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Unformatted text preview: ECE521 Linear Systems Fall 2009 Homework 2: Due Septemper 24, in class To the extent possible please type your homeworks. Note: You are free to use Matlab for calculations. However Matlab is not necessary for obtaining answers to any of the problems. Problem 1 : Consider a 2 matrix A ( t ) which is continuous for all real t but is not invertible for any t . 1. As an example of such A ( t ), show that A ( t ) = a ( t ) b ( t ) T where a ( t ) , b ( t ) ∈ R 2 are continu- ous for all real t is not invertible for any t . 2. Is it necessary the case that integraltext T A ( τ ) dτ is not invertible for any t ? Explain. Problem 2: 1. Let columns of X i , i = 1 , 2, span invariant subspaces of A . Do the columns of X = [ X 1 , X 2 ] also span an invariant subspace of A ? 2. Let S = 1 2 9 10 0 1 2- 3 0 0 1 2 0 0 0 1 Find a generalized eigenvector of S of maximal order. Show your work....
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