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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.
A semidefinite (or semi-definite) quadratic form is defined in much the same way, except that "always positive" and "always negative" are replaced by "never negative" and "never positive", respectively.
If = then this means the Hessian is positive semidefinite (or if the domain is the real line, it means that ″ ()), which implies the function is convex, and perhaps strictly convex, but not strongly convex.
If is positive definite, then the eigenvalues are all positive reals, so the chosen diagonal of also consists of positive reals. Hence the eigenvalues of Q U 1 2 Q ∗ {\displaystyle QU^{\frac {1}{2}}Q^{*}} are positive reals, which means the resulting matrix is the principal root of A {\displaystyle A} .
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:
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 ...