Assume you have a non-symmetric real square matrix all of whose eigenvalues are real. Can anything be said about it? Is it unitarily equivalent to a symmetric matrix? 


EDIT: Is it at least *similar* to a symmetric matrix? 



Matrices with real spectrum
Well there is the following. Consider
$$
 \begin{equation}
 A = \left( \begin{array}{cc}
 0 & 1 \\
 0 & 0 \\
 \end{array}
 \right)
 \end{equation}
$$
The matrix $A$ is non-symmetric with eigenvalues $\{0\}$. If $A$ were similar to a symmetric matrix $M$, then $A$ would be diagonalizable (because $M$ is), and we would have $A = 0$.


We can also look at
$$
 \begin{equation}
 B = \left( \begin{array}{cc}
 1 & 1 \\
 0 & 1 \\
 \end{array}
 \right)
 \end{equation}
$$
The eigenvalues of $B$ are $\{1\}$. If $B$ were similar to a symmetric matrix, then $B$ would equal the identity.


