Fundamentals of Engineering Analysis EGR 1302 - Matrix Multiplication, Types

© 2005 Baylor UniversitySlide 1

Fundamentals of Engineering AnalysisEGR 1302 - Matrix Multiplication, Types

© 2005 Baylor UniversitySlide 2

Matrix Multiplication

Define “Conformable”

To multiply A * B, the matrices must be conformable.

Given matrices: A m x n and B n x p

The number of “Columns” n of A, must equal the number of “Rows” n of BWhich defines the order of the multiplication

A=

2 x 3

For A: m=2, n=3 B=

3 x 4

For B: n=3, p=4

Note that B * A is Undefined (not allowed) because p = m

For A * B, n=n; i.e. 3=3, so A*B is “conformable”

© 2005 Baylor UniversitySlide 3

Order of Multiplication

The order in which a multiplication is expressed is important.

We use the terms “pre-multiply” or “post-multiply” to stipulate the order.

Given A * B = C, we say that “B” is “pre-multiplied” by “A”(we could also say that A is post-multiplied by B).

In other words, Matrix Multiplication is NOT Commutative(except in special cases)

Because matrices must be conformable for multiplication; in general

A * B = B * A

© 2005 Baylor UniversitySlide 4

Matrix Multiplication

232221

131211

aaaaaa

A

m x n2 x 3

232221

131211

ccaccc

Cm x p

333231

232221

131211

bbbbbbbbb

B

n x p3 x 3

=

A * B = C is Conformable

Is a Row on Column operation

The Product C will be a 2 x 3

© 2005 Baylor UniversitySlide 5

Matrix Multiplication

232221

131211

aaaaaa

333231

232221

131211

bbbbbbbbb

232221

131211

ccaccc* =

31132112111111 bababac

C11 is made up of Row 1 from A, and Column 1 from B

Note the “sum of products” form

C12 is made up of Row 1 from A, and Column 2 from B

32132212121112 bababac

Remember: 31132112111111 bababac

© 2005 Baylor UniversitySlide 6

Matrix Multiplication

232221

131211

aaaaaa

A

3231

2221

1211

bbbbbb

B

A * B =

333231

232221

131211

ccccccccc

C

2221

1211

cccc

C

B * A = 3 x 3

2 x 2

© 2005 Baylor UniversitySlide 7

112310

231021

* =9 5

1 7

Matrix Multiplication

A * B =

© 2005 Baylor UniversitySlide 8

Matrix Multiplication

4 3 -1

2 1 1

4 -1 11

112310

231021

B * A =

* =

Work this out yourself, before proceeding,

To make sure you understand the method of matrix multiplication.

© 2005 Baylor UniversitySlide 9

Linear Systems as Sum of Products

ax1 + bx2 + cx3 = d

Sum of Products form

[ a b c ] - a 1 x 3 row vector

x1

x2

x3

- a 3 x 1 column vector

[ a b c ] * = [ d ] - a 1 x 1 scalar – i.e.;

x1

x2

x3

ax1 + bx2 + cx3 = d

© 2005 Baylor UniversitySlide 10

Conformability and Order of Matrix Multiplication

Given: A5x4 B4x5 C6x4

A * B = D5x5

B * A = E4x4

A * C = not conformableC * A = not conformableC * B = F6x5

A * B * C = not conformableC * B * A = G6x4

© 2005 Baylor UniversitySlide 11

Properties of a Zero Matrix

123

011

212121

* =

0000

In Algebra, x * 0 = 0, but if x = 0, and y = 0, then x * y = 0

In Matrix Algebra, even if A = 0, and B = 0, A * B can be [0]

Note that:

123

011

212121

* =

257257257

© 2005 Baylor UniversitySlide 12

Matrix Form of Linear Equations

Distributive Property: A(B+C) = AB + AC

Associative Property: A(BC) = (AB)C

111 cba

3

2

1

xxx 312111 xcxbxa

1312111 dxcxbxa

2322212 dxcxbxa

3332313 dxcxbxa

Then can become

333

222

111

cbacbacba

3

2

1

xxx

3

2

1

ddd

A x d

* = Any Order

?How do wesolve thissystem

of equations

© 2005 Baylor UniversitySlide 13

Special Matrices

The Transpose Matrix

Rule: The Row becomes the Column, and the Column becomes the Row

232221

131211

aaaaaa

A

2313

2212

2111

aaaaaa

AT

A is a 2x3, so AT will be a 3x2

333231

232221

131211

bbbbbbbbb

B

332313

322212

312111

bbbbbbbbb

BTFor a 3x3

© 2005 Baylor UniversitySlide 14

Properties of the Transpose Matrix

112101TA

111021

A

221013

B

202113

TB

111021

221

013

232221435

A*B=

AT*BT = ?

BT*AT =

202113

112101

224323215

(AB)T= BT*AT

© 2005 Baylor UniversitySlide 15

Additional Properties of the Transpose

If A+B and A*B are allowed (are conformable), then

(A+B)T = AT + BT

(AB)T = BTAT

© 2005 Baylor UniversitySlide 16

The Symmetric Matrix

212123235

A = AT

Must be Square: n x n

jiij aa

A + AT must also be Symmetric

The Diagonal

© 2005 Baylor UniversitySlide 17

The Diagonal Matrix

300020005

Must be Square: n x nAll off-diagonal elements

Are Zero

333322221111 ,, bababa

If A and B areDiagonal

200040001

300020005

+A+B

will be Diagonal

500060006

=

200040001

300020005

If A and B areDiagonal * A*B

will be Diagonal

600080005

=

© 2005 Baylor UniversitySlide 18

The Identity Matrix

100010001

Must be Square: n x nAnd must be Diagonal

Can be any Order Notation: IN The Unity term

A*I = AI*A = A

A does not have to be square

Amxn * In = A or Im * Amxn = A

© 2005 Baylor UniversitySlide 19

Powers of Matrices

A * A = A2 for Square Matrices Only

A * A2 = A3 … and so on

If A is Diagonal … A2 = a112, a22

2, a332

200010003

200010003

400010009

=*

© 2005 Baylor UniversitySlide 20

Matrix Math on the TI-89 Calculator

My Philosophy for using Calculators(and Computers …)

Be aware of theOrder of Magnitude

Sign Errors are easy to miss

Double check your work

If you understand the solution methodology,You will understand the answer.

© 2005 Baylor UniversitySlide 21

103231012

A

412150

B

A*B – not conformable B*A = ?

Matrix Math on the TI-89 Calculator

© 2005 Baylor UniversitySlide 22

412150

B

Matrix Math on the TI-89 Calculator (cont.)

© 2005 Baylor UniversitySlide 23

Matrix Math on the TI-89 Calculator (cont.)

© 2005 Baylor UniversitySlide 24

Using the Matrix Editor on the TI-89

© 2005 Baylor UniversitySlide 25

Questions?