© 2005 baylor university slide 1 fundamentals of engineering analysis egr 1302 - matrix...
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© 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
aaa
aaaA
m x n2 x 3
232221
131211
cca
cccC
m x p
333231
232221
131211
bbb
bbb
bbb
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
aaa
aaa
333231
232221
131211
bbb
bbb
bbb
232221
131211
cca
ccc* =
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
aaa
aaaA
3231
2221
1211
bb
bb
bb
B
A * B =
333231
232221
131211
ccc
ccc
ccc
C
2221
1211
cc
ccC
B * A = 3 x 3
2 x 2
© 2005 Baylor UniversitySlide 7
112
310
23
10
21
* =9 5
1 7
Matrix Multiplication
A * B =
© 2005 Baylor UniversitySlide 8
Matrix Multiplication
4 3 -1
2 1 1
4 -1 11
112
310
23
10
21
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 conformable
C * A = not conformable
C * B = F6x5
A * B * C = not conformable
C * B * A = G6x4
© 2005 Baylor UniversitySlide 11
Properties of a Zero Matrix
123
011
21
21
21
* =
00
00
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
21
21
21* =
257
257
257
© 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
x
x
x 312111 xcxbxa
1312111 dxcxbxa
2322212 dxcxbxa
3332313 dxcxbxa
Then can become
333
222
111
cba
cba
cba
3
2
1
x
x
x
3
2
1
d
d
d
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
aaa
aaaA
2313
2212
2111
aa
aa
aa
AT
A is a 2x3, so AT will be a 3x2
333231
232221
131211
bbb
bbb
bbb
B
332313
322212
312111
bbb
bbb
bbb
BTFor a 3x3
© 2005 Baylor UniversitySlide 14
Properties of the Transpose Matrix
112
101TA
11
10
21
A
221
013B
20
21
13TB
11
10
21
221
013
232
221
435
A*B=
AT*BT = ?
BT*AT =
20
21
13
112
101
224
323
215
(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
212
123
235
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
300
020
005
Must be Square: n x nAll off-diagonal elements
Are Zero
333322221111 ,, bababa
If A and B areDiagonal
200
040
001
300
020
005
+A+B
will be Diagonal
500
060
006
=
200
040
001
300
020
005If A and B areDiagonal * A*B
will be Diagonal
600
080
005
=
© 2005 Baylor UniversitySlide 18
The Identity Matrix
100
010
001Must 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
200
010
003
200
010
003
400
010
009
=*
© 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
103
231
012
A
412
150B
A*B – not conformable B*A = ?
Matrix Math on the TI-89 Calculator
© 2005 Baylor UniversitySlide 22
412
150B
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?