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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 ...
Square roots that are not the all-zeros matrix come in pairs: if R is a square root of M, then −R is also a square root of M, since (−R)(−R) = (−1)(−1)(RR) = R 2 = M. A 2×2 matrix with two distinct nonzero eigenvalues has four square roots. A positive-definite matrix has precisely one positive-definite square root.
The principal square root of a real positive semidefinite matrix is real. [3] The principal square root of a positive definite matrix is positive definite; more generally, the rank of the principal square root of A is the same as the rank of A. [3] The operation of taking the principal square root is continuous on this set of matrices. [4]
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.
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,
Positive definite matrices are matrices for which all eigenvalues are positive. They can be decomposed as A = L L T {\displaystyle \mathbf {A} =\mathbf {L} \mathbf {L} ^{\mathsf {T}}} using the Cholesky decomposition , where L {\displaystyle \mathbf {L} } is a lower triangular matrix.
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 ...
The matrix exponential of a real symmetric matrix is positive definite. Let be an n×n real symmetric matrix and a column vector. Using the elementary properties of the matrix exponential and of symmetric matrices, we have: