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.

To speed up computations for a stack of matrices, especially for a large set of comparatively small matrices, use page-wise matrix functions. See Page-Wise Matrix Functions for more information.

Functions

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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
`svdsketch`	Compute SVD of low-rank matrix sketch
`svdappend`	Revise SVD after appending data (Since R2023b)
`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`	Generalized Schur (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`	QR 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
`expmv`	Matrix exponential multiplied by vector (Since R2023b)
`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 specified 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 for inversion
`condest`	1-norm condition number estimate
`rcond`	Reciprocal condition number
`condeig`	Condition number with respect to eigenvalues
`det`	Matrix determinant
`null`	Null space of matrix
`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

Page-Wise Matrix Functions

`pagemldivide`	Page-wise left matrix divide (Since R2022a)
`pagemrdivide`	Page-wise right matrix divide (Since R2022a)
`pagelsqminnorm`	Page-wise minimum-norm least-squares solution to linear equation (Since R2024a)
`pageinv`	Page-wise matrix inverse (Since R2022a)
`pagepinv`	Page-wise Moore-Penrose pseudoinverse (Since R2024a)
`pageeig`	Page-wise eigenvalues and eigenvectors (Since R2023a)
`pagesvd`	Page-wise singular value decomposition (Since R2021b)
`pagemtimes`	Page-wise matrix multiplication
`pagetranspose`	Page-wise transpose
`pagectranspose`	Page-wise complex conjugate transpose
`pagenorm`	Page-wise matrix or vector norm (Since R2022b)

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).
Matrix Exponentials
This example shows three of the 19 ways to compute the exponential of a matrix.
Determine Whether Matrix Is Symmetric Positive Definite
This topic explains how to use the chol and eig functions to determine whether a matrix is symmetric positive definite (a symmetric matrix with all positive eigenvalues).
LAPACK in MATLAB
LAPACK provides a foundation of routines for linear algebra functions and matrix computations in MATLAB.
Page-Wise Matrix Functions
Simplify code and speed up computations involving stacks of matrices, especially large sets of comparatively small matrices.

Featured Examples

Basic Matrix Operations

Basic techniques and functions for working with matrices in the MATLAB® language.

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Image Compression with Low-Rank SVD

Use svdsketch to compress an image while preserving important features of the image.

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Teaching Resources

Applied Linear Algebra

Learn linear algebra by applications in stoichiometry, steganography, static force analysis, and machine learning.

Matrix Methods of Linear Algebra

Learn concepts of matrix operations and use them to modify greyscale images, analyze linear circuits, and compute Markov chains.