linear-algebra.rst

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 :ref:`stdlib-linalg` 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
Triangular	Upper/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				XY	inv, sqrtm, expm
Triangular			XY	XY	inv, det
SymTridiagonal	X	X	XZ	XY	eigmax/min
Tridiagonal	X	X	XZ	XY
Bidiagonal	X	X	XZ	XY
Diagonal	X	X	XY	XY	inv, det, logdet, /
UniformScaling	X	X	XYZ	XYZ	/

Legend:

X	An optimized method for matrix-matrix operations is available
Y	An optimized method for matrix-vector operations is available
Z	An optimized method for matrix-scalar operations is available

Matrix factorizations

Matrix type	LAPACK	eig	eigvals	eigvecs	svd	svdvals
Hermitian	HE		ABC
Triangular	TR
SymTridiagonal	ST	A	ABC	AD
Tridiagonal	GT
Bidiagonal	BD				A	A
Diagonal	DI		A

Legend:

A	An optimized method to find all the characteristic values and/or vectors is available	e.g. eigvals(M)
B	An optimized method to find the ilth through the ihth characteristic values are available	eigvals(M, il, ih)
C	An optimized method to find the characteristic values in the interval [vl, vh] is available	eigvals(M, vl, vh)
D	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.