quiz3_1806_s05

quiz3_1806_s05 - Explain why any two choices of a lead to...

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18.06 Professor Strang Quiz 3 May 4, 2005 Grading 1 Your PRINTED name is: 2 3

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1( 3 7 p t s . ) (a) (16 points) Find the three eigenvalues and all the real eigenvectors of A . It is a symmetric Markov matrix with a repeated eigenvalue . 1 4 1 4 2 4 1 4 2 4 1 4 2 4 1 4 1 4 A = . (b) (9 points) Find the limit of A k as k →∞ . (You may work with A = S Λ S 1 without computing every entry.) (c) (6 points) Choose any positive numbers r, s, t so that A rI is positive deﬁnite A sI is indeﬁnite A tI is negative deﬁnite (d) (6 points) Suppose this A equals B T B . What are
2( 4 1 p t s . ) (a) (14 points) Complete this 2 by 2 matrix A (depending on a )so that its eigenvalues are λ =1 and λ = 1: a 1 A = (b) (9 points) How do you know that A has two independent eigenvectors ? (c) (9 points) Which choices of a give orthogonal eigenvectors

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Unformatted text preview: Explain why any two choices of a lead to matrices A that are similar (with the same Jordan form). 3 3 (22 pts.) Suppose the 3 by 3 matrix A has independent eigenvectors in Ax 1 = 1 x 1 , Ax 2 = 2 x 2 , Ax 3 = 3 x 3 . (Those s might not be dierent.) (a) (11 points) Describe the general form of every solution u ( t ) to the dierential equation du = Au . (The answer e At u (0) does not use the dt s and x s.) (b) (11 points) Starting from any vector u 0 in R 3 , suppose u k +1 = Au k . What are the conditions on the x s and s to guarantee that u k (as k ) ? Why ? 4...
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This note was uploaded on 01/14/2011 for the course EECS 18.06 taught by Professor Strang during the Spring '05 term at University of Michigan.

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quiz3_1806_s05 - Explain why any two choices of a lead to...

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