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Positive-definite and positive-semidefinite real matrices are at the basis of [[convex optimization]], since, given a [[function of several real variables]] that is twice [[differentiable function|differentiable]], then if its [[Hessian matrix]] (matrix of its second partial derivatives) is positive-definite at a point {{mvar|p}}, then the function is [[convex function|convex]] near {{mvar|p}}, and, conversely, if the function is convex near {{mvar|p}}, then the Hessian matrix is positive-semidefinite at {{mvar|p}}.
The set of positive definite matrices is an [[Open set|open]] [[convex cone]], while the set of positive semi-
Some authors use more general definitions of definiteness, including some non-symmetric real matrices, or non-Hermitian complex ones.
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