linearalgebra.md

Operation	Function	Definition
Matrix Creation and Manipulation
Create a Matrix	A = [1, 2, 3; 4, 5, 6; 7, 8, 9];	Create a 3x3 matrix A.
Transpose	B = A';	Transpose matrix A to get B.
Matrix Multiplication	C = A * B;	Multiply matrices A and B to get C.
Element-Wise Multiplication	D = A .* B;	Multiply matrices A and B element-wise to get D.
Matrix Operations
Determinant	det_A = det(A);	Compute the determinant of matrix A.
Inverse	inv_A = inv(A);	Compute the inverse of matrix A.
Matrix Rank	rank_A = rank(A);	Calculate the rank of matrix A.
Eigenvalues and Eigenvectors	[V, D] = eig(A);	Compute eigenvalues and eigenvectors of matrix A.
Matrix Exponentiation	eA = expm(A);	Compute the matrix exponential of A.
Solving Linear Systems
Linear System Ax = b	[x, flag] = linsolve(A, b);	Solve the linear system Ax = b for x.
Least Squares Solution	[x, residual] = lsqminnorm(A, b);	Find the least squares solution to Ax = b.
Matrix Factorizations
LU Decomposition	[L, U, P] = lu(A);	Perform LU decomposition with pivoting.
QR Decomposition	[Q, R] = qr(A);	Compute QR decomposition of matrix A.
Singular Value Decomposition (SVD)	[U, S, V] = svd(A);	Perform Singular Value Decomposition of A.
Vector Operations
Vector Dot Product	dot_product = dot(v1, v2);	Calculate the dot product of two vectors.
Vector Cross Product	cross_product = cross(v1, v2);	Compute the cross product of two vectors.
Vector Norm	norm_v = norm(v);	Calculate the Euclidean norm (magnitude) of vector v.
Vector Projection	projection = dot(v1, v2) / norm(v2) * v2;	Compute the projection of v1 onto v2.
Special Matrices
Identity Matrix	I = eye(n);	Create an identity matrix of size n.
Zero Matrix	Z = zeros(m, n);	Create a zero matrix of size m x n.
Diagonal Matrix	D = diag([1, 2, 3]);	Create a diagonal matrix from a vector.
Matrix Manipulation Functions
Reshaping a Matrix	B = reshape(A, m, n);	Reshape matrix A into a m x n matrix B.
Extracting a Submatrix	submatrix = A(2:4, 1:2);	Extract a submatrix from A.