Unformatted text preview: h is varied. This is intuitive, since h measures error in x 2 , but the control cannot aﬀect x 2 . Exercise 3. Since det BC = det B det C , χ A ( s ) = det( sIA ) = det P det( sIPAP1 ) det P1 = det( sIPAP1 ) = χ PAP1 ( s ) , whence A and ¯ A = PAP1 have the same eigenvalues. Exercise 4. A = n X j =1 λ j w j v T j = ⇒ A k = n X j =1 λ k j w j v T j = ⇒ e At = ∞ X k =0 t k k ! A k = ∞ X k =0 n X j =1 ( λ j t ) k k ! w j v T j = n X j =1 e λ j t w j v T j . Exercise 5. A ( A j1 b ) = A j b , which is either contained in the given list of vectors or, by CayleyHamilton, linearly dependent on them....
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 Fall '10
 ClaireTomlin
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