1 correspondencebetween matrices and linear maps, matrix multiplication

corresponds tocompositionof maps:[15]

if a k-by-m matrix B represents another linear

map g: RmR

k, then the composition gfis represented by BA since

(gf)(x) = g(f(x)) = g(Ax) = B(Ax) = (BA)x.

The last equality follows from the above-mentioned associativity of matrix multiplication.

Therank of a matrixA is the maximum number oflinearly independentrow vectors of the

matrix, which is the same as the maximum number of linearly independent column

vectors.[16]

Equivalently it is thedimensionof theimageof the linear map represented

by A.[17]

Therank-nullity theoremstates that the dimension of thekernelof a matrix plus the

rank equals the number of columns of the matrix.[18]

Square matrices[edit source|editbeta]

Main article:Square matrix

Asquare matrixis a matrix with the same number of rows and columns. An n-by-n matrix is

known as a square matrix of ordern. Any two square matrices of the same order can be

added and multiplied. The entries aiiform themain diagonalof a square matrix. They lie on

the imaginary line which runs from the top left corner to the bottom right corner of the matrix.

Main types[edit source|editbeta]

Name Example with n = 3
http://en.wikipedia.org/wiki/2_%C3%97_2_real_matriceshttp://en.wikipedia.org/wiki/2_%C3%97_2_real_matriceshttp://en.wikipedia.org/wiki/2_%C3%97_2_real_matriceshttp://en.wikipedia.org/wiki/Shear_mappinghttp://en.wikipedia.org/wiki/Shear_mappinghttp://en.wikipedia.org/wiki/Shear_mappinghttp://en.wikipedia.org/wiki/Squeeze_mappinghttp://en.wikipedia.org/wiki/Squeeze_mappinghttp://en.wikipedia.org/wiki/Squeeze_mappinghttp://en.wikipedia.org/wiki/Scaling_(geometry)http://en.wikipedia.org/wiki/Scaling_(geometry)http://en.wikipedia.org/wiki/Rotation_matrixhttp://en.wikipedia.org/wiki/Rotation_matrixhttp://en.wikipedia.org/wiki/Bijectionhttp://en.wikipedia.org/wiki/Bijectionhttp://en.wikipedia.org/wiki/Bijectionhttp://en.wikipedia.org/wiki/Function_compositionhttp://en.wikipedia.org/wiki/Function_compositionhttp://en.wikipedia.org/wiki/Function_compositionhttp://en.wikipedia.org/wiki/Matrix_(mathematics)#cite_note-15http://en.wikipedia.org/wiki/Matrix_(mathematics)#cite_note-15http://en.wikipedia.org/wiki/Matrix_(mathematics)#cite_note-15http://en.wikipedia.org/wiki/Rank_of_a_matrixhttp://en.wikipedia.org/wiki/Rank_of_a_matrixhttp://en.wikipedia.org/wiki/Rank_of_a_matrixhttp://en.wikipedia.org/wiki/Linear_independencehttp://en.wikipedia.org/wiki/Linear_independencehttp://en.wikipedia.org/wiki/Linear_independencehttp://en.wikipedia.org/wiki/Matrix_(mathematics)#cite_note-16http://en.wikipedia.org/wiki/Matrix_(mathematics)#cite_note-16http://en.wikipedia.org/wiki/Hamel_dimensionhttp://en.wikipedia.org/wiki/Hamel_dimensionhttp://en.wikipedia.org/wiki/Hamel_dimensionhttp://en.wikipedia.org/wiki/Image_(mathematics)http://en.wikipedia.org/wiki/Image_(mathematics)http://en.wikipedia.org/wiki/Image_(mathematics)http://en.wikipedia.org/wiki/Matrix_(mathematics)#cite_note-17http://en.wikipedia.org/wiki/Matrix_(mathematics)#cite_note-17http://en.wikipedia.org/wiki/Matrix_(mathematics)#cite_note-17http://en.wikipedia.org/wiki/Rank-nullity_theoremhttp://en.wikipedia.org/wiki/Rank-nullity_theoremhttp://en.wikipedia.org/wiki/Rank-nullity_theoremhttp://en.wikipedia.org/wiki/Kernel_(matrix)http://en.wikipedia.org/wiki/Kernel_(matrix)http://en.wikipedia.org/wiki/Kernel_(matrix)http://en.wikipedia.org/wiki/Matrix_(mathematics)#cite_note-18http://en.wikipedia.org/wiki/Matrix_(mathematics)#cite_note-18http://en.wikipedia.org/wiki/Matrix_(mathematics)#cite_note-18http://en.wikipedia.org/w/index.php?title=Matrix_(mathematics)&action=edit&section=11http://en.wikipedia.org/w/index.php?title=Matrix_(mathematics)&action=edit&section=11http://en.wikipedia.org/w/index.php?title=Matrix_(mathematics)&action=edit&section=11http://en.wikipedia.org/w/index.php?title=Matrix_(mathematics)&veaction=edit&section=11http://en.wikipedia.org/w/index.php?title=Matrix_(mathematics)&veaction=edit&section=11http://en.wikipedia.org/w/index.php?title=Matrix_(mathematics)&veaction=edit&section=11http://en.wikipedia.org/w/index.php?title=Matrix_(mathematics)&veaction=edit&section=11http://en.wikipedia.org/wiki/Square_matrixhttp://en.wikipedia.org/wiki/Square_matrixhttp://en.wikipedia.org/wiki/Square_matrixhttp://en.wikipedia.org/wiki/Square_matrixhttp://en.wikipedia.org/wiki/Square_matrixhttp://en.wikipedia.org/wiki/Square_matrixhttp://en.wikipedia.org/wiki/Main_diagonalhttp://en.wikipedia.org/wiki/Main_diagonalhttp://en.wikipedia.org/wiki/Main_diagonalhttp://en.wikipedia.org/w/index