A matrix is positive semidefinite if, for any non-zero vector \(x\), the quadratic form \(x^T A x \geq 0\) holds true, where \(A\) is the matrix in question. This property indicates that the matrix does not yield negative values when applied to any vector, suggesting that it reflects a scenario of minimized energy or optimal stability in systems. Understanding positive semidefinite matrices is crucial for determining optimality conditions, especially in unconstrained optimization problems where you want to find local minima or maxima.
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