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Unformatted text preview: ECE521 Linear Systems Fall 2007 Homework 2 Solutions 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 . The matrix A ( t ) has rank one so it cannot be invertible for any t . The other way of establishing this fact is by calculationg the determinant of A ( t ) which clearly is equal to zero. 2. Is it necessary the case that integraltext T A ( τ ) dτ is not invertible for any t ? Explain. The answer is no. As a counterexample consider a ( t ) = b ( t ) = [1 t ] T and calculat integraltext T A ( τ ) dτ . We have integraldisplay T A ( τ ) dτ = integraldisplay T parenleftbigg 1 τ τ τ 2 parenrightbigg dτ = parenleftbigg T 1 2 T 2 T 1 3 T 3 parenrightbigg which is nonsingular for all T negationslash = 0. 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 ? Say that X 1 ∈ R n × k 1 and X 2 ∈ R n × k 2 . Hence X ∈ R n × k 1 + k 2 . Let x ∈ span( X ). Then there exists α ∈ R n × k 1 + k 2 s.t. x = Xα . Let α = [ α T 1 α T 2 ] T where α i ∈ R n × k i . Then x = [ X 1 X 2 ][ α T 1 α T 2 ] T = X 1 α 1 + X 2 α 2 = x 1 + x 2 where x i ∈ span( X i ), i = 1 , 2. Thus Ax i ∈ span( X i ). Consequently Ax = Ax 1 + Ax 2 ∈ span( X 1 ) ∪ span( X 2 ) = span( X ). 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. September 18, 2009 2 There are at least two ways of approching this problem. One is by selecting a vector that we think might be a generalized eigenvector of maximal order and then proving that it indeed is a generalized eigenvector. The other is by solving a sequence of linear systems ( S − λI 4 ) x i = x i 1 , i = 1 , ..., 4 , where λ = 1 is the eigenvalue of S with the multiplicity 4 and x = [1 0 0 0] T being the only eigenvector of S ....
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