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Functions
- CholeskyDecomposition
- CompanionMatrix
- CoreNilpotentDecomposition
- DiagonalMatrix
- Eigensystem
- EigenvalueDecomposition
- Eigenvalues
- Eigenvectors
- FrobeniusDecomposition
- HermiteDecomposition
- HermiteReduce
- HessenbergDecomposition
- JordanDecomposition
- JordanMatrix
- KarhunenLoeveDecomposition
- LatticeReduce
- LowerTriangularMatrix
- LUDecomposition
- OrthogonalMatrix
- PolynomialHermiteDecomposition
- PolynomialHermiteReduce
- PolynomialSmithDecomposition
- PolynomialSmithReduce
- PrincipalComponents
- QRDecomposition
- SchurDecomposition
- SingularValueDecomposition
- SingularValueList
- SmithDecomposition
- SmithReduce
- TargetStructure
- UnitaryMatrix
- UpperTriangularMatrix
- Related Guides
- Tech Notes
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-
Functions
- CholeskyDecomposition
- CompanionMatrix
- CoreNilpotentDecomposition
- DiagonalMatrix
- Eigensystem
- EigenvalueDecomposition
- Eigenvalues
- Eigenvectors
- FrobeniusDecomposition
- HermiteDecomposition
- HermiteReduce
- HessenbergDecomposition
- JordanDecomposition
- JordanMatrix
- KarhunenLoeveDecomposition
- LatticeReduce
- LowerTriangularMatrix
- LUDecomposition
- OrthogonalMatrix
- PolynomialHermiteDecomposition
- PolynomialHermiteReduce
- PolynomialSmithDecomposition
- PolynomialSmithReduce
- PrincipalComponents
- QRDecomposition
- SchurDecomposition
- SingularValueDecomposition
- SingularValueList
- SmithDecomposition
- SmithReduce
- TargetStructure
- UnitaryMatrix
- UpperTriangularMatrix
- Related Guides
- Tech Notes
-
Functions
Matrix Decompositions
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.
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
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
HermiteReduce ▪ SmithReduce ▪ LatticeReduce ▪ PolynomialHermiteReduce ▪ PolynomialSmithReduce
Structured Matrices »
TargetStructure — specify efficient structured matrix output from matrix decompositions
UpperTriangularMatrix ▪ LowerTriangularMatrix ▪ DiagonalMatrix ▪ OrthogonalMatrix ▪ UnitaryMatrix ▪ JordanMatrix ▪ CompanionMatrix