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

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 = *

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

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

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

Multiplications F=A*B (2col, 2 rows) 1x7+2x8 1x9+2x10 1x11+2x12 23 29 35 3x7+4x8 3x9+4x10 3x11+4x12 53 67 81 5x7+6x8 5x9+6x10 5x11+6x12 83 105 127

Multiplications F=B*A (3cols, 3rows) 7x1+9x3+11x5 7x2+9x4+11x6 89 116 8x1+10x3+12x5 8X2+10x4+12x6 98 128

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

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

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

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)

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

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!

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

For example: (Must be non zero for inv)

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

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

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

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)

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

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

Or multiple vectors:

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

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

Use help to discover how to use these work