.php?title=Matrix_(mathematics)&action=edit&section=12http://en.wikipedia.org/w/index.php?title=Matrix_(mathematics)&action=edit&section=12http://en.wikipedia.org/w/index.php?title=Matrix_(mathematics)&action=edit&section=12http://en.wikipedia.org/w/index.php?title=Matrix_(mathematics)&veaction=edit&section=12http://en.wikipedia.org/w/index.php?title=Matrix_(mathematics)&veaction=edit&section=12http://en.wikipedia.org/w/index.php?title=Matrix_(mathematics)&veaction=edit&section=12http://en.wikipedia.org/w/index.php?title=Matrix_(mathematics)&veaction=edit&section=12http://en.wikipedia.org/wiki/File:Rotation_by_pi_over_6.svghttp://en.wikipedia.org/wiki/File:Scaling_by_1.5.svghttp://en.wikipedia.org/wiki/File:Squeeze_r=1.5.svghttp://en.wikipedia.org/wiki/File:Flip_map.svghttp://en.wikipedia.org/wiki/File:VerticalShear_m=1.25.svghttp://en.wikipedia.org/wiki/File:Rotation_by_pi_over_6.svghttp://en.wikipedia.org/wiki/File:Scaling_by_1.5.svghttp://en.wikipedia.org/wiki/File:Squeeze_r=1.5.svghttp://en.wikipedia.org/wiki/File:Flip_map.svghttp://en.wikipedia.org/wiki/File:VerticalShear_m=1.25.svghttp://en.wikipedia.org/wiki/File:Rotation_by_pi_over_6.svghttp://en.wikipedia.org/wiki/File:Scaling_by_1.5.svghttp://en.wikipedia.org/wiki/File:Squeeze_r=1.5.svghttp://en.wikipedia.org/wiki/File:Flip_map.svghttp://en.wikipedia.org/wiki/File:VerticalShear_m=1.25.svghttp://en.wikipedia.org/wiki/File:Rotation_by_pi_over_6.svghttp://en.wikipedia.org/wiki/File:Scaling_by_1.5.svghttp://en.wikipedia.org/wiki/File:Squeeze_r=1.5.svghttp://en.wikipedia.org/wiki/File:Flip_map.svghttp://en.wikipedia.org/wiki/File:VerticalShear_m=1.25.svghttp://en.wikipedia.org/wiki/File:Rotation_by_pi_over_6.svghttp://en.wikipedia.org/wiki/File:Scaling_by_1.5.svghttp://en.wikipedia.org/wiki/File:Squeeze_r=1.5.svghttp://en.wikipedia.org/wiki/File:Flip_map.svghttp://en.wikipedia.org/wiki/File:VerticalShear_m=1.25.svghttp://en.wikipedia.org/wiki/File:Rotation_by_pi_over_6.svghttp://en.wikipedia.org/wiki/File:Scaling_by_1.5.svghttp://en.wikipedia.org/wiki/File:Squeeze_r=1.5.svghttp://en.wikipedia.org/wiki/File:Flip_map.svghttp://en.wikipedia.org/wiki/File:VerticalShear_m=1.25.svghttp://en.wikipedia.org/wiki/File:Rotation_by_pi_over_6.svghttp://en.wikipedia.org/wiki/File:Scaling_by_1.5.svghttp://en.wikipedia.org/wiki/File:Squeeze_r=1.5.svghttp://en.wikipedia.org/wiki/File:Flip_map.svghttp://en.wikipedia.org/wiki/File:VerticalShear_m=1.25.svghttp://en.wikipedia.org/wiki/File:Rotation_by_pi_over_6.svghttp://en.wikipedia.org/wiki/File:Scaling_by_1.5.svghttp://en.wikipedia.org/wiki/File:Squeeze_r=1.5.svghttp://en.wikipedia.org/wiki/File:Flip_map.svghttp://en.wikipedia.org/wiki/File:VerticalShear_m=1.25.svghttp://en.wikipedia.org/wiki/File:Rotation_by_pi_over_6.svghttp://en.wikipedia.org/wiki/File:Scaling_by_1.5.svghttp://en.wikipedia.org/wiki/File:Squeeze_r=1.5.svghttp://en.wikipedia.org/wiki/File:Flip_map.svghttp://en.wikipedia.org/wiki/File:VerticalShear_m=1.25.svghttp://en.wikipedia.org/wiki/File:Rotation_by_pi_over_6.svghttp://en.wikipedia.org/wiki/File:Scaling_by_1.5.svghttp://en.wikipedia.org/wiki/File:Squeeze_r=1.5.svghttp://en.wikipedia.org/wiki/File:Flip_map.svghttp://en.wikipedia.org/wiki/File:VerticalShear_m=1.25.svghttp://en.wikipedia.org/w/index.php?title=Matrix_(mathematics)&veaction=edit&section=12http://en.wikipedia.org/w/index.php?title=Matrix_(mathematics)&action=edit&section=12http://en.wikipedia.org/wiki/Main_diagonalhttp://en.wikipedia.org/wiki/Square_matrixhttp://en.wikipedia.org/wiki/Square_matrixhttp://en.wikipedia.org/w/index.php?title=Matrix_(mathematics)&veaction=edit&section=11http://en.wikipedia.org/w/index.php?title=Matrix_(mathematics)&action=edit&section=11http://en.wikipedia.org/wiki/Matrix_(mathematics)#cite_note-18http://en.wikipedia.org/wiki/Kernel_(matrix)http://en.wikipedia.org/wiki/Rank-nullity_theoremhttp://en.wikipedia.org/wiki/Matrix_(mathematics)#cite_note-17http://en.wikipedia.org/wiki/Image_(mathematics)http://en.wikipedia.org/wiki/Hamel_dimensionhttp://en.wikipedia.org/wiki/Matrix_(mathematics)#cite_note-16http://en.wikipedia.org/wiki/Linear_independencehttp://en.wikipedia.org/wiki/Rank_of_a_matrixhttp://en.wikipedia.org/wiki/Matrix_(mathematics)#cite_note-15http://en.wikipedia.org/wiki/Function_compositionhttp://en.wikipedia.org/wiki/Bijectionhttp://en.wikipedia.org/wiki/Rotation_matrixhttp://en.wikipedia.org/wiki/Scaling_(geometry)http://en.wikipedia.org/wiki/Squeeze_mappinghttp://en.wikipedia.org/wiki/Squeeze_mappinghttp://en.wikipedia.org/wiki/Shear_mappinghttp://en.wikipedia.org/wiki/Shear_mappinghttp://en.wikipedia.org/wiki/2_%C3%97_2_real_matrices


