Necessary and sufficient condition for all the eigenvalues of a real matrix to be non-negative According to wikipedia, a totally positive matrix (https://en.wikipedia.org/wiki/Totally_positive_matrix) always have non-negative eigenvalues. But, is it necessary for a matrix to be totally positive to have non-negative eigenvalues? 
I'm looking for the necessary and sufficient condition for a real square matrix (not necessarily symmetric) to have non-negative eigenvalues.  

No it is not necessary. Just consider the matrix 
$$
A=
\begin{bmatrix}
2 & -1\\
-1 & 2
\end{bmatrix}
$$ 
which is not totally positive but it is still has non-negative eigenvalues $\{1,3\}.$
