What Is a Matrix?

COMP 116August 29, 2007

Outline: What is a matrix?Outline: What is a matrix?

A support for something growingA table of numbers

A stack (column) of row vectorsCreating, Indexing, Operating

A row of column vectorsA collection of data in columns

Plotting

A system of equations or polynomialsA transformation for a vector space

A linear operator on a vector spaceA change of variables or coordinates

www.whatisthematrix.comwww.whatisthematrix.com

An interconnected network or enclosure in which something grows

A matrix is: a table of numbers

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”

A table of numbersA table of numbers

Create With brackets & semicolon[1 2 3; 4:6;7, 8, 9; 10 11 12]

Commas optionalReturn in place of semicolonColon or functions to generate rows

Indexing a table of numbers

Indexing a table of numbers

With row, column scalars m(3,2) % 3rd row, 2nd column

With vectorsm(3, 2:4) % 3rd row, cols 2:4m(2:3,2:3) % 2x2 submatrix

Colon abbreviationm(3,:) % all of row 3m(:, 2) % all of column 2

Colon/end m(2:end-1,2:end-1) % internal

An example: Build a 5x5An 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 & indexingm = 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 & loopsma = zeros(5);for i = 1:5, ma(i,i) = 1/2; endfor i = 1:4, ma(i,i+1) = 1/4; endma

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

An Example: Build a 5x5An Example: Build a 5x5 % More esoteric: I wanted to see how little typing I could do.

%Adding identity matricesm = zeros(5)m(1:4,2:5) = eye(4)/4m = m + eye(5)/2

%Same thing only shorterm = eye(5)/2 m(1:4,2:5) = m(1:4,2:5) + 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

A table of numbers: operations

A table of numbers: operations

Size of a matrix m = rand(3,4);size(m) % [#rows, #columns]

size(m,1) % #rowssize(m,2) % #columns[r,c] = size(m); % get both

m' % Transposesize(m')

Arithmetic & Logic Operations

Arithmetic & Logic Operations

A + B: operate on same sizeArithmetic & Logic: +, -, ==, <, >, &, |Array operators: .*, ./, .^

Dimensions must agree!scalar + matrix: promote scalar to matrix5+A, 5==A, 5.*A, (5*A == 5.*A)

A matrix is: a stack of row vectors

A matrix is: a stack of row vectors

Example: average hi/lo temperatures for twelve months in two cities

parisHi=[55 55 59 64 68 75 81 81 77 70 63 55];parisLo=[39 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

A matrix is: a row of column vectors

A matrix is: a row of column vectors

temp' % Transpose: rows become columnstt = [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 colontt(9,:) % September temperaturestt(parisHi>rioHi,:) % rows w/ paris warmer

A collection of data in columns

A collection of data in columnsplot(temp') % temperature plots

legend(‘parisHi‘, ‘parisLo‘, ‘rioHi‘, ‘rioLo‘)

Many other functions operate on columns:

sum, prod, mean, std, diff, cumsum,…

Aside: row variantssum(temp, 2)

% sum rows to give 4x1 vectorsum(temp')'

% same

A matrix is: a system of equations

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:

apple

banana

carrot

food L 1.29 0.84 0.42

harris T 1.24 0.86 0.41

grocery 1.09 0.98 0.45

A system of equationsA 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 = 1.29*a + 0.84*b + 0.42*c

harrisT = 1.24*a + 0.86*b + 0.41*c

grocery = 1.09*a + 0.98*b + 0.45*c

A system of equationsA 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 @ store

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡→

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

grocery

harrisT

foodL

c

b

a

*

0.45 0.98 1.09

0.41 0.86 1.24

0.42 0.84 1.29

Matrix multiplicationMatrix multiplication

C = A*BInner dimensions must agree:

jxk * kxm gives jxm matrix

C(j,m) = A(j, :) * B(:, m)C(j,m) = sum(A(j,:) .* B(:,m)') % same

A B C

Matrix multiplicationMatrix multiplication

MATLAB: C = A*BC program

double A[Ma][Na], B[Mb][Nb], C[Mc][Nc];/* Check Na == Mb, Mc == Ma, Nc == Nb

for (j = 0; j < Ma; j++)for (m = 0; m < Nb; m++) C[j][m] = 0;

for (k = 0; k < Na; k++)C[j][m] = C[j][m] + A[j][k]*B[k][m];

Matrix multiplicationMatrix multiplication

item prices * item quantities = store totals

Matrix * vector = vectorMatrix * matrix = matrix

A^2 == A*A, so A must be square

A B C

Solving a system of equations

Solving a system of equations

Square matrices: same number of equations as unknowns (variables)

Generally a unique solution x to Ax=b, when A and b are known:

x = A\b; % Matrix left division

Continuing the food example: Are there quantities [a;b;c] that will cost [10;10;11]?

Solving a system of equations

Solving a system of equations

x = A\b satisfies Ax=b, 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

>> x = prices \ [10;10;11]

x = 0.7512 6.3146 8.8732

>> prices * xans = 10 10 11

Static equilibrium (balanced forces)

Static equilibrium (balanced forces)

Triangular bridge ABCA fixed10N load on BC 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.

