Hw8 - transforms that relates the following 2 block matrices p-B AB P and p B A P 5 Find all possible solutions u n(for n g that satisfy the

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ECE210A November 15, 2010 Homework 8 Due date: November 22, 2010 1. Reading assignment . Read chapters 4 and 5 of the class notes. 2. Find all the eigenvalues and eigenvectors of the n × n matrix: 0 1 0 0 0 0 0 1 ε 0 0 . Hint : Look for eigenvectors of the form 1 α α 2 . 3. Let A and B be two square matrices. Show that λ p A C 0 B P = λ ( A ) λ ( B ) . That is, the eigenvalues of the matrix on the left are union of the eigenvalues of A and B . 4. Let A be an m × p matrix and B be an p × m matrix, with p l m . Show that λ ( A B ) = λ ( B A ) ∪ { 0 , , 0 } m - p copies . That is, the eigenvalues of AB are the eigenvalues of B A along with m - p copies of 0. Hint : One possible approach is to ±nd a similarity tansformation (using block Gauss
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Unformatted text preview: transforms) that relates the following 2 block matrices: p-B AB P , and p B A P . 5. Find all possible solutions u [ n ] (for n g ) that satisfy the equation u [ n + 1] = ( λI + Z ) u [ n ] + α n e, where Z = 1 1 , and e = 1 1 . Simplify as much as possible. 6. Find all possible solutions u ( t ) (for t g ) that satisfy the equation d dt u ( t ) = ( λI + Z ) u ( t ) + e αt e, where e and Z are as de±ned in the previous problem. Simplify as much as possible. 1...
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This note was uploaded on 01/14/2011 for the course ECE 210a taught by Professor Chandrasekara during the Fall '08 term at UCSB.

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