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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.
This characteristic allows spectral matrices to be fully diagonalizable, meaning they can be decomposed into simpler forms using eigendecomposition. This decomposition process reveals fundamental insights into the matrix's structure and behavior, particularly in fields such as quantum mechanics, signal processing, and numerical analysis. [6]
Two-sided Jacobi SVD algorithm—a generalization of the Jacobi eigenvalue algorithm—is an iterative algorithm where a square matrix is iteratively transformed into a diagonal matrix. If the matrix is not square the QR decomposition is performed first and then the algorithm is applied to the R {\displaystyle R} matrix.
For a symmetric matrix A, the vector vec(A) contains more information than is strictly necessary, since the matrix is completely determined by the symmetry together with the lower triangular portion, that is, the n(n + 1)/2 entries on and below the main diagonal. For such matrices, the half-vectorization is sometimes more useful than the ...
The complex Schur decomposition reads as follows: if A is an n × n square matrix with complex entries, then A can be expressed as [1] [2] [3] = for some unitary matrix Q (so that the inverse Q −1 is also the conjugate transpose Q* of Q), and some upper triangular matrix U.
In linear algebra, the Cholesky decomposition or Cholesky factorization (pronounced / ʃ ə ˈ l ɛ s k i / shə-LES-kee) is a decomposition of a Hermitian, positive-definite matrix into the product of a lower triangular matrix and its conjugate transpose, which is useful for efficient numerical solutions, e.g., Monte Carlo simulations.
If, for an arbitrary n × n matrix M, M has nonnegative entries, we write M ≥ 0. If M has only positive entries, we write M > 0. Similarly, if the matrix M 1 − M 2 has nonnegative entries, we write M 1 ≥ M 2. Definition: A = B − C is a regular splitting of A if B −1 ≥ 0 and C ≥ 0. We assume that matrix equations of the form
Given an n × n square matrix A of real or complex numbers, an eigenvalue λ and its associated generalized eigenvector v are a pair obeying the relation [1] =,where v is a nonzero n × 1 column vector, I is the n × n identity matrix, k is a positive integer, and both λ and v are allowed to be complex even when A is real.l When k = 1, the vector is called simply an eigenvector, and the pair ...