sol7 - EE 221: Linear Systems HW #7 Solutions Sam Burden...

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EE 221 : Linear Systems HW #7 Solutions — Sam Burden — Nov 16, 2010 1 Note these solutions are overly terse and leave out some details; in your solutions, you should strive for greater clarity and rigor. Exercise 1. If A has 4 linearly independent eigenvectors, characteristic polynomial ( s - λ 1 ) 5 ( s - λ 2 ) 3 , the largest Jordan block associated with λ 1 has dimension 2, and the largest Jordan block associated with λ 2 has dimension 3, then its Jordan form is J = λ 1 1 0 0 0 0 0 0 0 λ 1 0 0 0 0 0 0 0 0 λ 1 1 0 0 0 0 0 0 0 λ 1 0 0 0 0 0 0 0 0 λ 1 0 0 0 0 0 0 0 0 λ 2 1 0 0 0 0 0 0 0 λ 2 1 0 0 0 0 0 0 0 λ 2 Hence with f ( s ) := cos e s , f ( J ) = 2 6 6 6 6 6 6 6 6 6 6 6 6 4 cos( e λ 1 ) - e λ 1 sin( e λ 1 ) 0 0 0 0 0 0 0 cos( e λ 1 ) 0 0 0 0 0 0 0 0 cos( e λ 1 ) - e λ 1 sin( e λ 1 ) 0 0 0 0 0 0 0 cos( e λ 1 ) 0 0 0 0 0 0 0 0 cos( e λ 1 ) 0 0 0 0 0 0 0 0 cos( e λ 2 ) - e λ 2 sin( e λ 2 ) - 1 2 e λ 2 sin( e λ 2 ) + e 2 λ 2 cos( e λ 2 ) 0 0 0 0 0 0 cos( e λ 2 ) - e λ 2 sin( e λ 2 ) 0 0 0 0 0 0 0 cos( e λ 2 ) 3 7 7 7 7 7 7 7 7 7 7 7 7 5 and f ( A ) is obtained from
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