Matrix Operations

## Learning Objectives - Create and manipulate matrices - Perform matrix arithmetic - Apply linear algebra operations - Use broadcasting for arrays ## Creating Matrices ### Basic Matrix Creation ```matlab % Direct entry (row separator: ; or newline) A = [1, 2, 3; 4, 5, 6; 7, 8, 9] % Using newline instead of semicolon B = [1, 2, 3 4, 5, 6 7, 8, 9] ``` ### Special Matrices ```matlab zeros(3) % 3x3 matrix of zeros zeros(3, 4) % 3x4 matrix of zeros ones(3) % 3x3 matrix of ones ones(2, 5) % 2x5 matrix of ones eye(3) % 3x3 identity matrix eye(3, 4) % 3x4 identity matrix rand(3) % 3x3 uniform random [0,1] randn(3) % 3x3 normal distribution magic(3) % 3x3 magic square ``` ### Range-Based Creation ```matlab % Colon operator 1:5 % [1, 2, 3, 4, 5] 0:0.5:2 % [0, 0.5, 1, 1.5, 2] 10:-1:7 % [10, 9, 8, 7] % linspace (linear spacing) linspace(0, 10, 5) % [0, 2.5, 5, 7.5, 10] % logspace (logarithmic spacing) logspace(0, 2, 3) % [1, 10, 100] ``` ## Matrix Indexing ### Single Element ```matlab A = [1, 2, 3; 4, 5, 6; 7, 8, 9]; A(1, 1) % 1 (row 1, col 1) A(3, 3) % 9 (row 3, col 3) A(5) % 5 (linear indexing: column-major) ``` ### Submatrices ```matlab A = [1, 2, 3; 4, 5, 6; 7, 8, 9]; A(1, :) % Row 1: [1, 2, 3] A(:, 2) % Column 2: [2; 5; 8] A(1:2, 1:2) % Top-left 2x2: [1, 2; 4, 5] A(end, :) % Last row: [7, 8, 9] A(:, end) % Last column: [3; 6; 9] ``` ### Logical Indexing ```matlab A = [1, 2, 3, 4, 5]; % Find elements greater than 3 idx = A > 3; % [0, 0, 0, 1, 1] A(idx) % [4, 5] % Direct logical indexing A(A > 3) % [4, 5] ``` ### Changing Elements ```matlab A = [1, 2; 3, 4]; A(1, 1) = 10; % A = [10, 2; 3, 4] A(2, :) = [5, 6]; % A = [10, 2; 5, 6] ``` ## Matrix Arithmetic ### Addition and Subtraction ```matlab A = [1, 2; 3, 4]; B = [5, 6; 7, 8]; C = A + B % [6, 8; 10, 12] C = A - B % [-4, -4; -4, -4] % Scalar operations C = A + 10 % [11, 12; 13, 14] ``` ### Matrix Multiplication ```matlab A = [1, 2; 3, 4]; B = [5, 6; 7, 8]; C = A * B % Matrix product: [19, 22; 43, 50] % Element-wise multiplication C = A .* B % [5, 12; 21, 32] ``` ### Matrix Division ```matlab A = [1, 2; 3, 4]; % Matrix right division (A * inv(B)) C = A / B % A * B^(-1) % Matrix left division (solves Ax = B) C = A \ B % A^(-1) * B % Element-wise division C = A ./ B % [0.2, 0.333; 0.428, 0.5] ``` ### Matrix Power ```matlab A = [1, 2; 3, 4]; % Matrix power (A * A) C = A ^ 2 % [7, 10; 15, 22] % Element-wise power C = A .^ 2 % [1, 4; 9, 16] ``` ## Linear Algebra Operations ### Transpose ```matlab A = [1, 2, 3; 4, 5, 6]; A' % Transpose: [1, 4; 2, 5; 3, 6] A.' % Same for real matrices (conjugate transpose for complex) ``` ### Determinant and Inverse ```matlab A = [1, 2; 3, 4]; det(A) % -2 inv(A) % [-2, 1; 1.5, -0.5] ``` ### Eigenvalues and Eigenvectors ```matlab A = [1, 2; 2, 1]; [V, D] = eig(A); % V: eigenvectors as columns % D: diagonal matrix of eigenvalues ``` ### Matrix Decompositions ```matlab A = [1, 2; 3, 4]; % LU decomposition [L, U] = lu(A); % QR decomposition [Q, R] = qr(A); % SVD (Singular Value Decomposition) [U, S, V] = svd(A); ``` ### Solving Linear Systems ```matlab % Solve Ax = b A = [1, 2; 3, 4]; b = [5; 11]; x = A \ b % Solution: [1; 2] x = inv(A) * b % Alternative method ``` ## Matrix Manipulation ### Concatenation ```matlab A = [1, 2]; B = [3, 4]; C = [A, B] % Horizontal: [1, 2, 3, 4] C = [A; B] % Vertical: [1, 2; 3, 4] ``` ### Reshaping ```matlab A = 1:12; reshape(A, 3, 4) % 3x4 matrix: [1, 4, 7, 10; 2, 5, 8, 11; 3, 6, 9, 12] reshape(A, 2, 6) % 2x6 matrix ``` ### Repmat and Reshape ```matlab A = [1, 2; 3, 4]; repmat(A, 2, 3) % Tile A: [1,2,1,2,1,2; 3,4,3,4,3,4; 1,2,1,2,1,2; 3,4,3,4,3,4] ``` ### Diagonal and Triangular ```matlab A = [1, 2, 3; 4, 5, 6; 7, 8, 9]; diag(A) % Main diagonal: [1; 5; 9] diag(A, 1) % First superdiagonal: [2; 6] diag(A, -1) % First subdiagonal: [4; 8] triu(A) % Upper triangular tril(A) % Lower triangular ``` ## Broadcasting ### Array-Matrix Operations ```matlab A = [1, 2, 3; 4, 5, 6]; v = [10, 20, 30]; % Row vector % Broadcasting: v added to each row B = A + v % [11, 22, 33; 14, 25, 36] ``` ### Array with Column Vector ```matlab A = [1, 2, 3; 4, 5, 6]; v = [10; 20]; % Column vector % Broadcasting: v added to each column B = A + v % [11, 12, 13; 24, 25, 26] ``` ### Scalar Broadcasting ```matlab A = [1, 2, 3; 4, 5, 6]; B = A + 10 % [11, 12, 13; 14, 15, 16] ``` ### Element-wise Functions ```matlab A = [0, pi/2, pi]; sin(A) % [0, 1, 0] - sin applied to each element exp(A) % [1, 4.81, 23.14] log(A(2)) % log(pi/2) ``` ## Matrix Information ```matlab A = [1, 2, 3; 4, 5, 6]; size(A) % [2, 3] length(A) % 3 (largest dimension) numel(A) % 6 ndims(A) % 2 ``` ## Summary - Create matrices with `[row1; row2; ...]` or specialized functions - Use `:` and indexing `A(row, col)` for access - `*` is matrix multiplication; `.*` is element-wise - `'` gives transpose; `^` is matrix power - Use `inv()`, `det()`, `eig()` for linear algebra - Broadcasting automatically expands compatible dimensions - Use `reshape()`, `repmat()`, `diag()` for manipulation