MATLAB Tutorial -- MIT 18.06 Linear Algebra

Introduction

MATLAB works constantly with matrices. A small number of basic commands lets you create vectors and matrices, change them, and operate with them.

Creating and Using Matrices -- Quick Example

% Create the 3x3 identity matrix
E = eye(3)

Produces:

[1 0 0]
[0 1 0]
[0 0 1]

% Pick out column 1
u = E(:, 1)

Produces:

[1]
[0]
[0]

% Reset the (3,1) entry to 5
E(3, 1) = 5

Produces:

[1 0 0]
[0 1 0]
[5 0 1]

% Multiply E and u
v = E * u

Produces:

[1]
[0]
[5]

Inverting a Matrix and Solving a Linear System

% Create a matrix: ones(3) + eye(3)
A = ones(3) + eye(3)

Produces:

[2 1 1]
[1 2 1]
[1 1 2]

% b is the third column of A
b = A(:, 3)

Produces:

[1]
[1]
[2]

% Compute the inverse
C = inv(A)

Produces (in decimal, normally; use format rat for fractions):

[ 0.75  -0.25  -0.25]
[-0.25   0.75  -0.25]
[-0.25  -0.25   0.75]

% Solve Ax = b -- the slow way
x = inv(A) * b

% Solve Ax = b -- the fast way (backslash, uses Gaussian elimination, never computes inverse)
x = A \ b

Both produce x = [0; 0; 1] (since b is the third column of A, the solution is the third standard basis vector).

The \ (backslash) operator uses Gaussian elimination when A is square and never computes the inverse matrix.

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Comments

% The comment symbol is the percent sign
% The symbols a and A are different: MATLAB is case-sensitive

• Type help slash for a description of how to use the backslash symbol. The word help can be followed by a MATLAB symbol, command name, or M-file name.
• The command name shown by help is uppercase, but must be lowercase in actual use.
• The backslash A \ b behaves differently when A is not square.
• To display all 16 digits, type format long. The default format short gives 4 digits after the decimal.
• A semicolon after a command suppresses display of the result. For example, A = ones(3); will not display the 3x3 matrix.
• Use the up-arrow cursor to return to previous commands.

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How to Input a Row or Column Vector

% Row vector (1x3 matrix)
u = [2 4 5]

% Column vector (3x1 matrix) -- semicolons separate rows
v = [2; 4; 5]

% Column vector -- transpose of a row vector
v = [2 4 5]'
v = u'

% Row vector with unit steps using the colon operator
w = 2:5    % produces w = [2 3 4 5]

% Row vector with custom step size
u = 1:2:7  % produces u = [1 3 5 7]

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How to Input a Matrix (a Row at a Time)

% Semicolon separates rows
A = [1 2 3; 4 5 6]

Produces:

[1 2 3]
[4 5 6]

% Multi-line input also works
A = [1 2 3
     4 5 6]

% Transpose
B = [1 2 3; 4 5 6]'   % A^T is A' in MATLAB

Produces:

[1 4]
[2 5]
[3 6]

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How to Create Special Matrices

% Diagonal matrix from a vector
diag(v)           % Diagonal matrix with vector v on its diagonal

% Toeplitz matrices
toeplitz(v)       % Symmetric constant-diagonal matrix with v as first row and first column
toeplitz(w, v)    % Constant-diagonal matrix with w as first column and v as first row

% Matrices of ones and zeros
ones(n)           % n x n matrix of ones
zeros(n)          % n x n matrix of zeros

% Identity matrix
eye(n)            % n x n identity matrix

% Random matrices
rand(n)           % n x n matrix with random entries between 0 and 1 (uniform distribution)
randn(n)          % n x n matrix with normally distributed entries (mean 0, variance 1)

% Rectangular matrices
ones(m, n)        % m x n matrix of ones
zeros(m, n)       % m x n matrix of zeros
rand(m, n)        % m x n matrix with random uniform entries

% Same shape as an existing matrix A
ones(size(A))
zeros(size(A))
eye(size(A))

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How to Change Entries in a Given Matrix A

% Reset a single entry
A(3, 2) = 7       % Sets the (3,2) entry to 7

% Reset an entire row
A(3, :) = v       % Sets the third row to vector v

% Reset an entire column
A(:, 2) = w       % Sets the second column to vector w

% The colon (:) stands for "all" (all columns or all rows)

% Exchange rows
A([2 3], :) = A([3 2], :)   % Swaps rows 2 and 3

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How to Create Submatrices of an m x n Matrix A

Command	Result
A(i, j)	The (i,j) entry (a scalar, i.e., a 1x1 matrix)
A(i, :)	The ith row (as a row vector)
A(:, j)	The jth column (as a column vector)
A(2:4, 3:7)	Rows 2 through 4 and columns 3 through 7 (as a 3x5 matrix)
A([2 4], :)	Rows 2 and 4, all columns (as a 2 x n matrix)
A(:)	One long column formed from all columns of A (an mn x 1 matrix)
triu(A)	Upper triangular part (sets entries below the main diagonal to zero)
tril(A)	Lower triangular part (sets entries above the main diagonal to zero)

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Matrix Multiplication and Inversion

Command	Description
A * B	Matrix product AB (if dimensions are compatible)
A .* B	Entry-by-entry (element-wise) product (if size(A) == size(B))
inv(A)	Inverse A^(-1) if A is square and invertible
pinv(A)	Pseudoinverse of A
A \ B	Left division -- computes inv(A) * B without explicitly inverting
x = A \ b	Solves Ax = b if inv(A) exists (uses Gaussian elimination)

Important: See help slash for details when A is a rectangular matrix.