A

B

C

1 2

3

Static equilibrium Static equilibrium

A is fixed, so we need no equation.(I.e., the ground cancels forces)

B may move: 2 eqnsBh: f1*cos() - f2*cos() = 0Bv: f1*sin() + f2*sin() - Load = 0

C may move horizontally: 1 eqnCh: f3 + f2*cos() = 0

Matix form: M*f=b

A

B

C

1 2

3

f1

f2

f3* =

0

Load

0

cos( ) -cos( ) 0

sin( ) sin( ) 0

0 cos() 1

Static equilibrium in MATLAB

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 = % 1&2 push up to balance Load, & thus push apart, % countered by contraction of 3.

A

B

C

1 2

3

5.7735

5.7735

-2.887

f1

f2

f3* =

0

Load

0

cos( ) -cos( ) 0

sin( ) sin( ) 0

0 cos() 1

Non-square systems of equations

Non-square systems of equations

Rectangular matricesUnderdetermined (fewer eqns than vars)Overdetermined (more eqns than vars)MATLAB solves Ax=b "in least squared sense"

Finds x that minimizes the squared residual: (A x – b)2

An exampleAn example

Data: friction in response to a force:

force 1 2 3 4 5 6 7 8 9 10friction 5.84 8.87 11.76 13.71 14.52 17.71 21.8 24.25 25.23 29.18

Looks like a line…Best line y=mx + c? Find m,c.Evaluate error.

Fit a lineFit a line

We have x and y:

force 1 2 3 4 5 6 7 8 9 10friction 5.84 8.87 11.76 13.71 14.52 17.71 21.8 24.25 25.23 29.18

We'd like m, c that5.84 = m*1+c8.87 = m*2+c11.76 = m*3+c13.71 = m*4+c14.52 = m*5+c

xy

Fit a lineFit a line

We'd like A*[m;c] = friction:1 12 13 14 15 16 17 18 19 1

10 1

m

c* =

5.848.87

11.7613.7114.5217.7121.8

24.2525.2329.18

Easy:A\frictiongives [m;c]

Fit a line in MATLABFit a line in MATLAB

A(:,1) = (1:10)';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

% 3.4707plot(1:10,b,'r.', 1:10,ans(1)*(1:10)+ans(2),'b-');

Error in fitting a lineError in fitting a line

mc = A\b; %A*mc will be close to b,

but not exact.

%Compute residual:resid = A*mc -b;

error = resid' * resid%error = 5.8308, which is min. over all lines%We use squared residual to measure

error…

Least squares repriseLeast 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 Ax=b, where

matrix A and column vector b are known, and x is the column vector of unknowns.

Error (scalar) is squared residualerror = (Ax-b)'*(Ax-b)

Example: average hi/lo temperatures for twelve months in two cities

parisHi=[55 55 59 64 68 75 81 81 77 70 63 55];parisLo=[39 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];

Fitting other models: Paris/Rio

Fitting other models: Paris/Rio

Model the annual temperature

Model the annual temperature

plot(temp')

Annual cycle: sine wave?

Fit best sine curve to data

Fitting a sine/cosine curveFitting a sine/cosine curve

for months m=1:12 (data)Consider variables A, B, CFit a curve of the form

temp = A + B cos(m/6) + C sin(m/6)

For ParisHi, we want55 = A + B cos(1/6) + C sin(1/6)55 = A + B cos(2/6) + C sin(2/6)59 = A + B cos(3/6) + C sin(3/6)64 = A + B cos(4/6) + C sin(4/6)

Write asM*[A;B;C]=temp?

=*

Fitting a sine & cosineFitting a sine & cosine

We'd like M*[A;B;C] = temp':1 cos(1* pi/6) sin(1* pi/6)1 cos(2* pi/6) sin(2* pi/6)1 cos(3* pi/6) sin(3* pi/6)1 cos(4* pi/6) sin(4* pi/6)1 cos(5* pi/6) sin(5* pi/6)1 cos(6* pi/6) sin(6* pi/6)1 cos(7* pi/6) sin(7* pi/6)1 cos(8* pi/6) sin(8* pi/6)1 cos(9* pi/6) sin(9* pi/6)1 cos(10* pi/6) sin(10* pi/6)1 cos(11* pi/6) sin(11* pi/6)1 cos(12* pi/6) sin(12* pi/6)

A

B

C

55 39 84 7355 41 85 7359 45 83 7264 46 80 6968 55 77 6675 61 76 6481 64 75 6381 64 76 6477 61 75 6570 54 77 6663 49 79 6855 41 82 71

Paris Rio

hi/lo hi/lo

* =Aph Apl Arh Arl

Bph Bpl Brh Brl

Cph Cpl Crh Crl

Fitting a sine & cosine in MATLAB

Fitting a sine & cosine in MATLAB

We'd like M*[A;B;C] = temp':m = 1:12; % monthsCON = ones(12,1); % columns of MCOS = cos(2*pi/12 * m)';SIN = sin(2*pi/12 * m)';

fit = [CON COS SIN] \ temp' % best fit

clf; plot(temp', '+'); % plot datahold on; plot([CON COS SIN]*fit); % plot fit

Error in fitting sine & cosine

Error in fitting sine & cosine

We'd like M*[A;B;C] = temp':

We have fit

resid = M*fit-temp';error = sum(resid .*resid)Gives four errors, one for each city

A space of facesA 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 space:A point (.5, -.2, .7) =

.5 x -.2 y +.7 z

A matrix is a set of rows or columns that define a vector space.

Space of facesSpace 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

Average studentAverage 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