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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
The definition of positive definite can be generalized by designating any complex matrix (e.g. real non-symmetric) as positive definite if {} > for all non-zero complex vectors , where {} denotes the real part of a complex number . [19] Only the Hermitian part (+) determines whether the matrix is positive definite, and is assessed in the ...
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 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 ...
In mathematics, positive semidefinite may refer to: Positive semidefinite function; Positive semidefinite matrix; Positive semidefinite quadratic form;
The operator is said to be positive-definite, and written >, if , >, for all {}. [ 1 ] Many authors define a positive operator A {\displaystyle A} to be a self-adjoint (or at least symmetric) non-negative operator.
Positive-definite functions on are intimately related to unitary representations of . Every unitary representation of G {\displaystyle G} gives rise to a family of positive-definite functions. Conversely, given a positive-definite function, one can define a unitary representation of G {\displaystyle G} in a natural way.
A real (1,1)-form is called semi-positive [1] (sometimes just positive [2]), respectively, positive [3] (or positive definite [4]) if any of the following equivalent conditions holds: is the imaginary part of a positive semidefinite (respectively, positive definite) Hermitian form.