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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 ...
A positive matrix is a matrix in which all the elements are strictly greater than zero. The set of positive matrices is the interior of the set of all non-negative matrices. While such matrices are commonly found, the term "positive matrix" is only occasionally used due to the possible confusion with positive-definite matrices, which are different.
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 generalized Rayleigh quotient can be reduced to the Rayleigh quotient (,) through the transformation = where is the Cholesky decomposition of the Hermitian positive-definite matrix B. For a given pair (x, y) of non-zero vectors, and a given Hermitian matrix H, the generalized Rayleigh quotient or sometimes two-sided Rayleigh quotient can be ...
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.
A matrix partitioned in sub-matrices called blocks. Block tridiagonal matrix: A block matrix which is essentially a tridiagonal matrix but with submatrices in place of scalar elements. Boolean matrix: A matrix whose entries are taken from a Boolean algebra. Cauchy matrix: A matrix whose elements are of the form 1/(x i + y j) for (x i), (y j ...
The Hessian matrix of a convex function is positive semi-definite. Refining this property allows us to test whether a critical point x {\displaystyle x} is a local maximum, local minimum, or a saddle point, as follows:
The above identities concerning the determinant of products and inverses of matrices imply that similar matrices have the same determinant: two matrices A and B are similar, if there exists an invertible matrix X such that A = X −1 BX. Indeed, repeatedly applying the above identities yields