1. EGR 106 – Week 4 – Math on Arrays
Linear algebraic operations:
Multiplication
Division
Row/column based operations
Rest of chapter 3

2. Array Multiplication (Linear Algebra)
In linear algebra, the matrix expression F = A * B means
Entries are dot products of rows of the first matrix with columns of the second
= *

3. For example: a b c d
a1xc1 + b1xc2 a1xd1 + b1xd2

4. Notes: The operation is generally not commutative A*B ≠ B*A
The number of columns of the 1st must match the number of rows of the 2nd
= *
n by m
n by k
k by m

5. For example, here multiplication works both ways, but is not commutative:
quite different!

6. Multiplications F=A*B (2col, 2 rows)
1x7+2x8 1x9+2x x11+2x12
3x7+4x x9+4x x11+4x12
5x7+6x8 5x9+6x x11+6x12

7. Multiplications F=B*A (3cols, 3rows)
7x1+9x3+11x x2+9x4+11x6
8x1+10x3+12x X2+10x4+12x6

8. And here it doesn’t work at all:
(2 cols, 3 rows)

9. Application of Multiplication
Application of matrix multiplication: n simultaneous equations in m unknowns (the x’s)
n rows, m columns

10. In matrix form this is A * x = b with
For example:
In matrix form this is A * x = b
with
Coefficients

11. column vectors (lower case)
In general: A * x = b
A is n by m
x is m by 1
b is n by 1
column vectors (lower case)

12. Usages – finding: v cable tensions in statics fluid flow in piping
heat flow in thermodynamics
e.g. v
currents in circuits
traffic flow
economics R1 R3 R2

13. Array Division Recall the command eye(n) This result is the array
multiplication identity matrix I
For any array A
A * I = I * A = A
must be properly sized!

14. Imagine that for square arrays A and B we have
A * B = B * A = I
then we call them inverses
A = B– B = A–1
In Matlab: A ^ or inv(A)
When does A–1 exist?
A is square
A has a non-zero determinant (det(A))

15. For example:
(Must be non zero for inv)

16. A–1 *A * x = A–1* b so x = A–1* b = I = x Solving A * x = b
Assume that A is square and det(A) ≠ 0
Multiply both sides by A–1 on the left
A–1 *A * x = A–1* b
so x = A–1* b
In Matlab, x = A \ b or x = inv(A)*b
= I
= x
backwards slash

17. For example: Check your work: 9x2+8x3=6 7x1+4x2+5x3=8 4x1+4x2+2x3=0 x1

18. Vector Based Operations
Some operations analyze a vector to yield a single value. For example:
sums the elements

19. Other operations for a vector A:
Minimum: min(A)
Maximum: max(A)
Median: median(A)
Mean or average: mean(A)
Standard deviation: std(A)
Product of the elements: prod(A)

20. Some operators yield two results:
min and max can yield both the value and its location
default is the first result

21. Some operators yield vector results
size(A) we’ve already seen
sort

22. Or multiple vectors:

23. Finally, when applied to an array, these operators perform their action on columns

24. the 2 means “use the 2nd dimension” i.e. spanning the columns
Unless you instruct it to work on rows!
the 2 means “use the 2nd dimension” i.e. spanning the columns

25. Use help to discover how to use these work