07Ma1bAnCayleyNotes

07Ma1bAnCayleyNotes - Chapter CH. The Cayley-Hamilton...

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Chapter CH. The Cayley-Hamilton Theorem. In this section V is a vector space over F = R or C of finite dimension n . Define M 0 n = M 0 n ( F ) to be the set of matrices A = ( a i,j ) M n such that a i, 1 = 0 for all i > 1. That is A is in M 0 n iff all entries in the first column of A , other than possibly the first entry, are 0. Remark 1 . Let A = ( a i,j ) M n . Then A M 0 n iff A = ± a α 0 A 0 where a = a ( A ) F , α = α ( A ) M 1 ,n - 1 is a row matrix, and A 0 M n - 1 . Lemma CH1. Let f ∈ L ( V ) , X = { x 1 ,... ,x n } a basis of V , and A = m X ( f ) . Then A M 0 n iff x 1 is an eigenvector for f . Proof. Let A = ( a i,j ). As A = m X ( f ), x 1 is an eigenvector for f iff f ( x 1 ) = a 1 , 1 x 1 iff a 1 ,j = 0 for j > 1 iff A M 0 n . Lemma CH2. Assume A M 0 n and let a = a ( A ) . Then char A ( x ) = ( x - a )char A 0 ( x ) . Proof. Let C = xI n - A = ( c i,j ) and C 0 = xI n - 1 - A 0 = ( c 0 i,j ). Then C = ± x - a - α 0 xI n - 1 - A 0 = ± x - a - α 0 C 0 Hence as in Problem 3 on HW5,
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07Ma1bAnCayleyNotes - Chapter CH. The Cayley-Hamilton...

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