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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.
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 .
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 ...
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.
is a positive semi-definite matrix. [citation needed] By definition, a positive semi-definite matrix, such as , is Hermitian; therefore f −x) is the ...
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:
This also leads to a proof of the above observation, that a positive-definite matrix has precisely one positive-definite square root: a positive definite matrix has only positive eigenvalues, and each of these eigenvalues has only one positive square root; and since the eigenvalues of the square root matrix are the diagonal elements of D 1/2 ...