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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 .
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 semidefinite may refer to: Positive semidefinite function; Positive semidefinite matrix; Positive semidefinite quadratic form;
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 ...
This implies that at a local minimum the Hessian is positive-semidefinite, and at a local maximum the Hessian is negative-semidefinite. For positive-semidefinite and negative-semidefinite Hessians the test is inconclusive (a critical point where the Hessian is semidefinite but not definite may be a local extremum or a saddle point).
A form is called strongly positive if it is a linear combination of products of semi-positive forms, with positive real coefficients. A real (p, p) -form η {\displaystyle \eta } on an n -dimensional complex manifold M is called weakly positive if for all strongly positive (n-p, n-p) -forms ζ with compact support, we have ∫ M η ∧ ζ ≥ 0 ...