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11hw6 - Homework Assignment#6 16.2 Let Q(x = xT Ax be a...

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Homework Assignment #6 16.2 Let Q ( x ) = x T A x be a quadratic form on R n . By evaluating Q on each of the coordinate axes in R n , prove that a necessary condition for a symmetric matrix to be positive definite (positive semidefinite) is that all the diagonal entries be positive (nonnegative). State and prove the corresponding result for negative and negative semidefinite matrices. Give an example to show that this necessary condition is not sufficient. Answer: The corresponding result is that a necessary condition for a symmetric matrix to be negative definite (negative semidefinite) is that all the diagonal entries be negative (nonpositive). All can be proven together by considering Q ( e i ) = e T i A e i = a ii where e i is the i t h standard basis vector. The sign of this is determined by the type of definiteness or semidefiniteness. If A is positive definite, then a ii > 0 for all i = 1 , . . . , n . If A is positive semidefinite, then a ii 0 for all i = 1 , . . . , n . If A is negative definite, then a ii < 0 for all i = 1 , . . . , n . If A is negative semidefinite, then a ii 0 for all i = 1 , . . . , n . As for examples, consider A 1 - 1 4 4 - 1 and A 2 0 4 4 0 . Then A ! satisifes the necessary conditions for negative definiteness, and A 2 satisfies the necessary condi- tions for negative semidefiniteness. But Q 1 (1 , 1) = 6 and Q 2 (1 , 1) = 8, showing that A 1 is not negative definite and that A 2 is not negative semidefinite. 16.6 Determine the definiteness of the following constrained quadratics. a ) Q ( x 1 , x 2 ) = x 2 1 + 2 x 1 x 2 - x 2 2 , subject to x 1 + x 2 = 0.
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