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  2. Cholesky decomposition - Wikipedia

    en.wikipedia.org/wiki/Cholesky_decomposition

    The Cholesky decomposition of a Hermitian positive-definite matrix A, is a decomposition of the form =, where L is a lower triangular matrix with real and positive diagonal entries, and L * denotes the conjugate transpose of L. Every Hermitian positive-definite matrix (and thus also every real-valued symmetric positive-definite matrix) has a ...

  3. Matrix decomposition - Wikipedia

    en.wikipedia.org/wiki/Matrix_decomposition

    Decomposition: = (right polar decomposition) or = ′ (left polar decomposition), where U is a unitary matrix and P and P' are positive semidefinite Hermitian matrices. Uniqueness: P {\displaystyle P} is always unique and equal to A ∗ A {\displaystyle {\sqrt {A^{*}A}}} (which is always hermitian and positive semidefinite).

  4. Non-negative matrix factorization - Wikipedia

    en.wikipedia.org/wiki/Non-negative_matrix...

    Non-negative matrix factorization (NMF or NNMF), also non-negative matrix approximation [1] [2] is a group of algorithms in multivariate analysis and linear algebra where a matrix V is factorized into (usually) two matrices W and H, with the property that all three matrices have no negative elements. This non-negativity makes the resulting ...

  5. Eigendecomposition of a matrix - Wikipedia

    en.wikipedia.org/wiki/Eigendecomposition_of_a_matrix

    The decomposition can be derived from the fundamental property of eigenvectors: = = =. The linearly independent eigenvectors q i with nonzero eigenvalues form a basis (not necessarily orthonormal) for all possible products Ax, for x ∈ C n, which is the same as the image (or range) of the corresponding matrix transformation, and also the ...

  6. Factorization - Wikipedia

    en.wikipedia.org/wiki/Factorization

    Matrix rings are non-commutative and have no unique factorization: there are, in general, many ways of writing a matrix as a product of matrices. Thus, the factorization problem consists of finding factors of specified types. For example, the LU decomposition gives a matrix as the product of a lower triangular matrix by an upper triangular matrix.

  7. QR decomposition - Wikipedia

    en.wikipedia.org/wiki/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 orthonormal matrix Q and an upper triangular matrix R. QR decomposition is often used to solve the linear least squares (LLS) problem and is the basis for a particular eigenvalue algorithm ...

  8. Polynomial matrix spectral factorization - Wikipedia

    en.wikipedia.org/wiki/Polynomial_Matrix_Spectral...

    Polynomial Matrix Spectral Factorization or Matrix Fejer–Riesz Theorem is a tool used to study the matrix decomposition of polynomial matrices. Polynomial matrices are widely studied in the fields of systems theory and control theory and have seen other uses relating to stable polynomials .

  9. Matrix factorization (algebra) - Wikipedia

    en.wikipedia.org/wiki/Matrix_factorization_(algebra)

    For a commutative ring and an element , a matrix factorization of is a pair of n-by-n matrices , such that =. This can be encoded more generally as a Z / 2 {\displaystyle \mathbb {Z} /2} - graded S {\displaystyle S} -module M = M 0 ⊕ M 1 {\displaystyle M=M_{0}\oplus M_{1}} with an endomorphism