An n×n complex matrix A is called positive definite if ℜ[x^* A x]>0 for all nonzero complex vectors x element C^n, where x^* denotes the conjugate transpose of the vector x. In the case of a real matrix A, equation (-1) reduces to x^T A x>0, where x^T denotes the transpose. Positive definite matrices are of both theoretical and computational importance in a wide variety of applications. They are used, for example, in optimization algorithms and in the construction of various linear regression models. A matrix m may be tested to determine if it is positive definite in the Wolfram Language using PositiveDefiniteMatrixQ[m].
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