problem5

problem5 - J of this matrix and write down cos e A...

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EE221A Linear System Theory Problem Set 5 Professor C. Tomlin Department of Electrical Engineering and Computer Sciences, UC Berkeley Fall 2007 Issued 10/23; Due 11/1 Problem 1. Suppose A C n × n is such that det( A ) = 0. Is det( e A ) = 0? Explain why or why not. Problem 2. You are told that A : R n R n and that R ( A ) N ( A ). Your friend from MIT says he can determine A up to a change of basis. Can he? Why or why not? Problem 3. A matrix A R 6 × 6 has minimal polynomial s 3 . Give bounds on the rank of A . Problem 4. A matrix A has minimal polynomial ( s - λ 1 ) 2 ( s - λ 2 ) 3 . Find cos( e A ) as a polynomial in A . Problem 5. In the preceding problem, assume that A has characteristic polynomial ( s - λ 1 ) 5 ( s - λ 2 ) 3 and that it has four linearly independent eigenvectors. Write down the Jordan form
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Unformatted text preview: J of this matrix and write down cos( e A ) explicitly. 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. Problem 8. Consider the linear system ˙ x = Ax + w ( t ) where w ( t ) is a T-periodic function, meaning that w ( T ) = w (0). There exists a T-periodic solution to this system, meaning there exists an x (0) such that x ( t + T ) = x ( t ). Find this solution by ±rst determining x (0). 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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