I'm trying to construct a normal matrix $A\in \mathbb R^{n\times n}$ such that all it's eigenvalues are complex but at least one of then also has a positive real part (well, at least two then, since the entries are real they'd come in pairs). Is that possible?
If the desired matrix exists, we'd find it among normal matrices which are neither hermitian nor skew-hermitian. E.g.
$$
\begin{pmatrix}
1 & 0 & 1 \\
1 & 1 & 0 \\
0 & 1 & 1 \\
\end{pmatrix}
$$
which is orthogonal up to scaling .
My attempt: obviously, $n$ needs to be even, so I've tried a companion matrix of $t^2+2t+2$
$$
\begin{pmatrix}
0 & -2 \\
1 & -2 \\
\end{pmatrix}
$$
(it's eigenvalues are $1\pm i$) which turned out not to be normal.
Best Answer
Just add a positive multiple of the identity to a rotation matrix, to make the eigenvalues have positive real part. $$ \begin{pmatrix}3&-1\\1&3\end{pmatrix}. $$