Fundamentals of Engineering Analysis ENGR Matrix Multiplication, Types

Transcription

1 Fundmentls of Engineering Anlysis ENGR - Mtri Multiplition, Types Spring Slide

2 Mtri Multiplition Define Conformle To multiply A * B, the mtries must e onformle. Given mtries: A m n nd B n p The numer of Columns n of A, must equl the numer of Rows n of B Whih defines the order of the multiplition A= For A: m=, n= B= 4 For B: n=, p=4 For A * B, n=n; i.e. =, so A*B is onformle Note tht B * A is Undefined (not llowed) euse p = m Slide

3 Order of Multiplition The order in whih multiplition is epressed is importnt. We use the terms pre-multiply or post-multiply to stipulte the order. Given A * B = C, we sy tht B is pre-multiplied y A (we ould lso sy tht A is post-multiplied y B). Beuse mtries must e onformle for multiplition; in generl A * B = B * A In other words, Mtri Multiplition is NOT Commuttive (eept in speil ses) Slide

4 Mtri Multiplition Is Row on Column opertion A m n B n p = A * B = C is Conformle C m p The Produt C will e Slide 4

5 Mtri Multiplition * = C is mde up of Row from A, nd Column from B Note the sum of produts form C is mde up of Row from A, nd Column from B Rememer: Slide 5

6 Mtri Multiplition A B A * B = C B * A = C Slide 6

7 Mtri Multiplition A * B = * = Slide 7

8 Mtri Multiplition B * A = * = Work this out yourself, efore proeeding, To mke sure you understnd the method of mtri multiplition. Slide 8

9 Liner Systems s Sum of Produts + + = d Sum of Produts form [ ] - row vetor - olumn vetor [ ] * = [ d ] - slr i.e.; + + = d Slide 9

10 Conformility nd Order of Mtri Multiplition Given: A 54 B 45 C 64 A * B = D 55 B * A = E 44 A * C = not onformle C * A = not onformle C * B = F 65 A * B * C = not onformle C * B * A = G 64 Slide

11 Slide Properties of Zero Mtri * = In Alger, * =, ut if =, nd y =, then * y = In Mtri Alger, even if A =, nd B =, A * B n e [] Note tht: * =

12 Slide Mtri Form of Liner Equtions Distriutive Property: A(B+C) = AB + AC Assoitive Property: A(BC) = (AB)C d d d Then n eome d d d A d * = Any Order? How do we solve this system of equtions

13 Speil Mtries The Trnspose Mtri Rule: The Row eomes the Column, nd the Column eomes the Row A A T A is, so A T will e For B B T Slide

14 Slide 4 Properties of the Trnspose Mtri T A A B T B 4 5 A*B= A T *B T =? B T *A T = 4 5 (AB) T = B T *A T

15 Additionl Properties of the Trnspose If A+B nd A*B re llowed (re onformle), then (A+B) T = A T + B T (AB) T = B T A T Slide 5

16 The Symmetri Mtri 5 Must e Squre: n n The Digonl ij ji A = A T A + A T must lso e Symmetri Slide 6

17 Slide 7 The Digonl Mtri 5 Must e Squre: n n All off-digonl elements Are Zero,, If A nd B re Digonl A+B will e Digonl = 4 5 If A nd B re Digonl * A*B will e Digonl =

18 The Identity Mtri Must e Squre: n n And must e Digonl Cn e ny Order Nottion: I N The Unity term I*A = A A*I = A A does not hve to e squre A mn * I n = A or I m * A mn = A Slide 8

19 Slide 9 Powers of Mtries A * A = A for Squre Mtries Only A * A = A nd so on If A is Digonl A =,, 4 9 = *

20 Mtri Mth on the TI-89 Clultor My Philosophy for using Clultors (nd Computers ) Be wre of the Order of Mgnitude Sign Errors re esy to miss Doule hek your work If you understnd the solution methodology, You will understnd the nswer. Slide

21 Slide A 4 5 B A*B not onformle B*A =? Mtri Mth on the TI-89 Clultor

22 Mtri Mth on the TI-89 Clultor (ont.) B 5 4 Slide

23 Mtri Mth on the TI-89 Clultor (ont.) Slide

24 Using the Mtri Editor on the TI-89 Slide 4

25 Questions? Slide 5