9/14

Diagonal matrix

Lower triangular matrix

Upper triangular matrix

Diagonal or triangular matrix[edit source|editbeta]

If all entries outside the main diagonal are zero, A is called adiagonal matrix. If only all

entries above (or below) the main diagonal are zero, A is called a lower (or

upper)triangular matrix.

Identity matrix[edit source|editbeta]

Theidentity matrixIn of size n is the n-by-n matrix in which all the elements on themain

diagonalare equal to 1 and all other elements are equal to 0, e.g.

It is a square matrix of ordern, and also a special kind ofdiagonal matrix. It is called

identity matrix because multiplication with it leaves a matrix unchanged:

AIn = ImA = A for any m-by-n matrix A.

Symmetr ic or skew-symm etr ic matr ix[edit source|editbeta]

A square matrix A that is equal to its transpose, i.e., A = AT, is asymmetric

matrix. If instead, A was equal to the negative of its transpose, i.e., A =A

T, then A is askew-symmetric matrix. In complex matrices, symmetry is often

replaced by the concept ofHermitian matrices, which satisfy A = A, where the

star orasteriskdenotes theconjugate transposeof the matrix, i.e., the

transpose of thecomplex conjugateofA.

By thespectral theorem, real symmetric matrices and complex Hermitian

matrices have aneigenbasis; i.e., every vector is expressible as alinear

combinationof eigenvectors. In both cases, all eigenvalues are real.[19]

This

theorem can be generalized to infinite-dimensional situations related to matrices

with infinitely many rows and columns, seebelow.
http://en.wikipedia.org/wiki/Diagonal_matrixhttp://en.wikipedia.org/wiki/Diagonal_matrixhttp://en.wikipedia.org/wiki/Lower_triangular_matrixhttp://en.wikipedia.org/wiki/Lower_triangular_matrixhttp://en.wikipedia.org/wiki/Upper_triangular_matrixhttp://en.wikipedia.org/wiki/Upper_triangular_matrixhttp://en.wikipedia.org/w/index.php?title=Matrix_(mathematics)&action=edit&section=13http://en.wikipedia.org/w/index.php?title=Matrix_(mathematics)&action=edit&section=13http://en.wikipedia.org/w/index.php?title=Matrix_(mathematics)&action=edit&section=13http://en.wikipedia.org/w/index.php?title=Matrix_(mathematics)&veaction=edit&section=13http://en.wikipedia.org/w/index.php?title=Matrix_(mathematics)&veaction=edit&section=13http://en.wikipedia.org/w/index.php?title=Matrix_(mathematics)&veaction=edit&section=13http://en.wikipedia.org/w/index.php?title=Matrix_(mathematics)&veaction=edit&section=13http://en.wikipedia.org/wiki/Diagonal_matrixhttp://en.wikipedia.org/wiki/Diagonal_matrixhttp://en.wikipedia.org/wiki/Diagonal_matrixhttp://en.wikipedia.org/wiki/Triangular_matrixhttp://en.wikipedia.org/wiki/Triangular_matrixhttp://en.wikipedia.org/wiki/Triangular_matrixhttp://en.wikipedia.org/w/index.php?title=Matrix_(mathematics)&action=edit&section=14http://en.wikipedia.org/w/index.php?title=Matrix_(mathematics)&action=edit&section=14http://en.wikipedia.org/w/index.php?title=Matrix_(mathematics)&action=edit&section=14http://en.wikipedia.org/w/index.php?title=Matrix_(mathematics)&veaction=edit&section=14http://en.wikipedia.org/w/index.php?title=Matrix_(mathematics)&veaction=edit&section=14http://en.wikipedia.org/w/index.php?title=Matrix_(mathematics)&veaction=edit&section=14http://en.wikipedia.org/w/index.php?title=Matrix_(mathematics)&veaction=edit&section=14http://en.wikipedia.org/wiki/Identity_matrixhttp://en.wikipedia.org/wiki/Identity_matrixhttp://en.wikipedia.org/wiki/Identity_matrixhttp://en.wikipedia.org/wiki/Main_diagonalhttp://en.wikipedia.org/wiki/Main_diagonalhttp://en.wikipedia.org/wiki/Main_diagonalhttp://en.wikipedia.org/wiki/Main_diagonalhttp://en.wikipedia.org/wiki/Diagonal_matrixhttp://en.wikipedia.org/wiki/Diagonal_matrixhttp://en.wikipedia.org/wiki/Diagonal_matrixhttp://en.wikipedia.org/w/index.php?title=Matrix_(mathematics)&action=edit&section=15http://en.wikipedia.org/w/index.php?title=Matrix_(mathematics)&action=edit&section=15http://en.wikipedia.org/w/index.php?title=Matrix_(mathematics)&action=edit&section=15http://en.wikipedia.org/w/index.php?title=Matrix_(mathematics)&veaction=edit&section=15http://en.wikipedia.org/w/index.php?title=Matrix_(mathematics)&veaction=edit&section=15http://en.wikipedia.org/w/index.php?title=Matrix_(mathematics)&veaction=edit&section=15http://en.wikipedia.org/w/index.php?title=Matrix_(mathematics)&veaction=edit&section=15http://en.wikipedia.org/wiki/Symmetric_matrixhttp://en.wikipedia.org/wiki/Symmetric_matrixhttp://en.wikipedia.org/wiki/Symmetric_matrixhttp://en.wikipedia.org/wiki/Symmetric_matrixhttp://en.wikipedia.org/wiki/Skew-symmetric_matrixhttp://en.wikipedia.org/wiki/Skew-symmetric_matrixhttp://en.wikipedia.org/wiki/Skew-symmetric_matrixhttp://en.wikipedia.org/wiki/Hermitian_matrixhttp://en.wikipedia.org/wiki/Hermitian_matrixhttp://en.wikipedia.org/wiki/Hermitian_matrixhttp://en.wikipedia.org/wiki/Asteriskhttp://en.wikipedia.org/wiki/Asteriskhttp://en.wikipedia.org/wiki/Asteriskhttp://en.wikipedia.org/wiki/Conjugate_transposehttp://en.wikipedia.org/wiki/Conjugate_transposehttp://en.wikipedia.org/wiki/Conjugate_transposehttp://en.wikipedia.org/wiki/Complex_conjugatehttp://en.wikipedia.org/wiki/Complex_conjugatehttp://en.wikipedia.org/wiki/Complex_conjugatehttp://en.wikipedia.org/wiki/Spectral_theoremhttp://en.wikipedia.org/wiki/Spectral_theoremhttp://en.wikipedia.org/wiki/Spectral_theoremhttp://en.wikipedia.org/wiki/Eigenbasishttp://en.wikipedia.org/wiki/Eigenbasishttp://en.wikipedia.org/wiki/Eigenbasishttp://en.wikipedia.org/wiki/Linear_combinationhttp://en.wikipedia.org/wiki/Linear_combinationhttp://en.wikipedia.org/wiki/Linear_combinationhttp://en.wikipedia.org/wiki/Linear_combinationhttp://en.wikipedia.org/wiki/Matrix_(mathematics)#cite_note-19http://en.wikipedia.org/wiki/Matrix_(mathematics)#cite_note-19http://en.wikipedia.org/wiki/Matrix_(mathematics)#cite_note-19http://en.wikipedia.org/wiki/Matrix_(mathematics)#Infinite_matriceshttp://en.wikipedia.org/wiki/Matrix_(mathematics)#Infinite_matriceshttp://en.wikipedia.org/wiki/Matrix_(mathematics)#Infinite_matriceshttp://en.wikipedia.org/wiki/Matrix_(mathematics)#Infinite_matriceshttp://en.wikipedia.org/wiki/Matrix_(mathematics)#cite_note-19http://en.wikipedia.org/wiki/Linear_combinationhttp://en.wikipedia.org/wiki/Linear_combinationhttp://en.wikipedia.org/wiki/Eigenbasishttp://en.wikipedia.org/wiki/Spectral_theoremhttp://en.wikipedia.org/wiki/Complex_conjugatehttp://en.wikipedia.org/wiki/Conjugate_transposehttp://en.wikipedia.org/wiki/Asteriskhttp://en.wikipedia.org/wiki/Hermitian_matrixhttp://en.wikipedia.org/wiki/Skew-symmetric_matrixhttp://en.wikipedia.org/wiki/Symmetric_matrixhttp://en.wikipedia.org/wiki/Symmetric_matrixhttp://en.wikipedia.org/w/index.php?title=Matrix_(mathematics)&veaction=edit&section=15http://en.wikipedia.org/w/index.php?title=Matrix_(mathematics)&action=edit&section=15http://en.wikipedia.org/wiki/Diagonal_matrixhttp://en.wikipedia.org/wiki/Main_diagonalhttp://en.wikipedia.org/wiki/Main_diagonalhttp://en.wikipedia.org/wiki/Identity_matrixhttp://en.wikipedia.org/w/index.php?title=Matrix_(mathematics)&veaction=edit&section=14http://en.wikipedia.org/w/index.php?title=Matrix_(mathematics)&action=edit&section=14http://en.wikipedia.org/wiki/Triangular_matrixhttp://en.wikipedia.org/wiki/Diagonal_matrixhttp://en.wikipedia.org/w/index.php?title=Matrix_(mathematics)&veaction=edit&section=13http://en.wikipedia.org/w/index.php?title=Matrix_(mathematics)&action=edit&section=13http://en.wikipedia.org/wiki/Upper_triangular_matrixhttp://en.wikipedia.org/wiki/Lower_triangular_matrixhttp://en.wikipedia.org/wiki/Diagonal_matrix


10/14

Invert ible matrix and its inverse[edit source|editbeta]

A square matrix A is calledinvertibleornon-singularif there exists a

matrix B such that

AB = BA = In.[20][21]

IfB exists, it is unique and is called theinverse matrixofA, denoted A1

.

Definite matrix[edit source|editbeta]

Positive definite matrix Indefinite matrix

Q(x,y) = 1/4x2

+ y2 Q(x,y) = 1/4x

2 1/4 y

2

Points such that Q(x,y)=1

(Ellipse).

Points such that Q(x,y)=1

(Hyperbola).

A symmetric nn-matrix is calledpositive-definite(respectively negative-

definite; indefinite), if for all nonzero vectors xRn

the associatedquadratic

formgiven by

Q(x) = xTAx

takes only positive values (respectively only negative values; both some negative and some positive

values).[22]

If the quadratic form takes only non-negative (respectively only non-positive) values, the

symmetric matrix is called positive-semidefinite (respectively negative-semidefinite); hence the matrix

is indefinite precisely when it is neither positive-semidefinite nor negative-semidefinite.

A symmetric matrix is positive-definite if and only if all its eigenvalues are positive.[23]

The table at the

right shows two possibilities for 2-by-2 matrices.

Allowing as input two different vectors instead yields thebilinear formassociated to A:

BA (x, y) = xTAy.

[24]

Orthogonal matr ix

An orthogonal matrixis asquare matrixwithrealentries whose columns and rows areorthogonalunit

vectors(i.e.,orthonormalvectors). Equivalently, a matrixA is

orthogonal if itstransposeis equal to itsinverse:

which entails
http://en.wikipedia.org/w/index.php?title=Matrix_(mathematics)&action=edit&section=16http://en.wikipedia.org/w/index.php?title=Matrix_(mathematics)&action=edit&section=16http://en.wikipedia.org/w/index.php?title=Matrix_(mathematics)&action=edit&section=16http://en.wikipedia.org/w/index.php?title=Matrix_(mathematics)&veaction=edit&section=16http://en.wikipedia.org/w/index.php?title=Matrix_(mathematics)&veaction=edit&section=16http://en.wikipedia.org/w/index.php?title=Matrix_(mathematics)&veaction=edit&section=16http://en.wikipedia.org/w/index.php?title=Matrix_(mathematics)&veaction=edit&section=16http://en.wikipedia.org/wiki/Invertible_matrixhttp://en.wikipedia.org/wiki/Invertible_matrixhttp://en.wikipedia.org/wiki/Invertible_matrixhttp://en.wikipedia.org/wiki/Matrix_(mathematics)#cite_note-20http://en.wikipedia.org/wiki/Matrix_(mathematics)#cite_note-20http://en.wikipedia.org/wiki/Matrix_(mathematics)#cite_note-20http://en.wikipedia.org/wiki/Inverse_matrixhttp://en.wikipedia.org/wiki/Inverse_matrixhttp://en.wikipedia.org/wiki/Inverse_matrixhttp://en.wikipedia.org/w/index.php?title=Matrix_(mathematics)&action=edit&section=17http://en.wikipedia.org/w/index.php?title=Matrix_(mathematics)&action=edit&section=17http://en.wikipedia.org/w/index.php?title=Matrix_(mathematics)&action=edit&section=17http://en.wikipedia.org/w/index.php?title=Matrix_(mathematics)&veaction=edit&section=17http://en.wikipedia.org/w/index.php?title=Matrix_(mathematics)&veaction=edit&section=17http://en.wikipedia.org/w/index.php?title=Matrix_(mathematics)&veaction=edit&section=17http://en.wikipedia.org/w/index.php?title=Matrix_(mathematics)&veaction=edit&section=17http://en.wikipedia.org/wiki/Positive_definite_matrixhttp://en.wikipedia.org/wiki/Positive_definite_matrixhttp://en.wikipedia.org/wiki/Indefinite_matrixhttp://en.wikipedia.org/wiki/Indefinite_matrixhttp://en.wikipedia.org/wiki/Ellipsehttp://en.wikipedia.org/wiki/Ellipsehttp://en.wikipedia.org/wiki/Ellipsehttp://en.wikipedia.org/wiki/Hyperbolahttp://en.wikipedia.org/wiki/Hyperbolahttp://en.wikipedia.org/wiki/Hyperbolahttp://en.wikipedia.org/wiki/Positive-definite_matrixhttp://en.wikipedia.org/wiki/Positive-definite_matrixhttp://en.wikipedia.org/wiki/Positive-definite_matrixhttp://en.wikipedia.org/wiki/Quadratic_formhttp://en.wikipedia.org/wiki/Quadratic_formhttp://en.wikipedia.org/wiki/Quadratic_formhttp://en.wikipedia.org/wiki/Quadratic_formhttp://en.wikipedia.org/wiki/Matrix_(mathematics)#cite_note-22http://en.wikipedia.org/wiki/Matrix_(mathematics)#cite_note-22http://en.wikipedia.org/wiki/Matrix_(mathematics)#cite_note-22http://en.wikipedia.org/wiki/Matrix_(mathematics)#cite_note-23http://en.wikipedia.org/wiki/Matrix_(mathematics)#cite_note-23http://en.wikipedia.org/wiki/Matrix_(mathematics)#cite_note-23http://en.wikipedia.org/wiki/Bilinear_formhttp://en.wikipedia.org/wiki/Bilinear_formhttp://en.wikipedia.org/wiki/Bilinear_formhttp://en.wikipedia.org/wiki/Matrix_(mathematics)#cite_note-24http://en.wikipedia.org/wiki/Matrix_(mathematics)#cite_note-24http://en.wikipedia.org/wiki/Matrix_(mathematics)#cite_note-24http://en.wikipedia.org/wiki/Matrix_(mathematics)#Square_matriceshttp://en.wikipedia.org/wiki/Matrix_(mathematics)#Square_matriceshttp://en.wikipedia.org/wiki/Matrix_(mathematics)#Square_matriceshttp://en.wikipedia.org/wiki/Real_numberhttp://en.wikipedia.org/wiki/Real_numberhttp://en.wikipedia.org/wiki/Real_numberhttp://en.wikipedia.org/wiki/Orthogonalhttp://en.wikipedia.org/wiki/Orthogonalhttp://en.wikipedia.org/wiki/Unit_vectorhttp://en.wikipedia.org/wiki/Unit_vectorhttp://en.wikipedia.org/wiki/Unit_vectorhttp://en.wikipedia.org/wiki/Unit_vectorhttp://en.wikipedia.org/wiki/Orthonormalityhttp://en.wikipedia.org/wiki/Orthonormalityhttp://en.wikipedia.org/wiki/Orthonormalityhttp://en.wikipedia.org/wiki/Transposehttp://en.wikipedia.org/wiki/Transposehttp://en.wikipedia.org/wiki/Transposehttp://en.wikipedia.org/wiki/Inverse_matrixhttp://en.wikipedia.org/wiki/Inverse_matrixhttp://en.wikipedia.org/wiki/Inverse_matrixhttp://en.wikipedia.org/wiki/File:Hyperbola2_SVG.svghttp://en.wikipedia.org/wiki/File:Ellipse_in_coordinate_system_with_semi-axes_labelled.svghttp://en.wikipedia.org/wiki/File:Hyperbola2_SVG.svghttp://en.wikipedia.org/wiki/File:Ellipse_in_coordinate_system_with_semi-axes_labelled.svghttp://en.wikipedia.org/wiki/File:Hyperbola2_SVG.svghttp://en.wikipedia.org/wiki/File:Ellipse_in_coordinate_system_with_semi-axes_labelled.svghttp://en.wikipedia.org/wiki/File:Hyperbola2_SVG.svghttp://en.wikipedia.org/wiki/File:Ellipse_in_coordinate_system_with_semi-axes_labelled.svghttp://en.wikipedia.org/wiki/File:Hyperbola2_SVG.svghttp://en.wikipedia.org/wiki/File:Ellipse_in_coordinate_system_with_semi-axes_labelled.svghttp://en.wikipedia.org/wiki/Inverse_matrixhttp://en.wikipedia.org/wiki/Transposehttp://en.wikipedia.org/wiki/Orthonormalityhttp://en.wikipedia.org/wiki/Unit_vectorhttp://en.wikipedia.org/wiki/Unit_vectorhttp://en.wikipedia.org/wiki/Orthogonalhttp://en.wikipedia.org/wiki/Real_numberhttp://en.wikipedia.org/wiki/Matrix_(mathematics)#Square_matriceshttp://en.wikipedia.org/wiki/Matrix_(mathematics)#cite_note-24http://en.wikipedia.org/wiki/Bilinear_formhttp://en.wikipedia.org/wiki/Matrix_(mathematics)#cite_note-23http://en.wikipedia.org/wiki/Matrix_(mathematics)#cite_note-22http://en.wikipedia.org/wiki/Quadratic_formhttp://en.wikipedia.org/wiki/Quadratic_formhttp://en.wikipedia.org/wiki/Positive-definite_matrixhttp://en.wikipedia.org/wiki/Hyperbolahttp://en.wikipedia.org/wiki/Ellipsehttp://en.wikipedia.org/wiki/Indefinite_matrixhttp://en.wikipedia.org/wiki/Positive_definite_matrixhttp://en.wikipedia.org/w/index.php?title=Matrix_(mathematics)&veaction=edit&section=17http://en.wikipedia.org/w/index.php?title=Matrix_(mathematics)&action=edit&section=17http://en.wikipedia.org/wiki/Inverse_matrixhttp://en.wikipedia.org/wiki/Matrix_(mathematics)#cite_note-20http://en.wikipedia.org/wiki/Matrix_(mathematics)#cite_note-20http://en.wikipedia.org/wiki/Invertible_matrixhttp://en.wikipedia.org/w/index.php?title=Matrix_(mathematics)&veaction=edit&section=16http://en.wikipedia.org/w/index.php?title=Matrix_(mathematics)&action=edit&section=16


