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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.


mldivide Solve systems of linear equations Ax = B for x
mrdivide Solve systems of linear equations xA = B for x
linsolve Solve linear system of equations
inv Matrix inverse
pinv Moore-Penrose pseudoinverse of matrix
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
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
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
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
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
norm Vector and matrix norms
normest 2-norm estimate
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


Matrices in the MATLAB Environment

Matrix creation and basic operations.

Systems of Linear Equations

Solving several types of systems of linear equations.


Eigenvalue and eigenvector computation.

Singular Values

Singular value decomposition (SVD).


Common matrix factorizations (Cholesky, LU, QR).

Inverses and Determinants

Matrix inverses and determinants.


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

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