problem7

# problem7 - (b What are the eigenvalues of e At(c Suppose...

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EE221A Linear System Theory Problem Set 7 Professor C. Tomlin Department of Electrical Engineering and Computer Sciences, UC Berkeley Fall 2011 Issued 11/3; Due 11/10 Problem 1. A has characteristic polynomial ( s - λ 1 ) 5 ( s - λ 2 ) 3 , it has four linearly independent eigenvectors, the largest Jordan block associated to λ 1 is of dimension 2, the largest Jordan block associated to λ 2 is of dimension 3. Write down the Jordan form J of this matrix and write down cos( e A ) explicitly. Problem 2. A matrix A R 6 × 6 has minimal polynomial s 3 . Give bounds on the rank of A . Problem 3: Jordan Canonical Form. Given A = - 3 1 0 0 0 0 0 0 - 3 1 0 0 0 0 0 0 - 3 0 0 0 0 0 0 0 - 4 1 0 0 0 0 0 0 - 4 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 (a) What are the eigenvalues of A ? How many linearly independent eigenvectors does A have? How many generalized eigenvectors?
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Unformatted text preview: (b) What are the eigenvalues of e At ? (c) Suppose this matrix A were the dynamic matrix of an LTI system. What happens to the state trajectory over time (magnitude grows, decays, remains bounded. ..)? Problem 4. You are told that A : R n → R n and that R ( A ) ⊂ N ( A ). Can you determine A up to a change of basis? Why or why not? Problem 6. Let A ∈ R n × n be non-singular. True or false: the nullspace of cos ( log ( A )) is an A-invariant subspace? Problem 7. Consider A ∈ R n × n , b ∈ R n . Show that span { b, Ab, . . . , A n-1 b } is an A-invariant subspace. 1...
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## This note was uploaded on 12/09/2011 for the course EE 221A taught by Professor Clairetomlin during the Fall '10 term at Berkeley.

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