What Is a Matrix

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What Is a Matrix?

• COMP 116
• August 29, 2007

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Outline What is a matrix?

• A support for something growing
• A table of numbers
• A stack (column) of row vectors
• Creating, Indexing, Operating
• A row of column vectors
• A collection of data in columns
• Plotting
• A system of equations or polynomials
• A transformation for a vector space
• A linear operator on a vector space
• A change of variables or coordinates

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www.whatisthematrix.com

• An interconnected network or enclosure in which
  something grows

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A matrix is a table of numbers

• Create
• With functions
• rand(3,4)ones(3,4)zeros(3)
• eye(4)
• diag(1,2,3,4)
• 1./hilb(4)
• Helpwin gallery
• Search for matrices

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A table of numbers

• Create
• With brackets semicolon1 2 3 467, 8, 9
  10 11 12
• Commas optional
• Return in place of semicolon
• Colon or functions to generate rows

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Indexing a table of numbers

• With row, column scalars
• m(3,2) 3rd row, 2nd column
• With vectors
• m(3, 24) 3rd row, cols 24
• m(23,23) 2x2 submatrix
• Colon abbreviation
• m(3,) all of row 3
• m(, 2) all of column 2
• Colon/end
• m(2end-1,2end-1) internal

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An example Build a 5x5

• Make 5x5 matrix with 1/2 on diagonal and 1/4
  just above
• with brackets
• m 1/2 1/4 0 0 0 0 1/2 1/4 0 0 0 0 1/2 1/4 0
  0 0 0 1/2 1/4
• Zeros indexing
• m zeros(5)
• m(1,1) 1/2 m(2,2) 1/2 m(3,3) 1/2 m(4,4)
  1/2 m(5,5) 1/2
• m(1,2) 1/4 m(2,3) 1/4 m(3,4) 1/4 m(4,5)
  1/4
• m
• Zeros loops
• ma zeros(5)
• for i 15, ma(i,i) 1/2 end
• for i 14, ma(i,i1) 1/4 end
• ma

0.50 0.25 0 0 0
0 0.50 0.25 0 0 0
0 0.50 0.25 0 0
0 0 0.50 0.25 0 0
0 0 0.50
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An Example Build a 5x5

• More esoteric I wanted to see how little
  typing I could do.
• Adding identity matrices
• m zeros(5)
• m(14,25) eye(4)/4
• m m eye(5)/2
• Same thing only shorter
• m eye(5)/2
• m(14,25) m(14,25) eye(4)/4
• Shortest combine eye and diag. Easiest to
  read, too.
• m eye(5)/2 diag(ones(1,4)/4, 1)

0.50 0.25 0 0 0
0 0.50 0.25 0 0 0
0 0.50 0.25 0 0
0 0 0.50 0.25 0 0
0 0 0.50
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A table of numbers operations

• Size of a matrix
• m rand(3,4)
• size(m) rows, columns
• size(m,1) rows
• size(m,2) columns
• r,c size(m) get both
• m' Transpose
• size(m')

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Arithmetic Logic Operations

• A B operate on same size
• Arithmetic Logic , -, , lt, gt, ,
• Array operators ., ./, .
• Dimensions must agree!
• scalar matrix promote scalar to matrix
• 5A, 5A, 5.A, (5A 5.A)

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A matrix is a stack of row vectors

• Example average hi/lo temperatures for twelve
  months in two cities
• parisHi55 55 59 64 68 75 81 81 77 70 63 55
• parisLo39 41 45 46 55 61 64 64 61 54 49 41
• rioHi 84 85 83 80 77 76 75 76 75 77 79 82
• rioLo 73 73 72 69 66 64 63 64 65 66 68 71
• temp parisHi parisLo rioHi rioLo
• size(temp)
• max(temp) Max in each month

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A matrix is a row of column vectors

• temp' Transpose rows become columns
• tt parisHi' parisLo' rioHi' rioLo' Same
• max(temp) max per month (12 cols)
• max(tt) max per city (4 cols)
• max(temp') same
• Indexing columns with colon
• tt(9,) September temperatures
• tt(parisHigtrioHi,) rows w/ paris warmer

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A collection of data in columns

• plot(temp') temperature plots
• legend(parisHi, parisLo, rioHi, rioLo)
• Many other functions operate on columns
• sum, prod, mean, std, diff, cumsum,
• Aside row variants
• sum(temp, 2) sum rows to give 4x1 vector
• sum(temp')' same

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A matrix is a system of equations

• A matrix is a compact way of writing related
  equations
• Buy a apples, b bananas, and c carrots, when the
  prices at three stores are

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A system of equations

• A matrix is a compact way of writing related
  equations
• Buy a apples, b bananas, and c carrots, when the
  prices at three stores are

food L 1.29a 0.84b 0.42c harrisT
1.24a 0.86b 0.41c grocery 1.09a
0.98b 0.45c
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A system of equations

• A matrix is a compact way of writing related
  equations
• Buy a apples, b bananas, and c carrots, when the
  prices at three stores are
• Prices quantity total cost _at_ store

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Matrix multiplication

• C AB
• Inner dimensions must agree jxk kxm gives jxm
  matrix
• C(j,m) A(j, ) B(, m)
• C(j,m) sum(A(j,) . B(,m)') same

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Matrix multiplication

• MATLAB C AB
• C program
• double AMaNa, BMbNb, CMcNc
• / Check Na Mb, Mc Ma, Nc Nb
• for (j 0 j lt Ma j) for (m 0 m lt Nb
  m) Cjm 0 for (k 0 k lt
  Na k)
• Cjm Cjm AjkBkm

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Matrix multiplication

• item prices item quantities store totals
• Matrix vector vector
• Matrix matrix matrix
• A2 AA, so A must be square

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Solving a system of equations

• Square matrices same number of equations as
  unknowns (variables)
• Generally a unique solution x to Axb, when A and
  b are known
• x A\b Matrix left division
• Continuing the food example Are there quantities
  abc that will cost 101011?

