Symmetric Matrix Eigenvector Proof

[chancellorpho]

Eigenvalue and eigenvector for a symmetric matrix

Homework Statement

Let A be a n by n real matrix with the property that the transpose of A equals A. Show that if Ax = lambda x, for some non-zero vector x in C(n) then lambda is real, and the real part of x is an eigenvector of A.

Homework Equations

The Attempt at a Solution

Since transpose of A equals A, A must be a symmetric matrix. But beyond that, I don't know where to start. Any help would be appreciated!

 

[chancellorpho]

Can anyone offer any insight?

 

[Cyosis]

Start out with (\boldsymbol{v},A \boldsymbol{v}). In case this notation is unknown to you it's supposed to represent the complex inner product.