Search results
Results from the WOW.Com Content Network
A function is a valid covariance function if and only if [2] this variance is non-negative for all possible choices of N and weights w 1, ..., w N. A function with this property is called positive semidefinite .
Given a formal Laurent series = =, the corresponding Hankel operator is defined as [2]: [] [[]]. This takes a polynomial [] and sends it to the product , but discards all powers of with a non-negative exponent, so as to give an element in [[]], the formal power series with strictly negative exponents.
The Gram matrix is positive semidefinite, and every positive semidefinite matrix is the Gramian matrix for some set of vectors. The fact that the Gramian matrix is positive-semidefinite can be seen from the following simple derivation:
One can define positive-definite functions on any locally compact abelian topological group; Bochner's theorem extends to this context. Positive-definite functions on groups occur naturally in the representation theory of groups on Hilbert spaces (i.e. the theory of unitary representations).
In mathematics (specifically linear algebra, operator theory, and functional analysis) as well as physics, a linear operator acting on an inner product space is called positive-semidefinite (or non-negative) if, for every (), , and , , where is the domain of .
Current events; Random article; About Wikipedia; Contact us; Contribute Help; ... In mathematics, positive semidefinite may refer to: Positive semidefinite function;
Semidefinite programming (SDP) is a subfield of mathematical programming concerned with the optimization of a linear objective function (a user-specified function that the user wants to minimize or maximize) over the intersection of the cone of positive semidefinite matrices with an affine space, i.e., a spectrahedron. [1]
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