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Wishart matrices are n × n random matrices of the form H = X X *, where X is an n × m random matrix (m ≥ n) with independent entries, and X * is its conjugate transpose. In the important special case considered by Wishart, the entries of X are identically distributed Gaussian random variables (either real or complex).
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.
Let X be a p × p symmetric matrix of random variables that is positive semi-definite. Let V be a (fixed) symmetric positive definite matrix of size p × p. Then, if n ≥ p, X has a Wishart distribution with n degrees of freedom if it has the probability density function
Any square matrix can uniquely be written 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}} denote the space of n × n {\displaystyle n\times n} matrices.
The conjugate gradient method can be applied to an arbitrary n-by-m matrix by applying it to normal equations A T A and right-hand side vector A T b, since A T A is a symmetric positive-semidefinite matrix for any A. The result is conjugate gradient on the normal equations (CGN or CGNR). A T Ax = A T b
Central normal complex random vectors that are circularly symmetric are of particular interest because they are fully specified by the covariance matrix . The circularly-symmetric (central) complex normal distribution corresponds to the case of zero mean and zero relation matrix, i.e. μ = 0 {\displaystyle \mu =0} and C = 0 {\displaystyle C=0} .
Generator matrix: In Coding theory, a matrix whose rows span a linear code: Gramian matrix: The symmetric matrix of the pairwise inner products of a set of vectors in an inner product space: Hessian matrix: The square matrix of second partial derivatives of a function of several variables: Householder matrix
In numerical linear algebra, the Arnoldi iteration is an eigenvalue algorithm and an important example of an iterative method.Arnoldi finds an approximation to the eigenvalues and eigenvectors of general (possibly non-Hermitian) matrices by constructing an orthonormal basis of the Krylov subspace, which makes it particularly useful when dealing with large sparse matrices.