$3\times 3$ nonnegative matrices with purely imaginary eigenvalues

Question

This question arised from the more generic "does there exist entrywise nonnegative matrices with some non-zero purely immaginary eigenvalues?"

The answer is no for $2\times 2$ matrices (one can actually write all such kind of matrices and check it, or simply from Perron-Frobenius theorem).

Though, the answer is yes for $4\times 4$ matrices, since the companion matrix of $(x^2+1)(x^2-x-1) = x^4-x^3-x-1$ is nonnegative.

Now I am asking if there's a smaller example with $3\times 3$ matrices. What I know is that

• the real eigenvalue must be nonnegative from the trace
• the determinant must be nonnegative from the previous property
• the real eigenvalue is bigger or equal in magnitude than the rest from Perron-Frobenius
• no companion matrix with such properties is nonnegative

Answer 1

Yes. Let $H$ be a Housholder reflection matrix whose first row/column is positive, such as $$ H=\pmatrix{\frac{1}{\sqrt{3}}&\frac{1}{\sqrt{3}}&\frac{1}{\sqrt{3}}\\ \frac{1}{\sqrt{3}}&\frac{2-\sqrt{3}}{3-\sqrt{3}}&\frac{-1}{3-\sqrt{3}}\\ \frac{1}{\sqrt{3}}&\frac{-1}{3-\sqrt{3}}&\frac{2-\sqrt{3}}{3-\sqrt{3}}}. $$ Since $H=H^{-1}$, when $t\ne0$ is real, the matrix $$ A=H\pmatrix{1&0&0\\ 0&0&-t\\ 0&t&0}H $$ has a conjugate pair of purely imaginary eigenvalues $\pm it$. Since $A$ is entrywise positive when $t=0$, it is also entrywise positive when $t\ne0$ is sufficiently small.