section9 - -1 = n X i =1 1 s-λ i e i v T i where ( e i ,v...

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EE221A Section 9 10/28/11 1 Administrivia Midterm avg 28, median 29, std dev 4.9 (out of 34) HW6 is out, due next Fri (Nov 4) 2 Cayley-Hamilton Theorem Recall: Characteristic polynomial of A : ˆ χ A ( s ) := det ( sI - A ) = s n + d 1 s n - 1 + ··· + d n Characteristic equation : ˆ χ A ( s ) = 0 Eigenvalues of A are roots of the char. poly/solutions of the char. equation. Cayley-Hamilton Theorem : Every matrix A satisfies its own characteristic equation, i.e. A n + d 1 A n - 1 + ··· + d n - 1 A + d n I = 0 n × n Important corollary : A m , m n , is a linear combination of { I,A,. ..,A n - 1 } . Exercise 1. Let A = ± 1 - 2 3 - 4 ² . Compute A - 1 using the C-H theorem. Exercise 2. Let A = ± 1 2 0 1 ² . Compute A 200 using the C-H theorem. 3 Dyadic expansion Exercise 3. Assume that A has n distinct eigenvalues λ 1 ,...,λ n . Show that: a) e At = n X i =1 e λ i t e i v T i b) ( sI - A )
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Unformatted text preview: -1 = n X i =1 1 s-λ i e i v T i where ( e i ,v i ,λ i ) are the right and left eigenvectors and corresponding eigenvalues of A . Exercise 4. Let A = ± 1 0 2 3 ² . Find an expression for A n . 1 4 A-invariant subspaces 2 4 A-invariant subspaces Consider the linear map A : V → V . A subspace M ⊂ V is said to be A-invariant if, A ( x ) ∈ M for all x ∈ M . Exercise 5. Show that the following subspaces of V are A-invariant: (a) N ( A ) (b) R ( A ) (c) N ( A-λI ) , where λ ∈ σ ( A ) Exercise 6. Let x ∈ R n . Show that the vector space Ω := span { x,Ax,A 2 x,. ..,A n-1 x } is A-invariant....
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This note was uploaded on 12/09/2011 for the course EE 221A taught by Professor Clairetomlin during the Fall '10 term at University of California, Berkeley.

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section9 - -1 = n X i =1 1 s-λ i e i v T i where ( e i ,v...

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