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Download as PDF; Printable version; ... In mathematics, positive semidefinite may refer to: Positive semidefinite function ... Statistics; Cookie statement; Mobile view;
In statistics, the covariance matrix of a multivariate probability distribution is always positive semi-definite; and it is positive definite unless one variable is an exact linear function of the others. Conversely, every positive semi-definite matrix is the covariance matrix of some multivariate distribution.
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 .
This implies that at a local minimum the Hessian is positive-semidefinite, and at a local maximum the Hessian is negative-semidefinite. For positive-semidefinite and negative-semidefinite Hessians the test is inconclusive (a critical point where the Hessian is semidefinite but not definite may be a local extremum or a saddle point).
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 ...
Some authors use the name square root or the notation A 1/2 only for the specific case when A is positive semidefinite, to denote the unique matrix B that is positive semidefinite and such that BB = B T B = A (for real-valued matrices, where B T is the transpose of B).
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.
The symmetric matrix is positive (semi) definite and is the Hessian matrix of the function () at the extremum point . This can be thought of intuitively as: "The distance between two infinitesimally close points on a statistical differential manifold is the informational difference between them."