Linear algebra

Matrix factorizations

Matrix factorizations (a.k.a. matrix decompositions) compute the factorization of a matrix into a product of matrices, and are one of the central concepts in linear algebra.

The following table summarizes the types of matrix factorizations that have been implemented in Julia. Details of their associated methods can be found in the Linear Algebra section of the standard library documentation.

Cholesky Cholesky factorization
CholeskyPivoted Pivoted Cholesky factorization
LU LU factorization
LUTridiagonal LU factorization for Tridiagonal matrices
UmfpackLU LU factorization for sparse matrices (computed by UMFPack)
QR QR factorization
QRCompactWY Compact WY form of the QR factorization
QRPivoted Pivoted QR factorization
Hessenberg Hessenberg decomposition
Eigen Spectral decomposition
SVD Singular value decomposition
GeneralizedSVD Generalized SVD

Special matrices

Matrices with special symmetries and structures arise often in linear algebra and are frequently associated with various matrix factorizations. Julia features a rich collection of special matrix types, which allow for fast computation with specialized routines that are specially developed for particular matrix types.

The following tables summarize the types of special matrices that have been implemented in Julia, as well as whether hooks to various optimized methods for them in LAPACK are available.

Hermitian Hermitian matrix
UpperTriangular Upper triangular matrix
LowerTriangular Lower triangular matrix
Tridiagonal Tridiagonal matrix
SymTridiagonal Symmetric tridiagonal matrix
Bidiagonal Upper/lower bidiagonal matrix
Diagonal Diagonal matrix
UniformScaling Uniform scaling operator

Elementary operations

Matrix type + - * \ Other functions with optimized methods
Hermitian       MV inv(), sqrtm(), expm()
UpperTriangular     MV MV inv(), det()
LowerTriangular     MV MV inv(), det()
SymTridiagonal M M MS MV eigmax(), eigmin()
Tridiagonal M M MS MV  
Bidiagonal M M MS MV  
Diagonal M M MV MV inv(), det(), logdet(), /()
UniformScaling M M MVS MVS /()


M (matrix) An optimized method for matrix-matrix operations is available
V (vector) An optimized method for matrix-vector operations is available
S (scalar) An optimized method for matrix-scalar operations is available

Matrix factorizations

Matrix type LAPACK eig() eigvals() eigvecs() svd() svdvals()
Hermitian HE   ARI      
UpperTriangular TR A A A    
LowerTriangular TR A A A    
SymTridiagonal ST A ARI AV    
Tridiagonal GT          
Bidiagonal BD       A A
Diagonal DI   A      


A (all) An optimized method to find all the characteristic values and/or vectors is available e.g. eigvals(M)
R (range) An optimized method to find the ilth through the ihth characteristic values are available eigvals(M, il, ih)
I (interval) An optimized method to find the characteristic values in the interval [vl, vh] is available eigvals(M, vl, vh)
V (vectors) An optimized method to find the characteristic vectors corresponding to the characteristic values x=[x1, x2,...] is available eigvecs(M, x)

The uniform scaling operator

A UniformScaling operator represents a scalar times the identity operator, λ*I. The identity operator I is defined as a constant and is an instance of UniformScaling. The size of these operators are generic and match the other matrix in the binary operations +, -, * and \. For A+I and A-I this means that A must be square. Multiplication with the identity operator I is a noop (except for checking that the scaling factor is one) and therefore almost without overhead.