2- 1

Chapter 2 Matrices

• Definition of a matrix

3231

2221

1211

A columns) 2 rows, (3matrix 23 (a)

aa

aa

aa

rcrr

c

c

bbb

bbb

bbb

21

22221

11211

Bmatrix c r (b)

2- 2

6

4

24

-34

24

34

1

2

21

1

3

21

1

3

21

E

PLd

L

E

wLd

L

E

wLd

L

A system of 3 equations:

Represented by a matrix:

1

21

31

3

6

411

24

341

24

314

E

PL

L

E

wL

L

E

wL

L

2- 3

Types of Matrices• Square matrix: # of rows = # of columns

• upper triangular matrix strictly upper triangular matrix

333231

232221

131211

aaa

aaa

aaa

0000

000

00

0

55

4544

353433

25242322

1514131211

a

aa

aaa

aaaa

aaaaa

00000

0000

000

00

0

45

3534

252423

15141312

a

aa

aaa

aaaa

2- 4

• lower triangular matrix strictly lower triangular matrix

• diagonal matrix

0

00

000

0000

5554535251

44434241

333231

2221

11

aaaaa

aaaa

aaa

aa

a

0

00

000

0000

00000

54535251

434241

3231

21

aaaa

aaa

aa

a

000

0 00

0 00

0 00

n

1

2

1

2- 5

• banded matrix

a square matrix with elements of zero except for the principal d

iagonal and values in the positions adjacent to the diagonal.

• tridiagonal matrix

000

00

00

00

000

5554

454443

343332

232221

1211

aa

aaa

aaa

aaa

aa

2- 6

• unit matrix: 1 on the principal diagonal

• null matrix: All elements are zero.

10000

01000

00100

00010

00001

Ι

00000

00000

00000

00000

00000

O

2- 7

• symmetric matrix: a square matrix in which

• skew-symmetric matrix: a square matrix in which for all i a

nd j

jiij aa

1.000.640.27-

0.641.000.23-

0.27-0.23-1.00

jiij aa

2- 8

• transpose of matrix A: AT

• (AT) T = A

jiTij aa

5.63.84.61.77.7

55188531235

60132283195140

5.65560

3.8188132

4.653283

1.712195

7.735140

TAA

2- 9

Matrix Operations

• Matrix equality

• Matrix addition and subtractionC = A + B = B + A (commutative)

C = A - B

ijijij bac

ijijij bac

jiba ijij and allfor if BA

2- 10

• Example: Matrix addition and subtraction

11109

531

642

A

112

487

320

B

121111

9118

962

BAC

1097

156

322

BAD

2- 11

Matrix Multiplication

• One example

8143

55 35

4(9)8(3)3(7)4(3)8(2)3(5)

1(9)6(3)4(7))3(1)2(6)5(4

93

32

75

483

164

CST

2- 12

Rules of Matrix Multiplication

1. # of columns in A = # of rows in B

2. # of rows in C = # of rows in A

3. # of columns in C = # of columns in B

4.

BAC

m

kkjikij bac

1

2- 13

5. Matrix multiplication is not commutative

6. Matrix multiplication is associativeABBA

)())( CBACBA

2- 14

Example: Matrix Multiplication

11109

531

642

A

112

487

320

B

7810992

203131

284240

BAE

282114

1269258

433629

ABF

ABBA

2- 15

Matrix Multiplication by a Scalar

ijij sabs AB

10

11109

531

642

sA

11010090

503010

604020

AB s

An example:

2- 16

Matrix Inversion

where A-1 is the inverse of A, and I is the unit matrix

IAA 1

1 c

0 c

0 c

1 c

22221221

21221121

22121211

21121111

aca

aca

aca

aca

equations ussimultaneo following by the

determined be can inverse the,)( and 2 If2n

cn ij 1A

2- 17

Example: Matrix Inversion

10

01

75

32

2221

1211

cc

cc

75

32A

1 73

0 52

0 73

1 52

2221

2221

1211

1211

cc

cc

cc

cc

25

37

get we

1A

2- 18

Matrix Singularity

• If the inverse of a matrix A exists, then A is said to be nonsingular.

• If the inverse of a matrix A does not exist, then A is said to be singular.

• If matrix A is singular, then the linear system of simultaneous equations represented by A has no unique solution.

2- 19

There are an infinite number of solutions if 2a = b.

There is no feasible solution if 2a b.

