1. Not finding help here? Sign up for a free 30min tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

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

    Deveno

    User Avatar
    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).
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook




Similar Discussions: Composition of endomorphisms have same eigenvalues
Loading...