# Question on fundamental theorem of symmetric matrices

I'm a little confused on the wording of this statement in my book:

A is orthogonally diagonalizable and has real eigenvalues if and only if A is symmetric.

They're not saying that only symmetric matrices can have all real eigenvalues, right?

Can matrices that are not symmetric have all real eigenvalues?

• The adverb "orthogonally" is the crucial part here. – user228113 Mar 11 '17 at 23:02
• consider $\begin{pmatrix} a & b \\ 0 & c\end{pmatrix}$ with $a,b,c\in \Bbb R$. This matrix is not symmetric if $b \neq 0$ but its eigenvalues are $a,c$. – Surb Mar 11 '17 at 23:04