EGR 106 Week 4 Math on Arrays

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EGR 106 Week 4 Math on Arrays

• Linear algebraic operations
• Multiplication
• Division
• Row/column based operations
• Rest of chapter 3

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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

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• For example

a
b
c
d
a1xc1 b1xc2 a1xd1 b1xd2
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• Notes
• The operation is generally not commutative
• AB ? BA
• The number of columns of the 1st must match the
  number of rows of the 2nd

n by k
k by m
n by m
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• For example, here multiplication works both ways,
  but is not commutative

quite different!
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Multiplications FAB (2col, 2 rows)

• 1x72x8 1x92x10 1x112x12
• 23 29 35
• 3x74x8 3x94x10 3x114x12
• 53 67 81
• 5x76x8 5x96x10 5x116x12
• 83 105 127

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Multiplications FBA (3cols, 3rows)

• 7x19x311x5 7x29x411x6
• 89 116
• 8x110x312x5 8X210x412x6
• 98 128

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• And here it doesnt work at all

(2 cols, 3 rows)
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Application of Multiplication

• Application of matrix multiplication n
  simultaneous equations in m unknowns (the xs)

n rows, m columns
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• For example
• In matrix form this is A x b
• with

Coefficients
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• In general A x b
• A is n by m
• x is m by 1
• b is n by 1
• 
  

column vectors (lower case)
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• Usages finding
• cable tensions in statics
• fluid flow in piping
• heat flow in thermodynamics
• e.g.
• v

• currents in circuits
• traffic flow
• economics

R1
R3
R2
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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!
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• Imagine that for square arrays A and B we have
• A B B A I
• then we call them inverses
• A B1 B A1
• In Matlab A -1 or inv(A)
• When does A1 exist?
• A is square
• A has a non-zero determinant (det(A))

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• For example

(Must be non zero for inv)
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• Solving A x b
• Assume that A is square and det(A) ? 0
• Multiply both sides by A1 on the left
• A1 A x A1 b
• so x A1 b
• In Matlab, x A \ b or x inv(A)b

I
x
backwards slash
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9x28x36 7x14x25x38 4x14x22x30

• For example
• Check your work

x1 x2 x3
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Vector Based Operations

• Some operations analyze a vector to yield a
  single value. For example

sums the elements
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• 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)

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

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• Some operators yield vector results
• size(A) weve already seen
• sort

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• Or multiple vectors

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

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• Unless you instruct it to work on rows!

the 2 means use the 2nd dimension i.e. spanning
the columns
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• Use help to discover how to use these work