Thus matrix A is singular.

bXX

aXX

21

21

64

32

64

32 Let

A

10

01

64

32for solution No

2221

1211

cc

cc

2- 20

• trace of a square matrix = sum of diagonal elements

• matrix augmentation: addition of a column or columns

to the initial matrix

n

iiiatr

1

)(A

100

010

001

432

141

132

432

141

132

aAA

2- 21

• matrix partition

2221

1211

AA

AAA

432

141

132

A

4 32

1

1

41

32

2221

1211

AA

A A

2- 22

Vectors• Column vector

• Row vector

• Vectors of two ordinates

132

2

1

2

2- 23

• orthogonal vectors

Two vectors are said to be orthogonal if their product is equal to zero.

If two vector are orthogonal, they are perpendicular to each other in the n-dimensional space.

01

32 example,For 3

2

2- 24

50

1

2lengthvector .n

ii )v(

• normalized vectors

A vector is normalized by dividing each element by its length.

A normalized vector has a length 1.

Two vectors that are both normalized and orthogonal to each other are said to be orthonormal vectors.

2- 25

Example: Vectors

5] 3- [21 V

1

1

1

2

V

6.16438)5()3((2) of length 2221 V

732.13)1()1((-1) of length 2222 V

2- 26

38

5

38

3

38

21nV

3

1

3

1

3

12nV

Normalized vectors:

l.orthonorma are and

.orthogonal are and ,0 Since

21

11

nn VV

VVVV 22

2- 27

Determinants• A determinant of a matrix A is denoted by |A|.

• The determinant of a 22 matrix:

• The determinant of a 33 matrix:

bcadc d

a b

a a

a a a

a a

a a a

a a

a a a

a a a

a a a

a a a

3231

222113

3331

232112

3332

232211

333231

232221

131211

2- 28

• The minor of aij, denoted by Aij, is the matrix after removing row i and column j.

• The determinant of an nn matrix:

• The general expression for the determinant of an nn matrix:

||a1)(||a||a||a|| 1n1n

131211 1n131211 AAAAA

||)1(||)1(||)1(||)1(|| 333

222

111

ininni

iii

iii

iii aaaa AAAAA

2- 29

Example: Matrix Determinant

• with the first row and their minors:

11109

531

642

A

|||||||| 131211 131211 AAAA aaa

0)]9(3)10(1[6)]9(5)11(1[4)]10(52[3(11)

109

316

119

514

1110

532

11109

531

642

||

A

2- 30

• with the second column and their minors:

• Since |A|=0, A is a singular matrix; that is the inverse of A doest not exist.

|||||||| 232212 232212 AAAA aaa

0]610[10]5422[3]4511[4

51

6210

119

623

119

514

11109

531

642

||

A

11109

531

642

A

2- 31

Properties of Determinants

1. If the values in any row (column) are proportional to the corresponding values in another row(column), the determinant equals zero

0|| where,

353

2142

121

AA

0|| where,

653

4142

221

AA

2- 32

2. If all the elements in any row(column) equal zero, the determinant equals zero.

3. If all the elements of any row(column) are multiplied by a constant c, the value of the determinant is multiplied by c.

14)]4(2)5(3[2|| where,54

)2(2)3(2

54

46

AA

2- 33

4. The value of the determinant is not changed by adding any row (column) multiplied by a constant c to another row (column).

5. If any two rows (columns) are interchanged, the sign of the determinant is changed.

7)]4(2)5(3|| where,54

23

AA

7)4(3)5(1|| where,54

3-1-

BB

-73(5)-2(4)45

32 and 72(4)-3(5)

54

23

2- 34

6. The determinant of a matrix equals that of its transpose; that is, |A| = |AT|.

7. If a matrix A is placed in diagonal form, then the product of the elements on the diagonal equals the determinant of A.

74(2)-3(5)52

43 and 72(4)-3(5)

54

23

7)3

7(3

3

70

03||

3

70

03

3

70

23

72(4)3(5)|A| with,54

23

AA

A

2- 35

8. If a matrix A has a zero determinant, then A is a singular matrix; that is, the inverse of A does not exist.

2- 36

Rank of A Matrix

• A matrix of r rows and c columns is said to be of order r by c. If it is a square matrix, r by r, then the matrix is of order r.

• The rank of a matrix equals the order of highest-order nonsingular submatrix.

2- 37

3 square submatrices:

Each of these has a determinant of 0, so the rank is less than 2. Thus the rank of R is 1.

Example 1: Rank of Matrix

842

421 matrix,order 32 R

84

42 ,

82

41 ,

42

21321

RRR

2- 38

Since |A|=0, the rank is not 3. The following submatrix has a nonzero determinant:

Thus, the rank of A is 2.

Example 2: Rank of Matrix

11109

531

642

A

2)1(4)3(231

42