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Eigenvalues, detA

  • Thread starter dsyh
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  • #1
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Homework Statement


Let A be nxn matrix, suppose n has real eigenvalues,λ1,...,λn, repeated according to multipilicities. Prove that detA = λ1...λn.

Homework Equations




The Attempt at a Solution


I started by applying the definition, Av = λv, where v is an eigenvector. then I just dun know how to keep going.. is there anyone can help me out? or at least give me some hints..
thx..
 

Answers and Replies

  • #2
D H
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If matrices A and B are similar, what can you say about their determinants? Use this relationship to help you solve the problem.
 
  • #3
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DH offered an correct approach,but you need to know something about similarity and Jordan normal form or Schur's lemma
Here's another approach: consider the characteristic polynomial det(A-λI),by fundamental theorem of algebra, it can be factorized into (λ1-λ)(λ2-λ)........(λn-λ),then let λ=0 and see what will happen
 

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