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In mathematics, positive semidefinite may refer to: Positive semidefinite function; Positive semidefinite matrix; Positive semidefinite quadratic form;
The set of positive definite matrices is an open convex cone, while the set of positive semi-definite matrices is a closed convex cone. [ 2 ] Some authors use more general definitions of definiteness, including some non-symmetric real matrices, or non-Hermitian complex ones.
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.
is a positive semi-definite matrix. [citation needed] By definition, a positive semi-definite matrix, such as , is Hermitian; therefore f −x) is the ...
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. In other words, it may take on zero values for some non-zero vectors of V. An indefinite quadratic form takes on both positive and negative ...
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:
Conversely, every symmetric positive semi-definite matrix is a covariance matrix. To see this, suppose M {\displaystyle M} is a p × p {\displaystyle p\times p} symmetric positive-semidefinite matrix.
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. [1]