# Real eigenvalues, similar symmetric matrix

I know that symmetric matrices have real eigenvalues, and that non-symmetric matrices that are similar to symmetric matrices must also have real eigenvalues, but is the converse true?

That is, if a matrix has real eigenvalues, must there exist a similar matrix that is symmetric?

No. Counterexample: $$\pmatrix{0&1\\0&0}$$ is not similar to any symmetric matrix.

On the other hand, every diagonalizable matrix with real eigenvalues is similar to a symmetric matrix.

• Thanks for the counter-example. Indeed, I am indeed looking at diagonalizable matrices. Can you explain in more detail or sketch out a proof if it is short, or point me to a text that shows this? – science404 Aug 14 '15 at 21:38
• @science404 If a matrix is diagonalizable, it is similar to a diagonal matrix, which is symmetric – angryavian Aug 14 '15 at 21:38
• Of course, thanks! – science404 Aug 14 '15 at 21:39