Matrices, Linear Algebra, and the MATLAB Environment

% Row vector (semicolons separate rows; commas/spaces separate columns)
v = [1, 2, 3, 4, 5]        % 1x5 row vector

% Column vector
c = [1; 2; 3; 4; 5]        % 5x1 column vector

% 3x3 matrix
A = [1, 2, 3;
     4, 5, 6;
     7, 8, 9]

% Colon operator: start:step:end
x = 0:0.1:1                % 0, 0.1, 0.2 ... 1.0 (11 elements)
y = linspace(0, 2*pi, 100) % 100 evenly spaced from 0 to 2π

% Special matrices
I   = eye(4)               % 4x4 identity
Z   = zeros(3, 5)          % 3x5 all-zeros
O   = ones(2, 4)           % 2x4 all-ones
R   = rand(3)              % 3x3 uniform random [0,1]
N   = randn(3)             % 3x3 standard normal

% size, length, numel
disp(size(A))              % [3, 3]
disp(length(v))            % 5  (longest dimension)
disp(numel(A))             % 9  (total elements)

A = magic(4)   % 4x4 magic square

% Single element: A(row, col) — 1-indexed
disp(A(2, 3))          % row 2, column 3

% Entire row or column
row2 = A(2, :)         % all columns of row 2
col3 = A(:, 3)         % all rows of column 3

% Submatrix: rows 1-2, cols 2-4
sub = A(1:2, 2:4)

% Last row/column using end
last_row = A(end, :)
corner    = A(end-1:end, end-1:end)  % bottom-right 2x2

% Linear indexing (column-major order)
A(5)    % 5th element column-by-column

% Logical indexing
v = [3, -1, 4, -2, 5, -3];
pos = v(v > 0)              % [3, 4, 5]
v(v < 0) = 0                % set negatives to zero: [3,0,4,0,5,0]

% reshape
M = reshape(1:12, 3, 4)    % 3x4 matrix from 1 to 12

A = [1,2; 3,4];
B = [5,6; 7,8];

% Matrix multiplication (dot product rows × cols)
C = A * B           % [19,22; 43,50]

% Element-wise multiplication (same size required)
D = A .* B          % [5,12; 21,32]

% Element-wise division and power
E = A ./ B          % [0.2,0.333...; 0.428...,0.5]
F = A .^ 2          % [1,4; 9,16]

% Transpose
At = A'             % [1,3; 2,4]  (conjugate transpose for complex)
At = A.'            % same for real matrices

% Matrix inverse and pseudo-inverse
Ai  = inv(A)        % exact inverse (square, non-singular only)
Api = pinv(A)       % Moore-Penrose pseudo-inverse

% Determinant and rank
det(A)              % -2
rank(A)             % 2

% Norms
norm(A)             % 2-norm (largest singular value)
norm(A, 'fro')      % Frobenius norm

% Solve Ax = b (backslash — always prefer over inv(A)*b)
A = [2, 1, -1;
    -3, -1,  2;
    -2,  1,  2];
b = [8; -11; -3];
x = A \ b           % [2; 3; -1]

% Verify
norm(A*x - b)       % should be near zero (~1e-15)

% Least-squares solution (overdetermined: more equations than unknowns)
A2 = randn(10, 3);
b2 = randn(10, 1);
x2 = A2 \ b2        % minimum norm least-squares solution

% Eigenvalues and eigenvectors
[V, D] = eig(A)     % D = diagonal eigenvalue matrix, V = eigenvectors
eigenvalues = diag(D)

% Singular Value Decomposition: A = U * S * V'
[U, S, V] = svd(A)
singular_values = diag(S)

% Low-rank approximation (keep top 2 singular values)
k  = 2;
Ak = U(:,1:k) * S(1:k,1:k) * V(:,1:k)';
fprintf('Rank-%d approximation error: %.4f\n', k, norm(A - Ak, 'fro'))