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Definite quadratic forms lend themselves readily to [[optimization]] problems. Suppose the matrix quadratic form is augmented with linear terms, as
:<math>x^\mathsf{T} A \, x +
where {{mvar|b}} is an {{mvar|n}}×1 vector of constants. The [[first-order condition]]s for a maximum or minimum are found by setting the [[matrix derivative]] to the zero vector:
:<math> 2 A \, x +
giving
:<math> x = -\tfrac{1}{2}\,A^{-1}b \;,</math>
assuming {{mvar|A}} is [[nonsingular matrix|nonsingular]]. If the quadratic form, and hence {{mvar|A}}, is positive-definite, the [[second partial derivative test|second-order condition]]s for a minimum are met at this point. If the quadratic form is negative-definite, the second-order conditions for a maximum are met.
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