es 240: scientific and engineering computation. matrix inverse chapter 11: matrix inverse uchechukwu...

ES 240: Scientific and Engineering Computation. Matrix Inverse

Chapter 11: Matrix Inverse

Uchechukwu Ofoegbu

Temple University


Gaussian Jordan MethodGaussian Jordan Method

Involves complete pivoting so that the coefficient matrix becomes diagonal

Eliminates the need for substitution Also used in finding inverses


Gaussian Jordan MethodGaussian Jordan Method

Steps

1. Write down the matrix corresponding to the linear system.2. Make sure that the first entry in the first column is nonzero. Do

this by interchanging the first row with one of the rows below it, if necessary.

3. Pivot the matrix about the first entry in the first column.• Pivoting in this case also involves normalizing the diagonal entry

4. Make sure that the second entry in the second column is nonzero. Do this by interchanging the second row with one of the rows below it, if necessary.

5. Pivot the matrix about the second entry in the second column.6. Continue in this manner until you have the identity matrix on

the left hand side


Example – System of EquationsExample – System of Equations

Using the Gauss-Jordan method, solve the following system of equations– Using Matlab colon operations– Check with Matlab’s back slash operation

9321

124

2132

zyx

zyx

zyx


Infinitely Many SolutionsInfinitely Many Solutions

When a linear system cannot be completely diagonalized,

– Apply the Gaussian elimination method to as many columns as possible. Proceed from left to right, but do not disturb columns that have already been put into proper form.

– Variables corresponding to columns not in proper form can assume any value.

– The other variables can be expressed in terms of the variables of step 2.

– 4. This will give the general form of the solution.


Example Infinitely Many SolutionsExample Infinitely Many Solutions

Find all solutions of 2 2 4 8

2 2

5 2 2.

x y z

x y z

x y z

General Solution

z = any real number

x = 3 - 2z

y = 1

1 [1]2[2] ( 1)[1][3] 1 [1]

2 2 4 8 1 1 2 4

1 1 2 2 0 2 0 2

1 5 2 2 0 6 0 6

1 [2]2[1] ( 1)[2][3] 6 [2]

1 0 2 3

0 1 0 1

0 0 0 0


Inconsistent SystemInconsistent System

When using the Gaussian elimination method:

if a row of zeros occurs to the left of the vertical line and a nonzero number is to the right of the vertical line in the same row, then the system

has no solution and is said to be inconsistent.


Example Inconsistent SystemExample Inconsistent System

Find all solutions of 3

5

2 4 4 1.

x y z

x y z

x y z

Because of the last row, the system is inconsistent.

[2] ( 1)[1][3] 2 [1]

1 1 1 3 1 1 1 3

1 1 1 5 0 2 2 2

2 4 4 1 0 2 2 7

1 [2]2[1] (1)[2]

[3] 2 [2]

1 0 0 4

0 1 1 1

0 0 0 5


Gauss-Jordan Method for InversesGauss-Jordan Method for Inverses

Step 1: Write down the matrix A, and on its right write an identity matrix of the same size.

Step 2: Perform elementary row operations on the left-hand matrix so as to transform it into an identity matrix. These same operations are performed on the right-hand matrix.

Step 3: When the matrix on the left becomes an identity matrix, the matrix on the right is the desired inverse.


Example InversesExample Inverses

Find the inverse of

4 2 3

8 3 5 .

7 2 4

A

Step 1:

31 11 0 02 4 40 1 1 2 1 0

3 5 70 0 12 4 4

Step 2:

4 2 3 1 0 0

8 3 5 0 1 0

7 2 4 0 0 1


31 11 0 04 4 20 1 1 2 1 0

1 5 30 0 14 4 2

Example Inverses (2)Example Inverses (2)

1 0 0 2 2 1

0 1 0 3 5 4

0 0 1 5 6 4

1

2 2 1

3 5 4

5 6 4

A

Step 3:


Vector and Matrix NormsVector and Matrix Norms

A norm is a real-valued function that provides a measure of the size or “length” of multi-component mathematical entities such as vectors and matrices.

Vector norms and matrix norms may be computed differently.


Vector NormsVector Norms

For a vector {X} of size n, the p-norm is:

Important examples of vector p-norms include:

p 1:sum of the absolute values X1 xi

i1

n

p 2 :Euclidian norm (length) X2 X

e xi

2

i1

n

p :maximum magnitude X

1inmax xi

Xp x i

p

i1

n

1/ p


Matrix NormsMatrix Norms

Common matrix norms for a matrix [A] include:

n

jij

ni

n

i

n

jijf

n

iij

nj

aA

aA

aA

11

1 1

2

111

maxnorm sum-row

norm Frobenius

maxnorm sum-column


ExampleExample

Find the norms of the following:

54321A

43

21B


Ill-Conditioned ProblemsIll-Conditioned Problems

Linear equations that are solvable but solutions become inaccurate because of rounding errors

Small changes in coefficient matrix that will make big changes in solutions

Ill-conditioned problem is the nature of the problem itself

Use format long ( quad precision) for mild ill-condition problems


Ill-Conditioned ProblemsIll-Conditioned Problems

ill-conditioned problem will have– det(A) very small– cond(A) large

Check ill-conditioned– det(A)*det(inv(A) deviates from 1– inv(inv(A)) deviates from A

A*inv(A) deviates from I cond(A*inv(A)) deviates from 1


MATLAB CommandsMATLAB Commands

MATLAB has built-in functions to compute both norms and condition numbers:– norm(X,p)

• Compute the p norm of vector X, where p can be any number, inf, or ‘fro’ (for the Euclidean norm)

– norm(A,p)• Compute a norm of matrix A, where p can be 1, 2, inf, or ‘fro’ (for the

Frobenius norm)

– cond(X,p) or cond(A,p)• Calculate the condition number of vector X or matrix A using the norm

specified by p.


LabLab

Ex 11.2a) Determine the inverse of the coefficient matrix using the

Gauss-Jordan method step-by-step in Matlab

b) Solve the system of equations using the Gauss-Jordan method step-by-step in Matlab

c) Compare your answers to a and b with Matlab’s built-in functions (inv, back slash)

d) Determine the Frobenius norm of the coefficient matrixa) By handb) Using Matlab’s built-in ‘norm’ functionc) Compare your answers

es 240: scientific and engineering computation. matrix inverse chapter 11: matrix inverse uchechukwu...

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