11/14

where Iis theidentity matrix.

An orthogonal matrixA is necessarilyinvertible(with inverseA1

=AT),unitary(A

1=A*),

andnormal(A*A =AA*). Thedeterminantof any orthogonal matrix is either +1 or 1. Aspecial

orthogonal matrixis an orthogonal matrix withdeterminant+1. As alinear transformation, every

orthogonal matrix with determinant +1 is a purerotation, while every orthogonal matrix with

determinant -1 is either a purereflection, or a composition of reflection and rotation.

Thecomplexanalogue of an orthogonal matrix is aunitary matrix.

Determinant[edit source|editbeta]

Main article:Determinant

A linear transformation on R2given by the indicated matrix. The determinant of this matrix is 1, as the area of the

green parallelogram at the right is 1, but the map reverses theorientation, since it turns the counterclockwise orientation

of the vectors to a clockwise one.

The determinantdet(A) or |A| of a square matrix A is a number encoding certain properties of the

matrix. A matrix is invertibleif and only ifits determinant is nonzero. Itsabsolute valueequals the area

(in R2) or volume (in R

3) of the image of the unit square (or cube), while its sign corresponds to the

orientation of the corresponding linear map: the determinant is positive if and only if the orientation is

preserved.

The determinant of 2-by-2 matrices is given by

The determinant of 3-by-3 matrices involves 6 terms (rule of Sarrus). The more lengthyLeibniz

formulageneralises these two formulae to all dimensions.[25]

The determinant of a product of square matrices equals the product of their determinants:

det(AB) = det(A) det(B).[26]

Adding a multiple of any row to another row, or a multiple of any column to another column,

does not change the determinant. Interchanging two rows or two columns affects the

determinant by multiplying it by 1.[27]

Using these operations, any matrix can be transformed

to a lower (or upper) triangular matrix, and for such matrices the determinant equals the

