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A function is negative semi-definite if the inequality is reversed. ... then function f must be positive-definite to ensure the covariance matrix A is positive-definite.
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 ...
In mathematics, a definite quadratic form is a quadratic form over some real vector space V that has the same sign (always positive or always negative) for every non-zero vector of V. According to that sign, the quadratic form is called positive-definite or negative-definite .
In mathematics, every analytic function can be used for defining a matrix function that maps square matrices with complex entries to square matrices of the same size. This is used for defining the exponential of a matrix , which is involved in the closed-form solution of systems of linear differential equations .
For the reverse implication, it suffices to show that if has all non-negative principal minors, then for all t>0, all leading principal minors of the Hermitian matrix + are strictly positive, where is the nxn identity matrix. Indeed, from the positive definite case, we would know that the matrices + are strictly positive definite.
If the quadratic form f yields only non-negative values (positive or zero), the symmetric matrix is called positive-semidefinite (or if only non-positive values, then negative-semidefinite); hence the matrix is indefinite precisely when it is neither positive-semidefinite nor negative-semidefinite. A symmetric matrix is positive-definite if and ...
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:
In operator theory, a branch of mathematics, a positive-definite kernel is a generalization of a positive-definite function or a positive-definite matrix. It was first introduced by James Mercer in the early 20th century, in the context of solving integral operator equations. Since then, positive-definite functions and their various analogues ...