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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.
A symmetric real matrix A is called positive-definite if the associated quadratic form = has a positive value for every nonzero vector x in . If f ( x ) only yields negative values then A is negative-definite ; if f does produce both negative and positive values then A is indefinite . [ 30 ]
In mathematics, Sylvester’s criterion is a necessary and sufficient criterion to determine whether a Hermitian matrix is positive-definite. Sylvester's criterion states that a n × n Hermitian matrix M is positive-definite if and only if all the following matrices have a positive determinant: the upper left 1-by-1 corner of M,
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 .
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.
For a symmetric positive definite matrix the preconditioner is typically chosen to be symmetric positive definite as well. The preconditioned operator P − 1 A {\\displaystyle P^{-1}A} is then also symmetric positive definite, but with respect to the P {\\displaystyle P} -based scalar product .
Consequently, only square matrices can be symmetric. The entries of a symmetric matrix are symmetric with respect to the main diagonal. So if the entries are written as A = (a ij), then a ij = a ji, for all indices i and j. For example, the following 3×3 matrix is symmetric: