Composition of endomorphisms have same eigenvalues

[Tmaker]

Homework Statement

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."

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

 

[xaos]

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...

 

[Deveno]

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).