# Transpose Eigenvector Proof

1. Apr 7, 2009

### chancellorpho

Eigenvalue and eigenvector for a symmetric matrix

1. The problem statement, all variables and given/known data

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.

2. Relevant equations

3. 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!

Last edited: Apr 7, 2009
2. Apr 7, 2009

### chancellorpho

Can anyone offer any insight?

3. Apr 8, 2009

### 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.