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Quadratic Forms Tests In optimization problems it is necessary to test the Hessian matrix, H , for positive- definiteness, negative-definiteness, or indefinite form. What we are interested in is the following condition (positive-definiteness): yHy T > 00 , for all y . How do we go about determining if this condition is true? To illustrate so conditions where it is easy to determine this, consider yy y y T 12 1 2 20 02 ,, bg L N M O Q P F H G I K J >≠ for all y , which becomes y y y y T 1 2 1 2 2 2 220 0 0 , b g b g a f L N M O Q P F H G I K J =+> for . In this situation, it is easy to determine that the quadratic form is positive definite. Similarly the following form is negative definite, y y y y T 1 2 1 2 2 2 03 230 0 0 , af L N M O Q P F H G I K J =− < for . And one that is obviously not either y y y y T 1 2 1 2 2 2 23 0 0 0 , L N M O Q P F H G I K J + < > R S T U V W for . So if we are fortunate and the quadratic form has diagonal structure, then we can easily determine which of the three categories it is in. On the other hand, and much more frequently, we are faced with a non-diagonal (but always symmetric) quadratic form.

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