Proof of Eigenvector Property with Simple Linear Algebra

[Hockeystar]

Homework Statement

Let x be an eigenvector of A with eigenvalue $$\lambda$$ and suppose x is also an eigenvector of B, corresponding to the eigenvalue $$\lambda$$₂. Let C = A + B. Show that x is an eigenvector of C. What is the corresponding eigenvalue?

to the eigenvalue 2

Homework Equations

The Attempt at a Solution

{$$\lambda$$I - A} = {$$\lambda$$₂I - B}

C = 2{$$\lambda$$I - A}
C is just a linear combination of the first eigenvector so it's got the same eigenvector.

Is this enough to complete the proof?

Is the corresponding eigenvalue just twice the original eigenvalue?

 

[lanedance]

why not just start by examining the product:
Cx

 

[earlh]

Start like this: you know $$Ax = \lambda x$$ and $$Bx = \lambda_{2} x$$ from the definition of an eigenvector

Thus $$Cx = (A + B)x = ...$$ and go from there. I think it will be straightforward.