MA353_HW11_sol

# MA353_HW11_sol - Purdue University MA 353 Linear Algebra II...

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Purdue University MA 353: Linear Algebra II with Applications Homework 11, due Apr. 9, Some solutions Sec. 5.4:#16: Let T be a linear operator on a ﬁnite-dimensional vector space V . (a) Prove that if the characteristic polynomial of T splits, then so does the characteristic polynomial of the restriction of T to any T -invariant subspace of V . Proof. Let f ( t ) be the characteristic polynomial of T . Then since it splits, it is written as f ( t ) = ( - 1) n ( t - λ 1 )( t - λ 2 ) ··· ( t - λ n ) for some λ i , where n = dim V . (Here λ i are not necessarily dis- tinct.) If T W is the restriction of T to a T -invariant subspace W , and g ( t ) is the characteristic polynomial of T W , then by Theorem 5.22, we know that g ( t ) divides f ( t ). Namely, g ( t ) is a factor of f ( t ), i.e. g ( t ) = ( - 1) k ( t - μ 1 )( t - μ 2 ) ··· ( t - μ k ) where k = dim W and each μ i is λ j for some j . Hence in particular g ( t ) splits. ± (b) Deduce that if the characteristic polynomial of T splits, then any nontrivial T -invariant subspace of V contains an eigenvector of T . Proof. Let W be a nontrivial T -invariant subspace of V . By the previous part, the characteristic polynomial g ( t ) is written as g ( t ) = ( - 1) k ( t - μ 1 )( t - μ 2 ) ··· ( t - μ k ) where k = dim W and for each i we have μ i = λ j for some j . So for some j , λ j is an eigenvalue of T W . Namely for some nonzero w W , we have T W ( w ) = λ

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## This note was uploaded on 04/29/2010 for the course MA 353 taught by Professor Staff during the Spring '08 term at Purdue.

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MA353_HW11_sol - Purdue University MA 353 Linear Algebra II...

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