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  2. Positive semidefinite - Wikipedia

    en.wikipedia.org/wiki/Positive_semidefinite

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  3. Positive operator - Wikipedia

    en.wikipedia.org/wiki/Positive_operator

    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 .

  4. Positive-definite function - Wikipedia

    en.wikipedia.org/wiki/Positive-definite_function

    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.

  5. Definite quadratic form - Wikipedia

    en.wikipedia.org/wiki/Definite_quadratic_form

    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.

  6. Convex function - Wikipedia

    en.wikipedia.org/wiki/Convex_function

    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.

  7. Square root of a matrix - Wikipedia

    en.wikipedia.org/wiki/Square_root_of_a_matrix

    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} .

  8. Gram matrix - Wikipedia

    en.wikipedia.org/wiki/Gram_matrix

    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:

  9. Positive-definite kernel - Wikipedia

    en.wikipedia.org/wiki/Positive-definite_kernel

    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 ...