Linear Algebra

Linear equations, eigenvalues, singular values, decomposition, matrix operations, matrix structure

Linear algebra functions in MATLAB^® provide fast, numerically robust matrix calculations. Capabilities include a variety of matrix factorizations, linear equation solving, computation of eigenvalues or singular values, and more. For an introduction, see Matrices in the MATLAB Environment.

Functions

Linear Equations

mldivide	Solve systems of linear equations Ax = B for x
mrdivide	Solve systems of linear equations xA = B for x
decomposition	Matrix decomposition for solving linear systems
lsqminnorm	Minimum norm least-squares solution to linear equation
linsolve	Solve linear system of equations
inv	Matrix inverse
pinv	Moore-Penrose pseudoinverse
lscov	Least-squares solution in presence of known covariance
lsqnonneg	Solve nonnegative linear least-squares problem
sylvester	Solve Sylvester equation AX + XB = C for X

Eigenvalues and Singular Values

eig	Eigenvalues and eigenvectors
eigs	Subset of eigenvalues and eigenvectors
balance	Diagonal scaling to improve eigenvalue accuracy
svd	Singular value decomposition
svds	Subset of singular values and vectors
gsvd	Generalized singular value decomposition
ordeig	Eigenvalues of quasitriangular matrices
ordqz	Reorder eigenvalues in QZ factorization
ordschur	Reorder eigenvalues in Schur factorization
polyeig	Polynomial eigenvalue problem
qz	QZ factorization for generalized eigenvalues
hess	Hessenberg form of matrix
schur	Schur decomposition
rsf2csf	Convert real Schur form to complex Schur form
cdf2rdf	Convert complex diagonal form to real block diagonal form

Matrix Decomposition

lu	LU matrix factorization
ldl	Block LDL' factorization for Hermitian indefinite matrices
chol	Cholesky factorization
cholupdate	Rank 1 update to Cholesky factorization
qr	Orthogonal-triangular decomposition
qrdelete	Remove column or row from QR factorization
qrinsert	Insert column or row into QR factorization
qrupdate	Rank 1 update to QR factorization
planerot	Givens plane rotation

Matrix Operations

transpose	Transpose vector or matrix
ctranspose	Complex conjugate transpose
mtimes	Matrix Multiplication
mpower	Matrix power
sqrtm	Matrix square root
expm	Matrix exponential
logm	Matrix logarithm
funm	Evaluate general matrix function
kron	Kronecker tensor product
cross	Cross product
dot	Dot product

Matrix Structure

bandwidth	Lower and upper matrix bandwidth
tril	Lower triangular part of matrix
triu	Upper triangular part of matrix
isbanded	Determine if matrix is within specific bandwidth
isdiag	Determine if matrix is diagonal
ishermitian	Determine if matrix is Hermitian or skew-Hermitian
issymmetric	Determine if matrix is symmetric or skew-symmetric
istril	Determine if matrix is lower triangular
istriu	Determine if matrix is upper triangular

Matrix Properties

norm	Vector and matrix norms
normest	2-norm estimate
vecnorm	Vector-wise norm
cond	Condition number with respect to inversion
condest	1-norm condition number estimate
rcond	Reciprocal condition number
condeig	Condition number with respect to eigenvalues
det	Matrix determinant
null	Null space
orth	Orthonormal basis for range of matrix
rank	Rank of matrix
rref	Reduced row echelon form (Gauss-Jordan elimination)
trace	Sum of diagonal elements
subspace	Angle between two subspaces

Topics

Matrices in the MATLAB Environment

Matrix creation and basic operations.

Systems of Linear Equations

Solve several types of systems of linear equations.

Eigenvalues

Eigenvalue and eigenvector computation.

Singular Values

Singular value decomposition (SVD).

Factorizations

Common matrix factorizations (Cholesky, LU, QR).

Inverses and Determinants

Matrix inverses and determinants.

LAPACK in MATLAB

LAPACK provides a foundation of routines for linear algebra functions and matrix computations in MATLAB.