EGR 106 Lecture 4 Array Mathematics

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EGR 106 Lecture 4Array Mathematics

• Linear Algebraic Operations
• Additon/Subtraction
• Multiplication
• Division
• Element by Element Operations
• Row/Column Based Operations
• Textbook 3.1-3.8

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Element by Element Math Operations

• For arrays of identical sizes, addition is
  defined term by term
• the command F A B means
• F(r,c) A(r,c) B(r,c)
• for all row and column pairs r,c

element-by-element addition
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• For example
• Notes
• Arrays must be of identical sizes
• One can be a scalar (it is sized up)
• Subtraction is identical

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• The other basic math operations work element by
  element using the dot notation (with A,B the same
  sizes)
• multiplication
• F A . B ? F(r,c) A(r,c) B(r,c)
• division
• F A ./ B ? F(r,c) A(r,c) / B(r,c)
• exponentiation
• F A . B ? F(r,c) A(r,c) B(r,c)

note periods!
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• For example

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• One could be scalar a 1 2 3 b 2

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• Built-in functions also work element-by-element
• log and exp
• trigonometric
• etc.

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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

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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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• And here it doesnt work at all

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Application of Multiplication

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

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• For example
• In matrix form this is A x b
• with

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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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• Wide Spread Applications in Engineering currents
  in circuits traffic flow, static equilibrium
  problems, fluid flow in piping systems, heat flow
  in thermodynamics, computational simulation

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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

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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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• For example
• Check your work

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General Linear Equation Solving (not in the
book!)

• Problem types
• overdetermined
• underdetermined
• Solution methods
• Cramer's method
• Gaussian elimination
• inverse matrix
• others

• Solution situations
• non-singular
• one unique solution
• singular
• no solution
• many solutions
• Matlab does them all

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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