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  2. Positive semidefinite - Wikipedia

    en.wikipedia.org/wiki/Positive_semidefinite

    In mathematics, positive semidefinite may refer to: Positive semidefinite function; Positive semidefinite matrix; Positive semidefinite quadratic form;

  3. Positive operator - Wikipedia

    en.wikipedia.org/wiki/Positive_operator

    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.

  4. Definite matrix - Wikipedia

    en.wikipedia.org/wiki/Definite_matrix

    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 ...

  5. Positive-definite function - Wikipedia

    en.wikipedia.org/wiki/Positive-definite_function

    By definition, a positive semi-definite matrix, such as , is Hermitian; therefore f −x) is the ...

  6. Hessian matrix - Wikipedia

    en.wikipedia.org/wiki/Hessian_matrix

    The Hessian matrix of a convex function is positive semi-definite. Refining this property allows us to test whether a critical point x {\displaystyle x} is a local maximum, local minimum, or a saddle point, as follows:

  7. Semidefinite programming - Wikipedia

    en.wikipedia.org/wiki/Semidefinite_programming

    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.

  8. Inner product space - Wikipedia

    en.wikipedia.org/wiki/Inner_product_space

    If the positive-definiteness condition is replaced by merely requiring that , for all , then one obtains the definition of positive semi-definite Hermitian form. A positive semi-definite Hermitian form ⋅ , ⋅ {\displaystyle \langle \cdot ,\cdot \rangle } is an inner product if and only if for all x {\displaystyle x} , if x , x = 0 ...

  9. Gram matrix - Wikipedia

    en.wikipedia.org/wiki/Gram_matrix

    The Gram matrix is symmetric in the case the inner product is real-valued; it is Hermitian in the general, complex case by definition of an inner product. 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 ...