Does diagonalizable imply symmetric?

1. Apr 10, 2012

Scootertaj

In order to prove my PDE system is well-posed, I need to show that if a matrix is diagonalizable and has only real eigenvalues, then it's symmetric.

2. Relevant equations
I've found theorems that relate orthogonally diagonalizable and symmetric matrices, but is that sufficient?

3. The attempt at a solution
See above.

2. Apr 10, 2012

micromass

It's not true. Consider the matrix

$$\left(\begin{array}{cc} 1 & 2\\ 0 & 3\end{array}\right)$$

3. Apr 10, 2012

Scootertaj

Darn.. well I guess to find another way to show the PDE is well-posed.