Complex Eigenvectors: Showing A(ReV) and A(ImV)

[Shaunzio]

Homework Statement

Let A be a 2x2 matrix with complex eigenvalue (lambda)=a-bi, (b does not equal 0) and associated eigenvector v-> C^2. Show that A(Rev)=aRev +bImv and A(ImV)=-bRev +aImv. Where Re is real part and I am is imaginary part.

Homework Equations

The Attempt at a Solution

I am terrible at proofs and I am no idea how one would show this..

 

[Dick]

They should probably make it clear A is real matrix. Start by using v=Re(v)+i*Im(v) and expand A(v)=(a-ib)v in terms of Re(v) and Im(v). There really aren't any tricks here.