math102HW9

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

This preview shows pages 1–2. Sign up to view the full content.

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

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 &gt; 0, det B 2 = 3 &gt; 0 and det B 1 = 2 &gt; 0, so B is positive definite. det C 3 = 32 &gt; 0, det C 2 = 6 &gt; 0 and det C 1 = 5 &gt; 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....
View Full Document

## This note was uploaded on 04/03/2011 for the course MATH 102 taught by Professor Szypowsk during the Fall '08 term at UCSD.

### Page1 / 3

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

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online