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In mathematics, positive semidefinite may refer to: Positive semidefinite function; Positive semidefinite matrix; ... 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.
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 ...
By definition, a positive semi-definite matrix, such as ... In statistics, and especially Bayesian statistics, the theorem is usually applied to real functions.
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:
Disturbances that are farther apart from each other are given lower weight, while those with equal subscripts are given a weight of 1. This ensures that second term converges (in some appropriate sense) to a finite matrix. This weighting scheme also ensures that the resulting covariance matrix is positive semi-definite. [2]
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 .
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."