Definite quadratic form: Difference between revisions

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:<math> Q(x) = c_1{x_1}^2 + c_2{x_2}^2 </math>

where <math>~ x = [x_1, x_2] \in V </math> and {{mvar|c}}{{sub|1}} and {{mvar|c}}{{sub|2}} are constants. If {{nobr| {{mvar|c}}{{sub|1}} > 0 }} and {{nobr| {{mvar|c}}{{sub|12}} > 0 ,}} the quadratic form {{mvar|Q}} is positive-definite, so ''Q'' evaluates to a positive number whenever <math>\; [x_1,x_2] \neq [0,0] ~.</math> If one of the constants is positive and the other is 0, then {{mvar|Q}} is positive semidefinite and always evaluates to either 0 or a positive number. If {{nobr| {{mvar|c}}{{sub|1}} > 0 }} and {{nobr| {{mvar|c}}{{sub|2}} < 0 ,}} or vice versa, then {{mvar|Q}} is indefinite and sometimes evaluates to a positive number and sometimes to a negative number. If {{nobr| {{mvar|c}}{{sub|1}} < 0 }} and {{nobr| {{mvar|c}}{{sub|2}} < 0 ,}} the quadratic form is negative-definite and always evaluates to a negative number whenever <math>\; [x_1,x_2] \neq [0,0] ~.</math> And if one of the constants is negative and the other is 0, then {{mvar|Q}} is negative semidefinite and always evaluates to either 0 or a negative number.

In general a quadratic form in two variables will also involve a cross-product term in {{mvar|x}}{{sub|1}}·{{mvar|x}}{{sub|2}}:

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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 + 2 b^\mathsf{T} x \;,</math>

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 + 2 b = \vec 0 \;,</math>

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.