Matrix decompositions represent an algorithmic platform for modern scientific computing. Matrix decompositions are efficient ways to extract the essence for key matrix-related problems, on which a variety of solvers and new algorithms can be based. Matrix decompositions typically represent the performance-critical core, from which the much more varied applications can take off.

Wolfram Language now has a very complete list of high-performance matrix decompositions. Furthermore, since Wolfram has efficient, structured matrices, the return values from these decomposition functions will also be able to take advantage of the subsequent efficient computation with the structured matrices.

Linear Equation–Related Decompositions

LUDecomposition — n×n matrix into lower and upper triangular matrices

CholeskyDecomposition — symmetric positive definite n×n matrix into A=U.

LDLDecomposition — symmetric positive definite n×n matrix into

BunchKaufmanDecomposition — symmetric n×n matrix into

Least Squares–Related Decompositions

QRDecomposition — m×n matrix into orthogonal and upper triangular

SingularValueDecomposition — m×n matrix into orthogonal and diagonal A=U.Σ.

SingularValueList  ▪  PrincipalComponents  ▪  KarhunenLoeveDecomposition

Eigen Problem Related Decompositions

EigenvalueDecomposition — matrix into similarity and diagonal

SchurDecomposition — matrix into orthogonal similarity and triangular A=Q.T.

HessenbergDecomposition — matrix into orthogonal similarity and Hessenberg A=Q.H.

JordanDecomposition — matrix into similarity and block Jordan diagonal

FrobeniusDecomposition — matrix into similarity and block companion diagonal

CoreNilpotentDecomposition — matrix into similarity block diagonal

Eigensystem  ▪  Eigenvalues  ▪  Eigenvectors  ▪  JordanReduce  ▪  FrobeniusReduce  ▪  OrderedSchurDecomposition

Integer & Polynomial Matrix Decompositions

HermiteDecomposition — integer matrix to unimodular and triangular

SmithDecomposition — integer matrix to unimodular and diagonal

PolynomialHermiteDecomposition — matrix to unimodular and triangular

PolynomialSmithDecomposition — matrix into unimodular and diagonal

PopovDecomposition — matrix to unimodular and Popov matrix

HermiteReduce  ▪  SmithReduce  ▪  LatticeReduce  ▪  PolynomialHermiteReduce  ▪  PolynomialSmithReduce

Other Matrix Decompositions

RankDecomposition — m×n matrix into column space and row space matrices R

PolarDecomposition — m×n matrix into unitary and positive semidefinite P

Structured Matrices »

TargetStructure — specify efficient structured matrix output from matrix decompositions

UpperTriangularMatrix  ▪  LowerTriangularMatrix  ▪  DiagonalMatrix  ▪  OrthogonalMatrix  ▪  UnitaryMatrix  ▪  JordanMatrix  ▪  CompanionMatrix