product of the entries on the main diagonal; this provides a method to calculate the
http://en.wikipedia.org/wiki/Identity_matrixhttp://en.wikipedia.org/wiki/Identity_matrixhttp://en.wikipedia.org/wiki/Identity_matrixhttp://en.wikipedia.org/wiki/Invertible_matrixhttp://en.wikipedia.org/wiki/Invertible_matrixhttp://en.wikipedia.org/wiki/Invertible_matrixhttp://en.wikipedia.org/wiki/Unitary_matrixhttp://en.wikipedia.org/wiki/Unitary_matrixhttp://en.wikipedia.org/wiki/Unitary_matrixhttp://en.wikipedia.org/wiki/Normal_matrixhttp://en.wikipedia.org/wiki/Normal_matrixhttp://en.wikipedia.org/wiki/Normal_matrixhttp://en.wikipedia.org/wiki/Determinanthttp://en.wikipedia.org/wiki/Determinanthttp://en.wikipedia.org/wiki/Determinanthttp://en.wikipedia.org/wiki/Determinanthttp://en.wikipedia.org/wiki/Determinanthttp://en.wikipedia.org/wiki/Determinanthttp://en.wikipedia.org/wiki/Linear_transformationhttp://en.wikipedia.org/wiki/Linear_transformationhttp://en.wikipedia.org/wiki/Linear_transformationhttp://en.wikipedia.org/wiki/Rotation_(mathematics)http://en.wikipedia.org/wiki/Rotation_(mathematics)http://en.wikipedia.org/wiki/Rotation_(mathematics)http://en.wikipedia.org/wiki/Reflection_(mathematics)http://en.wikipedia.org/wiki/Reflection_(mathematics)http://en.wikipedia.org/wiki/Reflection_(mathematics)http://en.wikipedia.org/wiki/Complex_numberhttp://en.wikipedia.org/wiki/Complex_numberhttp://en.wikipedia.org/wiki/Complex_numberhttp://en.wikipedia.org/wiki/Unitary_matrixhttp://en.wikipedia.org/wiki/Unitary_matrixhttp://en.wikipedia.org/wiki/Unitary_matrixhttp://en.wikipedia.org/w/index.php?title=Matrix_(mathematics)&action=edit&section=21http://en.wikipedia.org/w/index.php?title=Matrix_(mathematics)&action=edit&section=21http://en.wikipedia.org/w/index.php?title=Matrix_(mathematics)&action=edit&section=21http://en.wikipedia.org/w/index.php?title=Matrix_(mathematics)&veaction=edit&section=21http://en.wikipedia.org/w/index.php?title=Matrix_(mathematics)&veaction=edit&section=21http://en.wikipedia.org/w/index.php?title=Matrix_(mathematics)&veaction=edit&section=21http://en.wikipedia.org/w/index.php?title=Matrix_(mathematics)&veaction=edit&section=21http://en.wikipedia.org/wiki/Determinanthttp://en.wikipedia.org/wiki/Determinanthttp://en.wikipedia.org/wiki/Determinanthttp://en.wikipedia.org/wiki/Orientation_(mathematics)http://en.wikipedia.org/wiki/Orientation_(mathematics)http://en.wikipedia.org/wiki/If_and_only_ifhttp://en.wikipedia.org/wiki/If_and_only_ifhttp://en.wikipedia.org/wiki/If_and_only_ifhttp://en.wikipedia.org/wiki/Absolute_valuehttp://en.wikipedia.org/wiki/Absolute_valuehttp://en.wikipedia.org/wiki/Absolute_valuehttp://en.wikipedia.org/wiki/Rule_of_Sarrushttp://en.wikipedia.org/wiki/Rule_of_Sarrushttp://en.wikipedia.org/wiki/Rule_of_Sarrushttp://en.wikipedia.org/wiki/Leibniz_formula_for_determinantshttp://en.wikipedia.org/wiki/Leibniz_formula_for_determinantshttp://en.wikipedia.org/wiki/Leibniz_formula_for_determinantshttp://en.wikipedia.org/wiki/Leibniz_formula_for_determinantshttp://en.wikipedia.org/wiki/Matrix_(mathematics)#cite_note-25http://en.wikipedia.org/wiki/Matrix_(mathematics)#cite_note-25http://en.wikipedia.org/wiki/Matrix_(mathematics)#cite_note-25http://en.wikipedia.org/wiki/Matrix_(mathematics)#cite_note-26http://en.wikipedia.org/wiki/Matrix_(mathematics)#cite_note-26http://en.wikipedia.org/wiki/Matrix_(mathematics)#cite_note-26http://en.wikipedia.org/wiki/Matrix_(mathematics)#cite_note-27http://en.wikipedia.org/wiki/Matrix_(mathematics)#cite_note-27http://en.wikipedia.org/wiki/Matrix_(mathematics)#cite_note-27http://en.wikipedia.org/wiki/File:Determinant_example.svghttp://en.wikipedia.org/wiki/File:Determinant_example.svghttp://en.wikipedia.org/wiki/File:Determinant_example.svghttp://en.wikipedia.org/wiki/File:Determinant_example.svghttp://en.wikipedia.org/wiki/File:Determinant_example.svghttp://en.wikipedia.org/wiki/File:Determinant_example.svghttp://en.wikipedia.org/wiki/File:Determinant_example.svghttp://en.wikipedia.org/wiki/File:Determinant_example.svghttp://en.wikipedia.org/wiki/Matrix_(mathematics)#cite_note-27http://en.wikipedia.org/wiki/Matrix_(mathematics)#cite_note-26http://en.wikipedia.org/wiki/Matrix_(mathematics)#cite_note-25http://en.wikipedia.org/wiki/Leibniz_formula_for_determinantshttp://en.wikipedia.org/wiki/Leibniz_formula_for_determinantshttp://en.wikipedia.org/wiki/Rule_of_Sarrushttp://en.wikipedia.org/wiki/Absolute_valuehttp://en.wikipedia.org/wiki/If_and_only_ifhttp://en.wikipedia.org/wiki/Orientation_(mathematics)http://en.wikipedia.org/wiki/Determinanthttp://en.wikipedia.org/w/index.php?title=Matrix_(mathematics)&veaction=edit&section=21http://en.wikipedia.org/w/index.php?title=Matrix_(mathematics)&action=edit&section=21http://en.wikipedia.org/wiki/Unitary_matrixhttp://en.wikipedia.org/wiki/Complex_numberhttp://en.wikipedia.org/wiki/Reflection_(mathematics)http://en.wikipedia.org/wiki/Rotation_(mathematics)http://en.wikipedia.org/wiki/Linear_transformationhttp://en.wikipedia.org/wiki/Determinanthttp://en.wikipedia.org/wiki/Determinanthttp://en.wikipedia.org/wiki/Normal_matrixhttp://en.wikipedia.org/wiki/Unitary_matrixhttp://en.wikipedia.org/wiki/Invertible_matrixhttp://en.wikipedia.org/wiki/Identity_matrix


12/14

determinant of any matrix. Finally, theLaplace expansionexpresses the determinant in terms

ofminors, i.e., determinants of smaller matrices.[28]

This expansion can be used for a

recursive definition of determinants (taking as starting case the determinant of a 1-by-1

matrix, which is its unique entry, or even the determinant of a 0-by-0 matrix, which is 1), that

can be seen to be equivalent to the Leibniz formula. Determinants can be used to

solvelinear systemsusingCramer's rule, where the division of the determinants of two

related square matrices equates to the value of each of the system's variables.[29]

ROW ECHELON

Inlinear algebra, amatrixis in echelon form if it has the shape resulting of aGaussian

elimination.Row echelon form means that Gaussian elimination has operated on the rows

and column echelon form means that Gaussian elimination has operated on the columns. In other

words, a matrix is in column echelon form if itstransposeis in row echelon form. Therefore only row

echelon forms are considered in the remainder of this article. The similar properties of column

echelon form are easily deduced by transposing all the matrices.

Specifically, a matrix is in row echelon form if

All non-zero rows (rows with at least one nonzero element) are above any rows of all zeroes (all

zero rows, if any, belong at the bottom of the matrix).

Theleading coefficient(the first nonzero number from the left, also called thepivot) of a nonzero

row is always strictly to the right of the leading coefficient of the row above it.

All entries in a column below a leading entry are zeroes (implied by the first two criteria).[1]

Some texts add the condition that the leading coefficient of any nonzero row must be 1.[2]

This is an example of a 35 matrix in row echelon form:

FROM WEB:http://en.wikipedia.org/wiki/Row_echelon_form.

