Search results
Results from the WOW.Com Content Network
Positive-definiteness arises naturally in the theory of the Fourier transform; it can be seen directly that to be positive-definite it is sufficient for f to be the Fourier transform of a function g on the real line with g(y) ≥ 0.
In mathematics, positive definiteness is a property of any object to which a bilinear form or a sesquilinear form may be naturally associated, which is positive-definite. See, in particular: Positive-definite bilinear form; Positive-definite function; Positive-definite function on a group; Positive-definite functional; Positive-definite kernel
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,
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 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 ...
A Hermitian diagonally dominant matrix with real non-negative diagonal entries is positive semidefinite. This follows from the eigenvalues being real, and Gershgorin's circle theorem. If the symmetry requirement is eliminated, such a matrix is not necessarily positive semidefinite. For example, consider
If the Hessian is negative definite (equivalently, has all eigenvalues negative) at a, then f attains a local maximum at a. If the Hessian has both positive and negative eigenvalues then a is a saddle point for f (and in fact this is true even if a is degenerate). In those cases not listed above, the test is inconclusive. [2]
In mathematics, positive semidefinite may refer to: Positive semidefinite function; Positive semidefinite matrix; Positive semidefinite quadratic form;