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Solving a system of equations

• x A\b satisfies Axb, so buy ¾ lb
  apples, 6.3 lb bananas, 8 7/8 lb carrots,to
  spend 10 at foodL or harrisT and 11 at the
  corner grocery

gtgt x prices \ 101011 x 0.7512
6.3146 8.8732 gtgt prices x ans 10
10 11
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Static equilibrium (balanced forces)

• Triangular bridge ABC
• A fixed
• 10N load on B
• C rolls
• Forces f1, f2, f3 on bars ( expand - contract)
• horizontal and vertical forces must balance at
  vertices (or they'd move collapse bridge)
• Let's see if we can find three equations for
  these three variables.

g
q
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Static equilibrium

• A is fixed, so we need no equation.(I.e., the
  ground cancels forces)
• B may move 2 eqns
• Bh f1cos(q) - f2cos(g) 0
• Bv f1sin(q) f2sin(g) - Load 0
• C may move horizontally 1 eqn
• Ch f3 f2cos(g) 0
• Matix form Mfb

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Static equilibrium in MATLAB

• Build the matrix and solve
• M cos(theta) -cos(gamma) 0
  sin(theta) sin(gamma) 0 0
  cos(gamma) 1
• b 0 Load 0
• forces M \ b
• forces 12 push up to balance Load, thus
  push apart, countered by contraction of 3.

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Non-square systems of equations

• Rectangular matrices
• Underdetermined (fewer eqns than vars)
• Overdetermined (more eqns than vars)
• MATLAB solves Axb "in least squared sense"
• Finds x that minimizes the squared
  residual (A x b)2

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An example

• Data friction in response to a force

• Looks like a line
• Best line ymx c?
• Find m,c.
• Evaluate error.

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Fit a line

• We have x and y

• We'd like m, c that
• 5.84 m1c
• 8.87 m2c
• 11.76 m3c
• 13.71 m4c
• 14.52 m5c

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Fit a line

• We'd like Amc friction

• EasyA\frictiongives mc

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Fit a line in MATLAB

• A(,1) (110)'
• A(,2) ones(10,1)
• b 5.84 8.87 11.76 13.71 14.52 17.71 21.80
  24.25 25.23 29.18'
• A\b
• 2.5121
• 3.4707
• plot(110,b,'r.', 110,ans(1)(110)ans(2),'b-')
  

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Error in fitting a line

• mc A\b
• Amc will be close to b,but not exact.
• Compute residual
• resid Amc -b
• error resid' resid
• error 5.8308, which is min. over all lines
• We use squared residual to measure error

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Least squares reprise

• Least squares requires
• a mathematical model giving a function to fit,
• parameters of the model that must be found,
• observed data used to calculate best parameters
• Linear least squares problems are those that can
  be written as Axb, where
• matrix A and column vector b are known, and
• x is the column vector of unknowns.
• Error (scalar) is squared residual
• error (Ax-b)'(Ax-b)

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Fitting other models Paris/Rio

• Example average hi/lo temperatures for twelve
  months in two cities
• parisHi55 55 59 64 68 75 81 81 77 70 63 55
• parisLo39 41 45 46 55 61 64 64 61 54 49 41
• rioHi 84 85 83 80 77 76 75 76 75 77 79 82
• rioLo 73 73 72 69 66 64 63 64 65 66 68 71
• temp parisHi parisLo rioHi rioLo

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Model the annual temperature

• plot(temp')
• Annual cycle sine wave?
• Fit best sine curve to data

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Fitting a sine/cosine curve

• for months m112 (data)
• Consider variables A, B, C
• Fit a curve of the formtemp A B cos(mp/6)
  C sin(mp/6)
• For ParisHi, we want
• 55 A B cos(1p/6) C sin(1p/6)
• 55 A B cos(2p/6) C sin(2p/6)
• 59 A B cos(3p/6) C sin(3p/6)
• 64 A B cos(4p/6) C sin(4p/6)

Write asMABCtemp?
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Fitting a sine cosine
Paris Rio hi/lo hi/lo

• We'd like MABC temp'

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Fitting a sine cosine in MATLAB

• We'd like MABC temp'
• m 112 months
• CON ones(12,1) columns of M
• COS cos(2pi/12 m)'
• SIN sin(2pi/12 m)'
• fit CON COS SIN \ temp' best fit
• clf plot(temp', '')
  plot data
• hold on plot(CON COS SINfit) plot fit

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Error in fitting sine cosine

• We'd like MABC temp'
• We have fit
• resid Mfit-temp'
• error sum(resid .resid)
• Gives four errors, one for each city

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A space of faces

• In assn1, you turned images into numbers and
  averaged them...
• The set of all faces you can make by linear
  combinations is a vector spaceA point (.5, -.2,
  .7)
• .5 x -.2 y .7 z
• A matrix is a set of rows or columns that define
  a vector space.

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Space of faces

• The following is just for fun
• The idea comes from Turk, M. and Pentland, A.
  (1991). Eigenfaces for recognition. Journal of
  Cognitive Neuroscience, 3(1)71--86.http//www.cs
  .ucsb.edu/mturk/research.htm

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Average student

• 67 images, each a 221 x 201 matrix,stored as a
  67x44421 matrix
• Align eyes, nose, mouth, ears
• Principal components analysis (PCA) finds
  greatest deviation from average
• Eigenvectors added or subtracted from average
  face