# Functions

• det(A) computes the determinant of the matrix A.
• lambda = eig(A) returns the eigenvalues of A in the vector lambda, and
• [V, lambda] = eig(A) also returns the eigenvectors in V but lambda is now a matrix whose diagonals contain the eigenvalues. This relationship holds true (within round off errors) A = V*lambda*inv(V).
• inv(A) computes the inverse of non-singular matrix A. Note that calculating the inverse is often 'not' necessary. See the next two operators as examples. Note that in theory A*inv(A) should return the identity matrix, but in practice, there may be some round off errors so the result may not be exact.
• A / B computes X such that $XB=A$ . This is called right division and is done without forming the inverse of B.
• A \ B computes X such that $AX=B$ . This is called left division and is done without forming the inverse of A.
• norm(A, p) computes the p-norm of the matrix (or vector) A. The second argument is optional with default value $p=2$ .
• rank(A) computes the (numerical) rank of a matrix.
• trace(A) computes the trace (sum of the diagonal elements) of A.
• expm(A) computes the matrix exponential of a square matrix. This is defined as
$I+A+{\frac {A^{2}}{2!}}+{\frac {A^{3}}{3!}}+\cdots$

Below are some more linear algebra functions. Use help to find out more about them.

• balance (eigenvalue balancing),
• cond (condition number),
• dmult (computes diag(x) * A efficiently),
• dot (dot product),
• givens (Givens rotation),
• kron (Kronecker product),
• null (orthonormal basis of the null space),
• orth (orthonormal basis of the range space),
• pinv (pseudoinverse),
• syl (solves the Sylvester equation).

# Factorizations

• R = chol(A) computes the Cholesky factorization of the symmetric positive definite matrix A, i.e. the upper triangular matrix R such that $R^{T}R=A$ .
• [L, U] = lu(A) computes the LU decomposition of A, i.e. L is lower triangular, U upper triangular and $A=LU$ .
• [Q, R] = qr(A) computes the QR decomposition of A, i.e. Q is orthogonal, R is upper triangular and $A=QR$ .

Below are some more available factorizations. Use help to find out more about them.

• qz (generalized eigenvalue problem: QZ decomposition),
• qzhess (Hessenberg-triangular decomposition),
• schur (Schur decomposition),
• svd (singular value decomposition),
• housh (Householder reflections),
• krylov (Orthogonal basis of block Krylov subspace).