Can anyone please tell me what can I know about a square matrix that has only non-positive eigenvalues.
In a text book I read it says that lets suppose matrix A has only non-positive eigenvalues then
$$ A \preccurlyeq \mathbf{0} $$
I thought it meant it each of its element is less than 0 but that is not true.
The notation you used to compare $A$ to $0$ is often used for symmetric matrices, and it means the matrix is negative semi-definite, which means $x^TAx \leq 0$ for all vectors $x$. This does not mean all entries are non-positive but it does mean that the diagonal is all non-positive.