Steps

1. 1

A Matrix in Row-Echelon form
http://en.wikipedia.org/wiki/Laplace_expansionhttp://en.wikipedia.org/wiki/Laplace_expansionhttp://en.wikipedia.org/wiki/Laplace_expansionhttp://en.wikipedia.org/wiki/Minor_(linear_algebra)http://en.wikipedia.org/wiki/Minor_(linear_algebra)http://en.wikipedia.org/wiki/Minor_(linear_algebra)http://en.wikipedia.org/wiki/Matrix_(mathematics)#cite_note-28http://en.wikipedia.org/wiki/Matrix_(mathematics)#cite_note-28http://en.wikipedia.org/wiki/Matrix_(mathematics)#cite_note-28http://en.wikipedia.org/wiki/Linear_systemhttp://en.wikipedia.org/wiki/Linear_systemhttp://en.wikipedia.org/wiki/Linear_systemhttp://en.wikipedia.org/wiki/Cramer%27s_rulehttp://en.wikipedia.org/wiki/Cramer%27s_rulehttp://en.wikipedia.org/wiki/Cramer%27s_rulehttp://en.wikipedia.org/wiki/Matrix_(mathematics)#cite_note-29http://en.wikipedia.org/wiki/Matrix_(mathematics)#cite_note-29http://en.wikipedia.org/wiki/Matrix_(mathematics)#cite_note-29http://en.wikipedia.org/wiki/Linear_algebrahttp://en.wikipedia.org/wiki/Linear_algebrahttp://en.wikipedia.org/wiki/Linear_algebrahttp://en.wikipedia.org/wiki/Matrix_(mathematics)http://en.wikipedia.org/wiki/Matrix_(mathematics)http://en.wikipedia.org/wiki/Matrix_(mathematics)http://en.wikipedia.org/wiki/Gaussian_eliminationhttp://en.wikipedia.org/wiki/Gaussian_eliminationhttp://en.wikipedia.org/wiki/Gaussian_eliminationhttp://en.wikipedia.org/wiki/Gaussian_eliminationhttp://en.wikipedia.org/wiki/Transposehttp://en.wikipedia.org/wiki/Transposehttp://en.wikipedia.org/wiki/Transposehttp://en.wikipedia.org/wiki/Leading_coefficient#Linear_algebrahttp://en.wikipedia.org/wiki/Leading_coefficient#Linear_algebrahttp://en.wikipedia.org/wiki/Leading_coefficient#Linear_algebrahttp://en.wikipedia.org/wiki/Pivot_elementhttp://en.wikipedia.org/wiki/Pivot_elementhttp://en.wikipedia.org/wiki/Pivot_elementhttp://en.wikipedia.org/wiki/Row_echelon_form#cite_note-1http://en.wikipedia.org/wiki/Row_echelon_form#cite_note-1http://en.wikipedia.org/wiki/Row_echelon_form#cite_note-1http://en.wikipedia.org/wiki/Row_echelon_form#cite_note-2http://en.wikipedia.org/wiki/Row_echelon_form#cite_note-2http://en.wikipedia.org/wiki/Row_echelon_form#cite_note-2http://en.wikipedia.org/wiki/Row_echelon_formhttp://en.wikipedia.org/wiki/Row_echelon_formhttp://en.wikipedia.org/wiki/Row_echelon_formhttp://www.wikihow.com/Image:Matrix-1.pnghttp://www.wikihow.com/Image:Matrix-1.pnghttp://www.wikihow.com/Image:Matrix-1.pnghttp://www.wikihow.com/Image:Matrix-1.pnghttp://www.wikihow.com/Image:Matrix-1.pnghttp://www.wikihow.com/Image:Matrix-1.pnghttp://en.wikipedia.org/wiki/Row_echelon_formhttp://en.wikipedia.org/wiki/Row_echelon_form#cite_note-2http://en.wikipedia.org/wiki/Row_echelon_form#cite_note-1http://en.wikipedia.org/wiki/Pivot_elementhttp://en.wikipedia.org/wiki/Leading_coefficient#Linear_algebrahttp://en.wikipedia.org/wiki/Transposehttp://en.wikipedia.org/wiki/Gaussian_eliminationhttp://en.wikipedia.org/wiki/Gaussian_eliminationhttp://en.wikipedia.org/wiki/Matrix_(mathematics)http://en.wikipedia.org/wiki/Linear_algebrahttp://en.wikipedia.org/wiki/Matrix_(mathematics)#cite_note-29http://en.wikipedia.org/wiki/Cramer%27s_rulehttp://en.wikipedia.org/wiki/Linear_systemhttp://en.wikipedia.org/wiki/Matrix_(mathematics)#cite_note-28http://en.wikipedia.org/wiki/Minor_(linear_algebra)http://en.wikipedia.org/wiki/Laplace_expansion


13/14

Know what Row-Echelon form is. The Row-Echelon form is where the leading (first non-zero)entry of each row, has only zeroes below it.

Our Example

So, starting with a Matrix of any size. Do not tamper with the top row, it is easier to do itsystematically. So look at the first entry in each row, and decide what factor of Row 1 needs tobe added/subtracted from each row to get zero as the first term. In the example to the right, wecan see that Row 2 - Row 1 would give us a zero, and Row 3 - 3*Row 1 would also.

2. 3

R2 - R1, and R3 - 3*R1

So, after computing the row operations in step 2, the matrix now looks like this. We canthen see what row operation would give us another zero in the bottom row, it is Row 3 - Row 2.

3. 4

Matrix in Row-Echelon form

Now our final Matrix looks like this, which is in row-echelon form as the first entry in eachrow, has only zeroes below it.

http://www.wikihow.com/Reduce-a-Matrix-to-Row-Echelon-Form.

ROW EQUIVALENT

Inlinear algebra, twomatricesare row equivalent if one can be changed to the other by a sequence

ofelementary row operations. Alternatively, two m n matrices are row equivalent if and only if they

have the samerow space. The concept is most commonly applied to matrices that representsystems

of linear equations, in which case two matrices of the same size are row equivalent if and only if the

correspondinghomogeneoussystems have the same set of solutions, or equivalently the matrices

have the samenull space.

