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Homework Help: Composition of endomorphisms have same eigenvalues

  1. Feb 14, 2012 #1
    1. The problem statement, all variables and given/known data

    For two endomorphisms ψ and φ on a vector space V over a field K, show that ψφ and φψ have the same eigenvalues. "Hint: consider the cases λ=0 and λ≠0 separately."

    3. The attempt at a solution

    I know that similar endomorphisms (φ and ψφ(ψ^-1)) have the same eigenvalues, so I have tried manipulating that expression with various choices for φ and ψ, but no luck. Other than that I just need a little help getting started
    1. The problem statement, all variables and given/known data

    2. Relevant equations

    3. The attempt at a solution
  2. jcsd
  3. Feb 14, 2012 #2
    if this is a linear? homomorphism and the values commute in a field...

    [edit] working on problem, i will assume not linear...

    [/2 edit] okay i see, you think will need the fact that the inverse maps have inverse eigenvalues...
    Last edited: Feb 14, 2012
  4. Feb 15, 2012 #3


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    Science Advisor

    hint: for λ ≠ 0, let v be an eigenvector of φψ, and consider ψφψ(v).

    this argument doesn't work if the eigenvalue is 0 (why?).

    all is not lost, however. note if 0 is an eigenvalue of φψ, this means φψ is singular.

    all you need to do is show that ψφ is likewise singular (hint: determinants).
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