Eigenvalues

[Jennifer1990]

1. The problem statement, all variables and given/known data
Suppose that v is an eigenvector of both A and B with corresponding eigenvalues lambda and mui respectively. Show that v is an eigenvector of A+B and of AB and determine the corresponding eigenvalues


2. Relevant equations



3. The attempt at a solution
Av = lambda*v
Bv = mui*v
this is all i can think of....can someone give me a hint abt the next step?

[Pengwuino]

Add the two equations together, tada! Of course, you'll need to exploit associativity... or linearity... man im always getting terms confused.

[Jennifer1990]

u mean like:
(Av +Bv) = lambda*v + mui*v
(A+B)v = (lambda + mui) v

[Pengwuino]

Yup! As for finding the eigenvalues of AB, simply multiply AB by v and remember that your eigenvalues are scalars that can move freely.