# Category: Matrix decompositions

Robust principal component analysis
Robust Principal Component Analysis (RPCA) is a modification of the widely used statistical procedure of principal component analysis (PCA) which works well with respect to grossly corrupted observati
Crout matrix decomposition
In linear algebra, the Crout matrix decomposition is an LU decomposition which decomposes a matrix into a lower triangular matrix (L), an upper triangular matrix (U) and, although not always needed, a
Symbolic Cholesky decomposition
In the mathematical subfield of numerical analysis the symbolic Cholesky decomposition is an algorithm used to determine the non-zero pattern for the factors of a symmetric sparse matrix when applying
Dynamic mode decomposition
Dynamic mode decomposition (DMD) is a dimensionality reduction algorithm developed by Peter Schmid in 2008.Given a time series of data, DMD computes a set of modes each of which is associated with a f
QR decomposition
In linear algebra, a QR decomposition, also known as a QR factorization or QU factorization, is a decomposition of a matrix A into a product A = QR of an orthogonal matrix Q and an upper triangular ma
Principal component analysis
Principal component analysis (PCA) is a popular technique for analyzing large datasets containing a high number of dimensions/features per observation, increasing the interpretability of data while pr
LU decomposition
In numerical analysis and linear algebra, lower–upper (LU) decomposition or factorization factors a matrix as the product of a lower triangular matrix and an upper triangular matrix (see matrix decomp
Weyr canonical form
In mathematics, in linear algebra, a Weyr canonical form (or, Weyr form or Weyr matrix) is a square matrix satisfying certain conditions. A square matrix is said to be in the Weyr canonical form if th
Modes of variation
In statistics, modes of variation are a continuously indexed set of vectors or functions that are centered at a mean and are used to depict the variation in a population or sample. Typically, variatio
Matrix decomposition
In the mathematical discipline of linear algebra, a matrix decomposition or matrix factorization is a factorization of a matrix into a product of matrices. There are many different matrix decompositio
Singular value decomposition
In linear algebra, the singular value decomposition (SVD) is a factorization of a real or complex matrix. It generalizes the eigendecomposition of a square normal matrix with an orthonormal eigenbasis
Rank factorization
In mathematics, given a field , nonnegative integers , and a matrix , a rank decomposition or rank factorization of A is a factorization of A of the form A = CF, where and , where is the rank of .
Lifting scheme
The lifting scheme is a technique for both designing wavelets and performing the discrete wavelet transform (DWT). In an implementation, it is often worthwhile to merge these steps and design the wave
Cholesky decomposition
In linear algebra, the Cholesky decomposition or Cholesky factorization (pronounced /ʃəˈlɛski/ shə-LES-kee) is a decomposition of a Hermitian, positive-definite matrix into the product of a lower tria
Block LU decomposition
In linear algebra, a Block LU decomposition is a matrix decomposition of a block matrix into a lower block triangular matrix L and an upper block triangular matrix U. This decomposition is used in num
Polar decomposition
In mathematics, the polar decomposition of a square real or complex matrix is a factorization of the form , where is a unitary matrix and is a positive semi-definite Hermitian matrix, both square and
Schur decomposition
In the mathematical discipline of linear algebra, the Schur decomposition or Schur triangulation, named after Issai Schur, is a matrix decomposition. It allows one to write an arbitrary complex square
RRQR factorization
An RRQR factorization or rank-revealing QR factorization is a matrix decomposition algorithm based on the QR factorization which can be used to determine the rank of a matrix. The singular value decom
Interpolative decomposition
In numerical analysis, interpolative decomposition (ID) factors a matrix as the product of two matrices, one of which contains selected columns from the original matrix, and the other of which has a s
Jordan normal form
In linear algebra, a Jordan normal form, also known as a Jordan canonical form (JCF),is an upper triangular matrix of a particular form called a Jordan matrix representing a linear operator on a finit
Polynomial matrix spectral factorization
Polynomial matrices are widely studied in the fields of systems theory and control theory and have seen other uses relating to stable polynomials. In stability theory, Spectral Factorization has been
Jordan–Chevalley decomposition
In mathematics, the Jordan–Chevalley decomposition, named after Camille Jordan and Claude Chevalley, expresses a linear operator as the sum of its commuting semisimple part and its nilpotent part. The
Eigendecomposition of a matrix
In linear algebra, eigendecomposition is the factorization of a matrix into a canonical form, whereby the matrix is represented in terms of its eigenvalues and eigenvectors. Only diagonalizable matric