matrices 1
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http://www2.esc.auckland.ac.nz/teaching/ENGSCI111FC/TRANSCRIPT
MatricesMM1 module 3, lecture 1
David Godfrey
Slide number 2
Matrices lecture 1 objectives
• After this lecture you should have a clear understanding of:• What matrices are;• Performing basic operations on matrices;• Special forms of matrices;• The matrix determinant;• How to calculate the inverse of a 2 2 matrix.
Slide number 3
What is a matrix?
• A matrix is a rectangular array of elements
• The elements may be of any type (e.g. integer, real, complex, logical, or even other matrices).
• In this course we will only consider matrices that have integer, real, or complex elements.
5 0 1 23 4 9 23 1 4 2
Slide number 4
Order of matrices…
• Order 4 3:
• Order 3 4:
3 columns
4 rows
5 0 1
2 3 4
9 2 6
3 1 4
4 columns
3 rows
5 0 1 2
3 4 9 2
3 1 4 2
Slide number 5
…Order of matrices
• Order 2 4:
• Order 1 6:
• Order 3 1:
1 columns
3 rows
3
1
5
6 columns1 rows
2 1 1 2 1 5
4 columns2 rows
23 0.5 4.3 12
8 2 8 1
Slide number 6
Specifying matrix elements
• aij denotes the element of the matrix A on the ith row and jth column.
A
column j
row i
5 0 12 3 4 9 2 63 1 4
• a12 = 0
• a21 = 2
• a23 = -4
• a32 = 2
• a41 = 3
• a43 = 4
Slide number 7
What are matrices used for?
• Transformations• Transitions• Linear equations
Slide number 8
Matrix operations: scalar multiplication
• Multiplying an m n matrix by a scalar results in an m n matrix with each of its elements multiplied by the scalar.
• e.g.
826
12418
864
2010
413
629
432
105
2
1239
18627
1296
3015
413
629
432
105
3
Slide number 9
Matrix operations: addition…
• Adding or subtracting an m n matrix by an m n matrix results in an m n matrix with each of its elements added or subtracted.
• e.g.
431
8112
113
136
024
213
541
231
413
629
432
105
417
436
971
334
024
213
541
231
413
629
432
105
Slide number 10
…Matrix operations: addition
• Note that matrices being added or subtracted must be of the same order.
• e.g.
invalid! 113
201
413
629
432
105
Slide number 11
Matrix operations: multiplication…
• Multiplying an m n matrix by an n p matrix
results in an m p matrix
wsrom
columnsp
wsron
columnsp
wsrom
columnsn
Slide number 12
…Matrix operations: multiplication…
• Example 1…
1 0 2
3 1 1
2 1
3 2
1 4
0
( 12) (0 3) (2 1) 0
1 0 2
3 1 1
2 1
3 2
1 4
0 7
( 1 1) (0 2) (2 4) 7
Slide number 13
…Matrix operations: multiplication…
• …Example 1
1 0 2
3 1 1
2 1
3 2
1 4
0 7
10
(32) (13) (11) 10
1 0 2
3 1 1
2 1
3 2
1 4
0 7
10 9
(3 1) (1 2) (1 4) 9
Slide number 14
…Matrix operations: multiplication…
• Example 2…
3)13()32(
3
21
13
41
32
8)23()12(
83
21
13
41
32
Slide number 15
…Matrix operations: multiplication…
• …Example 2
1)14()31(
1
83
21
13
41
32
9)24()11(
91
83
21
13
41
32
Slide number 16
…Matrix operations: multiplication…
• Example 3
• i.e. the number of columns in the first matrix must equal the number of rows in the second matrix!
invalid!
113
201
124
862
351
Slide number 17
…Matrix operations: multiplication…
• Example 4
4 2 1
5
6
(4 1) ( 2 5) 6
Slide number 18
…Matrix operations: multiplication…
• Example 5…
4)41(
424
5
1
2)21(
2424
5
1
Slide number 19
…Matrix operations: multiplication…
• …Example 5
20)45(
20
2424
5
1
10)25(
1020
2424
5
1
Slide number 20
…Matrix operations: multiplication
• Matrix multiplication is NOT commutative• In general, if A and B are two matrices then
A B ≠ B A• i.e. the order of matrix multiplication is
important!• e.g.
10
22
10
02
10
21
10
42
10
21
10
02
Slide number 21
Matrix operations: transpose…
• If B = AT, then bij = aji
• i.e. the transpose of an m n matrix is an n m matrix with the rows and columns swapped.
• e.g.
4641
1230
3925
413
629
432
105T
3925
3
9
2
5
T
Slide number 22
…Matrix operations: transpose
• (A B)T = BT AT
• Note the reversal of order.• Justification (not a proof):
e.g. if A is 3 2 and B is 2 4then AT is 2 3 and BT is 4 2so AT BT cannot be multipliedbut BT AT can be multiplied.
Slide number 23
Special matrices: row and column
• A 1 n matrix is called a row matrix.e.g.
• An m 1 matrix is called a column matrix.e.g.
1 columns
3 rows
3
1
5
6 columns1 rows
2 1 1 2 1 5
Slide number 24
Special matrices: square
• An n n matrix is called a square matrix.i.e. a square matrix has the same number of rows and columns.e.g.
