Does diagonalizable imply symmetric?

[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.

Homework Equations

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

The Attempt at a Solution

See above.

 

[micromass]

It's not true. Consider the matrix

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

 

[Scootertaj]

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