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Unformatted text preview: EE221A Linear System Theory Problem Set 2 Professor C. Tomlin Department of Electrical Engineering and Computer Sciences, UC Berkeley Fall 2007 Issued 9/18; Due 9/27 Problem 1: Useful properties of eigenvalues. Let A ∈ R n × m , B ∈ R m × n and let n ≥ m . Observe that AB has n eigenvalues, while BA has m eigenvalues. Prove that m of the eigenvalues of AB are precisely those of BA , while the remaining n- m eigenvalues of AB are at zero. Problem 2: Nullspaces. (a) Prove that if ˜ A is obtained from A by elementary row operations, then N ( A ) = N ( ˜ A ). (b) Prove that if ˜ A is obtained from A by elementary column operations, then R ( A ) = R ( ˜ A ). Problem 3: Linear Matrix Equations. Let A ∈ C m × n ,B ∈ C n × q ,C ∈ C m × n , and D ∈ C n × q . (a) When is the matrix equation AX = C solvable for X ∈ C n × n . When is the solution unique? (b) When is the matrix equation XB = D solvable for X ∈ C n × n . When is the solution unique?...
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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 University of California, Berkeley.
- Fall '10
- Electrical Engineering