Because elementary row operations are reversible, row equivalence is anequivalence relation. It iscommonly denoted by atilde(~).
http://www.wikihow.com/Reduce-a-Matrix-to-Row-Echelon-Formhttp://www.wikihow.com/Reduce-a-Matrix-to-Row-Echelon-Formhttp://en.wikipedia.org/wiki/Linear_algebrahttp://en.wikipedia.org/wiki/Linear_algebrahttp://en.wikipedia.org/wiki/Linear_algebrahttp://en.wikipedia.org/wiki/Matrix_(mathematics)http://en.wikipedia.org/wiki/Matrix_(mathematics)http://en.wikipedia.org/wiki/Matrix_(mathematics)http://en.wikipedia.org/wiki/Elementary_row_operationshttp://en.wikipedia.org/wiki/Elementary_row_operationshttp://en.wikipedia.org/wiki/Elementary_row_operationshttp://en.wikipedia.org/wiki/Row_spacehttp://en.wikipedia.org/wiki/Row_spacehttp://en.wikipedia.org/wiki/Row_spacehttp://en.wikipedia.org/wiki/Systems_of_linear_equationshttp://en.wikipedia.org/wiki/Systems_of_linear_equationshttp://en.wikipedia.org/wiki/Systems_of_linear_equationshttp://en.wikipedia.org/wiki/Systems_of_linear_equationshttp://en.wikipedia.org/wiki/Systems_of_linear_equations#Homogeneous_systemshttp://en.wikipedia.org/wiki/Systems_of_linear_equations#Homogeneous_systemshttp://en.wikipedia.org/wiki/Systems_of_linear_equations#Homogeneous_systemshttp://en.wikipedia.org/wiki/Null_spacehttp://en.wikipedia.org/wiki/Null_spacehttp://en.wikipedia.org/wiki/Null_spacehttp://en.wikipedia.org/wiki/Equivalence_relationhttp://en.wikipedia.org/wiki/Equivalence_relationhttp://en.wikipedia.org/wiki/Equivalence_relationhttp://en.wikipedia.org/wiki/Tildehttp://en.wikipedia.org/wiki/Tildehttp://en.wikipedia.org/wiki/Tildehttp://www.wikihow.com/Image:Matrix-4.pnghttp://www.wikihow.com/Image:Matrix-4.pnghttp://www.wikihow.com/Image:Matrix-3.pnghttp://www.wikihow.com/Image:Matrix-3.pnghttp://www.wikihow.com/Image:Matrix-2.pnghttp://www.wikihow.com/Image:Matrix-2.pnghttp://www.wikihow.com/Image:Matrix-4.pnghttp://www.wikihow.com/Image:Matrix-4.pnghttp://www.wikihow.com/Image:Matrix-3.pnghttp://www.wikihow.com/Image:Matrix-3.pnghttp://www.wikihow.com/Image:Matrix-2.pnghttp://www.wikihow.com/Image:Matrix-2.pnghttp://www.wikihow.com/Image:Matrix-4.pnghttp://www.wikihow.com/Image:Matrix-4.pnghttp://www.wikihow.com/Image:Matrix-3.pnghttp://www.wikihow.com/Image:Matrix-3.pnghttp://www.wikihow.com/Image:Matrix-2.pnghttp://www.wikihow.com/Image:Matrix-2.pnghttp://www.wikihow.com/Image:Matrix-4.pnghttp://www.wikihow.com/Image:Matrix-4.pnghttp://www.wikihow.com/Image:Matrix-3.pnghttp://www.wikihow.com/Image:Matrix-3.pnghttp://www.wikihow.com/Image:Matrix-2.pnghttp://www.wikihow.com/Image:Matrix-2.pnghttp://www.wikihow.com/Image:Matrix-4.pnghttp://www.wikihow.com/Image:Matrix-4.pnghttp://www.wikihow.com/Image:Matrix-3.pnghttp://www.wikihow.com/Image:Matrix-3.pnghttp://www.wikihow.com/Image:Matrix-2.pnghttp://www.wikihow.com/Image:Matrix-2.pnghttp://www.wikihow.com/Image:Matrix-4.pnghttp://www.wikihow.com/Image:Matrix-4.pnghttp://www.wikihow.com/Image:Matrix-3.pnghttp://www.wikihow.com/Image:Matrix-3.pnghttp://www.wikihow.com/Image:Matrix-2.pnghttp://www.wikihow.com/Image:Matrix-2.pnghttp://en.wikipedia.org/wiki/Tildehttp://en.wikipedia.org/wiki/Equivalence_relationhttp://en.wikipedia.org/wiki/Null_spacehttp://en.wikipedia.org/wiki/Systems_of_linear_equations#Homogeneous_systemshttp://en.wikipedia.org/wiki/Systems_of_linear_equationshttp://en.wikipedia.org/wiki/Systems_of_linear_equationshttp://en.wikipedia.org/wiki/Row_spacehttp://en.wikipedia.org/wiki/Elementary_row_operationshttp://en.wikipedia.org/wiki/Matrix_(mathematics)http://en.wikipedia.org/wiki/Linear_algebrahttp://www.wikihow.com/Reduce-a-Matrix-to-Row-Echelon-Form


14/14

There is a similar notion ofcolumn equivalence, defined by elementary column operations; two

matrices are column equivalent if and only if their transpose matrices are row equivalent. Two

rectangular matrices that can be converted into one another allowing both elementary row and

column operations are called simplyequivalent.

http://en.wikipedia.org/wiki/Row_equivalence.

Invertible matrixFrom Wikipedia, the free encyclopedia

Inlinear algebraan n-by-n (square)matrixA is called invertible (some authors

use nonsingularornondegenerate) if there exists an n-by-n matrix B such that

where In denotes the n-by-nidentity matrixand the multiplication used is ordinarymatrix multiplication.

If this is the case, then the matrix B is uniquely determined by A and is called the inverseofA,

denoted by A1

. It follows from the theory of matrices that if forfinite square matrices A and B, then also.

Non-square matrices (m-by-n matrices for which m n) do not have an inverse. However, in some cases

such a matrix may have aleft inverseorright inverse. IfA is m-by-n and therankofA is equal to n,

then A has a left inverse: an n-by-m matrix B such that BA = I. IfA has rank m, then it has a right inverse:

an n-by-m matrix B such that AB = I.

A square matrix that is not invertible is called singularordegenerate. A square matrix is singularif and

only ifitsdeterminantis 0. Singular matrices are rare in the sense that if you pick a random square matrix

over acontinuous uniform distributionon its entries, it willalmost surelynot be singular.

While the most common case is that of matrices over the realorcomplexnumbers, all these definitions can

be given for matrices over anycommutative ring. However, in this case the condition for a square matrix to

be invertible is that its determinant is invertible in the ring, which in general is a much stricter requirement

than being nonzero. The conditions for existence of left-inverse resp. right-inverse are more complicated

since a notion of rank does not exist over rings.

Matrix inversion is the process of finding the matrix B that satisfies the prior equation for a given invertible

matrix A.
http://en.wikipedia.org/wiki/Matrix_equivalencehttp://en.wikipedia.org/wiki/Matrix_equivalencehttp://en.wikipedia.org/wiki/Matrix_equivalencehttp://en.wikipedia.org/wiki/Row_equivalencehttp://en.wikipedia.org/wiki/Linear_algebrahttp://en.wikipedia.org/wiki/Linear_algebrahttp://en.wikipedia.org/wiki/Linear_algebrahttp://en.wikipedia.org/wiki/Matrix_(mathematics)http://en.wikipedia.org/wiki/Matrix_(mathematics)http://en.wikipedia.org/wiki/Matrix_(mathematics)http://en.wikipedia.org/wiki/Identity_matrixhttp://en.wikipedia.org/wiki/Identity_matrixhttp://en.wikipedia.org/wiki/Identity_matrixhttp://en.wikipedia.org/wiki/Matrix_multiplicationhttp://en.wikipedia.org/wiki/Matrix_multiplicationhttp://en.wikipedia.org/wiki/Matrix_multiplicationhttp://en.wikipedia.org/wiki/Inverse_element#Matriceshttp://en.wikipedia.org/wiki/Inverse_element#Matriceshttp://en.wikipedia.org/wiki/Inverse_element#Matriceshttp://en.wikipedia.org/wiki/Inverse_element#Matriceshttp://en.wikipedia.org/wiki/Inverse_element#Matriceshttp://en.wikipedia.org/wiki/Inverse_element#Matriceshttp://en.wikipedia.org/wiki/Rank_(linear_algebra)http://en.wikipedia.org/wiki/Rank_(linear_algebra)http://en.wikipedia.org/wiki/Rank_(linear_algebra)http://en.wikipedia.org/wiki/If_and_only_ifhttp://en.wikipedia.org/wiki/If_and_only_ifhttp://en.wikipedia.org/wiki/If_and_only_ifhttp://en.wikipedia.org/wiki/If_and_only_ifhttp://en.wikipedia.org/wiki/Determinanthttp://en.wikipedia.org/wiki/Determinanthttp://en.wikipedia.org/wiki/Determinanthttp://en.wikipedia.org/wiki/Continuous_uniform_distrib

all about matrices

Documents