Trace of a matrix and sign of the eigenvalues

Question

$A \in \mathbb R^{4 \times 4}$ is matrix whose diagonal elements are zero. Can I say the following?

1. If $A$ is neither skew-symmetric nor the zero matrix, then $A$ has at least one eigenvalue with positive real part.

2. In my case I have all the diagonal elements equal to zero. What if in general $\mbox{tr}(A)=0$?

Answer 1

Routh–Hurwitz (H) stability criterion:

This can also be used in general to find if a matrix $A\in \mathbb{R}^{n\times n}$ has eigenvalues with a positive real part.

Use RH criterion on the characteristic polynomial of $A$, that is, $det(sI-A)=0$.

Answer 2

Consider this matrix.

$\begin{bmatrix}0&1\\&0&1\\&&0&1\\&&&0\end{bmatrix}$