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In mathematics, a symmetric matrix with real entries is positive-definite if the real number is positive for every nonzero real column vector, where is the row vector transpose of . [1] More generally, a Hermitian matrix (that is, a complex matrix equal to its conjugate transpose) is positive-definite if the real number is positive for every nonzero complex column vector , where denotes the ...
Similarly in characteristic different from 2, each diagonal element of a skew-symmetric matrix must be zero, since each is its own negative. In linear algebra, a real symmetric matrix represents a self-adjoint operator [1] represented in an orthonormal basis over a real inner product space.
Conjugate gradient, assuming exact arithmetic, converges in at most n steps, where n is the size of the matrix of the system (here n = 2). In mathematics, the conjugate gradient method is an algorithm for the numerical solution of particular systems of linear equations, namely those whose matrix is positive-semidefinite.
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.
The derivation of the result hinges on a few basic observations: The real matrix / /, with (), is well-defined and skew-symmetric.; Any skew-symmetric real matrix can be block-diagonalized via orthogonal real matrices, meaning there is () such that = with a real positive-definite diagonal matrix containing the singular values of .
The particular system of equations was used by Morris to illustrate the concept of ill-conditioned system of equations. The matrix has been used as an example and for test purposes in many research papers and books over the years. John Todd has referred to as “the notorious matrix W of T. S. Wilson”. [1]
For elliptic partial differential equations this matrix is positive definite, which has a decisive influence on the set of possible solutions of the equation in question. [86] The finite element method is an important numerical method to solve partial differential equations, widely applied in simulating complex physical systems. It attempts to ...
A Stieltjes matrix is necessarily an M-matrix. Every n×n Stieltjes matrix is invertible to a nonsingular symmetric nonnegative matrix, though the converse of this statement is not true in general for n > 2. From the above definition, a Stieltjes matrix is a symmetric invertible Z-matrix whose eigenvalues have positive real parts. As it is a Z ...