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Numbers and Matrices Associated with A

Command	Description
det(A)	Determinant (if A is square)
rank(A)	Rank (number of pivots = dimension of row space and column space)
size(A)	Returns [m n] -- the dimensions of A
trace(A)	Trace = sum of diagonal entries = sum of eigenvalues
null(A)	A matrix whose n - r columns form an orthogonal basis for the nullspace of A
orth(A)	A matrix whose r columns form an orthogonal basis for the column space of A

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Examples

% Elementary elimination matrix
E = eye(4);
E(2, 1) = -3;
% E * A subtracts 3 times row 1 of A from row 2

% Augmented matrix
B = [A b];    % Creates the augmented matrix with b as an extra column

% Permutation matrix
E = eye(3);
P = E([2 1 3], :);   % Creates a permutation matrix

% Identity: triu(A) + tril(A) - diag(diag(A)) equals A

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Built-in M-files for Matrix Factorizations (All Important!)

% LU decomposition: PA = LU
[L, U, P] = lu(A)

% Eigenvalues
e = eig(A)              % Returns a vector containing the eigenvalues of A

% Eigenvalues and eigenvectors: AS = SE
[S, E] = eig(A)         % E is a diagonal eigenvalue matrix, S is the eigenvector matrix
                        % If A is not diagonalizable (too few eigenvectors), S is not invertible

% QR decomposition: A = QR
[Q, R] = qr(A)          % Q is an m x m orthogonal matrix, R is an m x n triangular matrix

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Creating M-files

M-files are text files ending with .m that MATLAB uses for functions and scripts.

• A script is a sequence of commands executed when the file name is typed at the MATLAB prompt. MATLAB's demos are examples of scripts (e.g., the demo called house).
• Most of MATLAB's built-in functions are actually M-files and can be viewed by typing type xxx where xxx is the function name.
• To write your own scripts or functions, create a new text file with any name ending in .m.
• A script file is simply a list of MATLAB commands.
• For an M-file to be a function, it must start with the keyword function, followed by the output variables in brackets, the function name, and the input variables.

Example 1: mult.m

function [C] = mult(A)
r = rank(A);
C = A' * A;

Save as mult.m. This function takes a matrix A and returns only the matrix product C. The variable r is not returned because it was not listed as an output variable. The semicolons suppress output to the MATLAB window -- useful when dealing with large matrices.

Example 2: properties.m

function [V, D, r] = properties(A)
% This function finds the rank, eigenvalues and eigenvectors of A
[m, n] = size(A);
if m == n
    [V, D] = eig(A);
    r = rank(A);
else
    disp('Error: The matrix must be square');
end

This function takes matrix A as input and returns the eigenvector matrix, eigenvalue matrix, and rank as outputs. The % is used for comments. The function checks if the input matrix is square.

• Typing properties(A) returns only the first output, V (the matrix of eigenvectors).
• You must type [V, D, r] = properties(A) to get all three outputs.

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Keeping a Diary of Your Work

diary('filename')    % Records everything in the MATLAB window to a text file
diary on             % Start recording
diary off            % Stop recording

Old diary files can be viewed using a text editor, printed with lpr in Unix, or viewed in MATLAB using the type filename command.

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Saving Your Variables and Matrices

The diary command saves commands and output, but does not save the content of variables and matrices.

whos                 % Lists all variables and their matrix sizes
save('xxx')          % Saves all variables into file xxx.mat (MATLAB adds .mat extension)
load xxx             % Loads variables from xxx.mat back into MATLAB

.mat files are different from .m files (which are scripts or functions).

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Graphics

Basic Plotting

plot(x, y)           % Plots points (xi, yi) connected by solid lines

• x and y must be vectors of the same length.
• If no x vector is given, MATLAB assumes x(i) = i. So plot(y) has equal spacing on the x-axis: points are (i, y(i)).

Line Style and Color

The type and color of the line can be changed by a third argument. The default is a solid black line.

% MATLAB 5: red + markers and dotted line
plot(x, y, 'r+:')

% MATLAB 4: dashed line
plot(x, y, '--')

% MATLAB 4: dotted line
plot(x, y, ':')

Discrete Points (No Lines)

plot(x, y, 'o')      % Circles
plot(x, y, '+')      % Plus signs
plot(x, y, '*')      % Stars
plot(x, y, '.')      % Dots

Multiple Graphs

% Two graphs on the same axes
plot(x, y, X, Y)

Logarithmic Scales

loglog(x, y)         % Both axes logarithmic
semilogy(x, y)       % y-axis logarithmic
semilogx(x, y)       % x-axis logarithmic

Axis Scaling and Labels

axis([a b c d])      % Scales graph to rectangle: a <= x <= b, c <= y <= d
title('Height of Satellite')
xlabel('Time in seconds')
ylabel('Height in meters')

Hold Command

hold                 % Keeps the current graph while plotting a new one
                     % Repeating hold clears the screen

Printing and Saving

print -Pprintername  % Send to printer
print -d filename    % Save graphics window to a file

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Based on the MATLAB tutorial from MIT 18.06 Linear Algebra, Spring 2010.