Not to be confused with matrix factorization of a polynomial.
Representation of a matrix as a product
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 decompositions; each finds use among a particular class of problems.
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be decomposed via the LU decomposition. The LU decomposition factorizes a matrix into a lower triangular matrix L and an upper triangular matrix U. The...
In linear algebra, the singular value decomposition (SVD) is a factorization of a real or complex matrix into a rotation, followed by a rescaling followed...
this way. When the matrix being factorized is a normal or real symmetric matrix, the decomposition is called "spectral decomposition", derived from the...
decomposition or factorization factors a matrix as the product of a lower triangular matrix and an upper triangular matrix (see matrixdecomposition)...
In mathematics, the polar decomposition of a square real or complex matrix A {\displaystyle A} is a factorization of the form A = U P {\displaystyle A=UP}...
decomposition, also known as a QR factorization or QU factorization, is a decomposition of a matrix A into a product A = QR of an orthonormal matrix Q...
Cholesky decomposition or Cholesky factorization (pronounced /ʃəˈlɛski/ shə-LES-kee) is a decomposition of a Hermitian, positive-definite matrix into the...
the Crout matrixdecomposition is an LU decomposition which decomposes a matrix into a lower triangular matrix (L), an upper triangular matrix (U) and,...
decomposition or Schur triangulation, named after Issai Schur, is a matrixdecomposition. It allows one to write an arbitrary complex square matrix as...
Singular value decomposition M = UΣVT, U and V orthogonal, Σ diagonal matrix Eigendecomposition of a symmetric matrix (decomposition according to the...
mathematics, and in particular modular representation theory, a decompositionmatrix is a matrix that results from writing the irreducible ordinary characters...
is unitarily similar to a diagonal matrix, as a consequence of the spectral theorem. Thus, U has a decomposition of the form U = V D V ∗ , {\displaystyle...
problems is a reason to favour matrixdecomposition methods like using the singular value decomposition. Some matrixdecomposition methods may be unstable,...
Hessenberg matrix has zero entries above the first superdiagonal. They are named after Karl Hessenberg. A Hessenberg decomposition is a matrixdecomposition of...
matrix A as BTB = A, as in the Cholesky factorization, even if BB ≠ A. This distinct meaning is discussed in Positive definite matrix § Decomposition...
This decomposition is closely related to the singular value decomposition of a matrix and is known as an 'Euler' or 'Bloch-Messiah' decomposition. If instead...
multivariate quality control, proper orthogonal decomposition (POD) in mechanical engineering, singular value decomposition (SVD) of X (invented in the last quarter...
Frobenius normal form Jordan matrix Jordan–Chevalley decompositionMatrixdecomposition Modal matrix Weyr canonical form Shilov defines the term Jordan...
algebra, the complete orthogonal decomposition is a matrixdecomposition. It is similar to the singular value decomposition, but typically somewhat cheaper...
as sum of a symmetric and a skew-symmetric matrix. This decomposition is known as the Toeplitz decomposition. Let Mat n {\displaystyle {\mbox{Mat}}_{n}}...
Spectral decomposition is any of several things: Spectral decomposition for matrix: eigendecomposition of a matrix Spectral decomposition for linear operator:...
the LU decomposition algorithm, an invertible matrix may be written as the product of a lower triangular matrix L and an upper triangular matrix U if and...
{\displaystyle A} . When the matrix being decomposed is Hermitian, the spectral decomposition is a special case of the Schur decomposition (see the proof in case...
constraint satisfaction, query optimization, and matrixdecomposition. The concept of tree decomposition was originally introduced by Rudolf Halin (1976)...
The left and right singular vectors in the singular value decomposition of a normal matrix A = U Σ V ∗ {\displaystyle \mathbf {A} =\mathbf {U} {\boldsymbol...
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