math102HW9

math102HW9 - Math 102 - Homework 9/10 (selected problems)...

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Unformatted text preview: Math 102 - Homework 9/10 (selected problems) David Lipshutz Problem 1. (Strang, 6.2: #2) Decide for or against the positive definiteness of A = 2- 1- 1- 1 2- 1- 1- 1 2 , B = 2- 1- 1- 1 2 1- 1 1 2 , C = 0 1 2 1 0 1 2 1 0 2 = 5 2 1 2 2 2 1 2 5 Proof. Since A is singular, A is not positive definite. det B 3 = 4 > 0, det B 2 = 3 > 0 and det B 1 = 2 > 0, so B is positive definite. det C 3 = 32 > 0, det C 2 = 6 > 0 and det C 1 = 5 > so C is positive definite. Problem 2. (Strang, 6.2: #4) Show from the eigenvalues that if A is positive definite, so is A 2 and so is A- 1 . Proof. If A is positive definite, then the eigenvalues of A are all positive. Any eigenvector of A , with eigenvalues , is an eigenvector of A 2 with eigenvalue 2 . A is diagonizable so A has a full set of eigenvectors, therefore all the eigenvectors of A are eigenvectors of A 2 , so A 2 is positive definite. Also, the eigenvalues ofis positive definite....
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This note was uploaded on 04/03/2011 for the course MATH 102 taught by Professor Szypowsk during the Fall '08 term at UCSD.

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math102HW9 - Math 102 - Homework 9/10 (selected problems)...

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