% This is a comment in an m-file.
% This m-file is an introduction to working with matrices; it is
% organized into cells.
clear % Clear all variables to start fresh
%% Using help
help \
%% This is a cell break so we can evaluate our m-file step by step
2+2 % result is saved to default variable 'ans'
%% Assignments to variables
x = 2+2 
%% Assign result to variable x
y = 2^2 + log(pi) * sin(x) / (1-exp(-x^2)) * erf(-0.01); % ';' suppresses output, use of built-in functions
%% Change floating point number display format
format long 
y
format long e
y
format short e
%% Complex numbers
%%
c1=exp(pi/4*i) 
c2=exp(pi/4i) % What is the difference?
c3=(1+3i)/(1-3i)
%% Vectors and Arrays
%%
x = [1 2 3] % Re-define x as a [1x3] matrix (i.e. a vector)
y = [1; 2; 3] % Re-define y as a [3x1] matrix
%%
b = x + [2 1 0] % Can add (or subtract) two matrices of same size
b = x + y % This doesn't work!
%%
size(x)
size(y)
%% Multiply two vectors of same size element by element
z = [3 2 -1];
a = x.*z 
%% Multiply by a scalar
b = 2*a
%% Matrices
%% Explicit defintion of a [2x3] matrix
d = [1 2 3 ; 4 5 6]
%% Transpose of a matrix
d.'
%% Building matrices from matrices
d = [x ; z] % [2x3] matrix
d = [x z] % Concatenation, [1x6] matrix
d = [[x ; z] [x ; z] ; [3 4 5 -3 2 1]] % A more complicated example!
%% Building a tridiagonal matrix
g = [2 4 6 8]
g1 = [-2 3 4]
g2 = [1 -2 7]
G = diag(g) + diag(g1,1) + diag(g1,-1)
%% Identity matrix
eye(3,3)
%% Random matrix (uniformly distributed)
d = rand(6,8)
%% Ones matrix
ones(3,3)
%% Zeros matrix
zeros(3,3)
%% Linearly space vectors
x = linspace(0,10,6)
x = [0:2:10]
%% Parts of matrices
element = d(3,4) % An element
row   = d(2,:) % An entire row
column = d(:,2) % An entire column
part1 = d(3:5, 4:5) % A sub-section
part2 = d(2,[2 3])  % Another sub-section
diagonal = diag(d)   % The diagonal
upper = triu(d) % Upper triangle
lower = tril(d) % Lower triangle
%% Finding the the last element of a matrix
x(end)
%% Deleting a row or column
dd = rand(5,7)
dd(2,:) = [] % Deletes second row of dd
dd(:,3:5) = [] % Deletes 3rd to 5th columns of dd
%% Operations on matrices
%% Matrix multiplication
e = rand(8,6);
matrix_mult = d*e 
%% Term by term multiplication
term_mult = d.*e.' 
%% Term by term division
term_div = d./e.'  
%% Element functions operate term by term
z = sqrt(d)*e 
%% Raising to a power
z_square = z.^2 % Term by term
z_times_z = z^2 % 
%% Inverse of a matrix
inv(z)
z*inv(z) % Check!
eps % Machine epsilon
%% Various built-in matrix functions
max(z)
%%
max(max(z))
%%
min(z)
%%
min(min(z))
%%
sum(z)
%%
sum(sum(z))
%% Finding out properties of current variables
whos % All variables
whos d % A particular variable
%% Clear a variable from memory
clear d
whos d
%% Clear all variables
clear
whos