4705
7350
0523
5031
211
010
210
21
321
Slide number 25
Special matrices: diagonal
• A square matrix is diagonal if non-zero elements only occur on the leading diagonal.i.e. aij = 0 for i ≠ je.g.
• Premultiplying a matrix by a diagonal matrix scales each row by the diagonal element.
• Postmultiplying a matrix by a diagonal matrix scales each column by the diagonal element.
4000
0300
0020
0001
200
010
000
20
021
Slide number 26
Special matrices: triangular
• A lower triangular matrix is a square matrix having all elements above the leading diagonal zero.e.g.
• An upper triangular matrix is a square matrix having all elements below the leading diagonal zero.e.g.
4705
0350
0023
0001
1 20 1
Slide number 27
Special matrices: null
• The null matrix, 0, behaves like 0 in arithmetic addition and subtraction.
• Null matrices can be of any order and have all of their elements zero.
0000
0000
0000
0000
00
00
00
00
00
0000
00
00
Slide number 28
Special matrices: identity…
• The identity matrix, I, behaves like 1 in arithmetic multiplication.
• Identity matrices are diagonal. They have 1s on the diagonal and 0s elsewhere.e.g.
• In the world of the matrix the identity truly is ‘the one’.
1000
0100
0010
0001
100
010
001
10
011
II
II
Slide number 29
…Special matrices: identity
• The identity matrix multiplied by any compatible matrix results in the same matrix.i.e. I A = Ae.g.
• Any matrix multiplied by a compatible identity matrix results in the same matrix.i.e. A I = Ae.g.
• Multiplication by the identity matrix is thus commutative.
51
13
51
13
10
01
51
13
10
01
51
13
Slide number 30
Determinant of a 22 matrix…
• Matrices can represent geometric transformations, such as scaling, rotation, shear, and mirroring.
• 2 2 matrices can represent geometric transformations in a 2–dimensional space, such as a plane.
• Determinants of 2 2 matrices give us information about how such transformations change the area of shapes.
• Determinants are also useful to define the inverse of a matrix.
Slide number 31
…Determinant of a 22 matrix…
• The determinant of a 2 2 matrix is the product of the 2 leading diagonal terms minus the product of the cross- diagonal.
• i.e. if A is a 2 2 matrix, then the determinant of A is denoted by det(A) = |A| = a11 a22 – a21 a12
• e.g.
det3 1
2 6
3 1
2 6 36 2 1 20
det 2 5
1 3
2 5
1 3 2 3 15 11
Slide number 32
…Determinant of a 22 matrix
• |A B| = |A| |B|• Note the order is not important.• Justification (not a proof):
We will shortly see that A and B can represent geometric transformations, and A B represents the combined transformation of B followed by A. The determinant represents the factor by which the area is changed, so the combined transformation changes area by a factor |A B|. Looking at the individual transformations, the area of the first is changed by a factor |B|, and the second by |A|. The overall transformation is thus changed by a factor |B| |A|, which is the same as |A| |B|.
Slide number 33
Inverse of a matrix
• In arithmetic multiplication the inverse of a number c is 1/c sincec 1/c = 1 and 1/c c = 1
• For matrices the inverse of a matrix A is denoted by A-1
• A A-1 = IA-1 A = Iwhere I is the identity matrix.
• Multiplication of a matrix by its inverse is thus commutative.
• We shall only consider the inverse of 2 2 matrices.
Slide number 34
Inverse of a 22 matrix…
• The inverse of a 2 2 matrix A is given by
• Note:The leading term is 1/determinant;The diagonal elements are swapped;The cross-diagonal elements change their sign.
a11 a12
a21 a22
1
1
a11 a12
a21 a22
a22 a12
a21 a11
Slide number 35
…Inverse of a 22 matrix…
• Example 1
• Note that A A-1 = I (right inverse)
and A-1 A = I (left inverse)
11
24
6
1
11
24
41
211
41
211
10
01
11
24
6
1
41
21
10
01
41
21
11
24
6
1
Slide number 36
…Inverse of a 22 matrix…
• Example 2
• Note that A A-1 = I (right inverse)
and A-1 A = I (left inverse)
2 2
2 3
1
1
2 2
2 3
3 2
2 2
1
10
3 2
2 2
0.3 0.2
0.2 0.2
10
01
2.02.0
2.03.0
32
22
10
01
32
22
2.02.0
2.03.0
Slide number 37
…Inverse of a 22 matrix…
• Example 3
• Note that the determinant is zero so the inverse does not exist for this matrix.
• Matrices with zero determinant can have no inverse.
• Such matrices are called singular.
2 2
3 3
1
1
2 2
3 3
3 2
3 2
1
0
3 2
2 2
invalid!
Slide number 38
…Inverse of a 22 matrix
• (A B)-1 = B-1 A-1
• Note the reversal of order.• Justification (not a proof):
B-1 A-1 A B = B-1 (A-1 A) B = B-1 B = Iso B-1 A-1 is the inverse of A Bi.e. (A B)-1 = B-1 A-1
Slide number 39
Matrices lecture 1 objectives
• After this lecture you should have a clear understanding of:• What matrices are;• Performing basic operations on matrices;• Special forms of matrices;• The matrix determinant;• How to calculate the inverse of a 2 2 matrix.