# Symmetric matrix eigenvalue and eigenvector

• Apr 7th 2009, 07:21 PM
chancellorphoenix
Symmetric matrix eigenvalue and eigenvector
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 Cn then lambda is real, and the real part of x is an eigenvector of A.

I know that since transpose of A equals A, A is a symmetric matrix, but beyond that, I'm stuck. Help would be appreciated.
• Apr 7th 2009, 09:44 PM
ThePerfectHacker
Quote:

Originally Posted by chancellorphoenix
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 Cn then lambda is real, and the real part of x is an eigenvector of A.

I know that since transpose of A equals A, A is a symmetric matrix, but beyond that, I'm stuck. Help would be appreciated.

The problem here is that I do not know what how you are expected to solve this problem. Perhaps, you are allowed to use the that Hermetian matrices have real eigenvalues. Now prove that symmetric real matrices are a subset of Hermetian matrices and so they must have real eigenvalues.