Solving Eigenvector Problems: A+B and AB with Corresponding Eigenvalues

[Jennifer1990]

Homework Statement

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

Homework Equations

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 I am 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.