Matrices

MSU CSE 260

Outline

• Introduction• Matrix Arithmetic:

– Sum, Product• Transposes and Powers of Matrices

– Identity matrix, Transpose, Symmetric matrices• Zero-one Matrices:

– Join, Meet, Boolean product• Exercise 2.6

Introduction

mnmm

ij

n

n

aaa

a

aaa

aaa

A

21

22221

11211

m rows

n columns

mn matrix

element in ith row, jth column

When m=n, A is called a square matrix.

Also written as A=aij

Definition A matrix is a rectangular array of numbers.

Matrix Equality

• Definition Let A and B be two matrices.A=B if they have the same number of rows and columns, and every element at each position in A equals element at corresponding position in B.

Matrix Addition, Subtraction

Let A = aij , B = bij be mn matrices. Then:A + B = aij + bij, and A - B = aij - bij

34

04

34

32

41

43

02

43

11

30

82

52

32

41

43

02

43

11

Matrix MultiplicationLet A be a mk matrix, and B be a kn matrix,

k

tkjikjijitjitij

ij

babababac

cAB

12211 ...

24232221

14131211

34333231

24232221

14131211

232221

131211

cccc

cccc

bbbb

bbbb

bbbb

aaa

aaa

12321322121211 cbababa

Matching Dimensions

24232221

14131211

34333231

24232221

14131211

232221

131211

cccc

cccc

bbbb

bbbb

bbbb

aaa

aaa

To multiply two matrices, inner numbers must match:

23 34

have to be equal

24 matrix

23 3424

Otherwise,not defined.

Multiplicative Properties

Note that just because AB is defined, BA may not be.Example If A is 34, B is 46, then AB=36, but BA is not defined (46 . 34).Even if both AB and BA are defined, they may not havethe same size. Even if they do, matrices do not commute.

23

34

35

23

11

12

12

11

BAAB

BA

match dimensions assuming

, however )()( BCACAB

Efficiency of Multiplication

24232221

14131211

34333231

24232221

14131211

232221

131211

cccc

cccc

bbbb

bbbb

bbbb

aaa

aaa

a11b12 + a12b22 + a13b32 = c12

Takes 3 multiplications, and 2 additions for each element.This has to be done 24 (=8) times (since product matrix is 24). So 243 multiplications are needed.•(mk) (kn) matrix product requires m.k.n multiplications.

23 34

Best Order?

Let A be a 2030 matrix, B 3040, C 4010. (AB)C or A(BC)?

(2030 3040) 4010

2030 (3040 4010)

operations24000403020

operations 8000104020

operations 12000104030 operations 6000103020

32000

18000

So, A(BC) is best in this case.

Identity Matrix

100

010

001

3I

The identity matrix has 1’s down the diagonal, e.g.:

fe

dc

ba

fe

dc

ba

0000

0000

0000

100

010

001

For a mn matrix A, Im A = A In

mm mn = mn nn

Inverse Matrix

Let A and B be nn matrices.If AB=BA=In then B is called the inverse of A,

denoted B=A-1.

Not all square matrices are invertible.

Use of Inverse to Solve Equations

r

q

p

z

y

x

ccc

bbb

aaa

321

321

321

r

q

p

zcycxc

zbybxb

zayaxa

321

321

321KAIX

KAAXA

KAX

1

11

r

q

p

ccc

bbb

aaa

z

y

x

31

21

11

31

21

11

31

21

11

Please note that a-1j is

NOT necessarily (aj)-1.

Transposes of Matrices

ija

jia

Flip across diagonal

654

321

63

52

41tA written

Transposes are used frequently in various algorithms.

Symmetric Matrix

AAt If A is called symmetric.

201

034

141is symmetric. Note, for A to besymmetric, is has to be square.

nI is trivially symmetric...

Examples

k

xxijx

k

xjxxi

k

x

xjt

ixttt

k

xxijxji

tij

t

k

xxjixij

ttt

baababAB

baccAB

bacAB

ABAB

111

1

)(

)(

Power Matrix

• For a nn square matrix A, the power matrix is defined as:

Ar = A A … A r times

• A0 is defined as In.

Zero-one Matrices

• All entries are 0 or 1.

• Operations are and .• Boolean product is defined using:

for multiplication, and

for addition.

Boolean Operations

011

010

010

101BA

010

000BABA meet"" called

011

111BABA join"" called

Terminology is from Boolean Algebra.Think“join” is “put together”, like union, and“meet” is “where they meet”, or intersect.

Boolean Product

110

011,

01

10

01

BA

nkbBkmaA ijij be and be , )()()(, 2211 kjikjijiijij bababaccBA

(Should be a ‘dot’)

011

110

011

01)00()11(

11)01()10(

01)00()11(

BA

Since this is “or’d”, youcan stop when you finda ‘1’

Boolean Product Properties

111

100

100

,

011

110

011

BA

100

111

100

BA

111

011

011

AB

• In general, A B B A• Example

Boolean Power

• A Boolean power matrix can be defined in exactly the same way as a power matrix. For a nn square matrix A, the power matrix is defined as:

A[r] = A A … Ar times

A[0] is defined as In.

Exercise 2.6