Condition: Non-Symmetric Matrix with real eigenvalues

Question

Is there a necessary condition for a non-symmetric real matrix $A\in\mathbb{R}^{n\times n}$ to have all its eigenvalues real and positive?

Answer 1

For a matrix $A$, all of its eigenvalus are real iff the $n\times n$ matrix $(\textrm{Tr} (A^{i+j})\,)_{ i,j=0}^{n-1}$ is positive semi-definite.

For complex numbers $\alpha_1$, $\ldots$, $\alpha_n$, we have: all of the numbers $\alpha_k$ are real if and only if the symmetric matrix $(\sum \alpha_k^{i+j})_{i,j=0}^{n-1}$ is real and positive semi-definite, check for eg An Introduction to Algebraic Geometry by Michel Coste.

Now, if you already know that all of the eigenvalue of $A$ are real, they will be $\ge 0$ if and only the characteristic polynomial of $A$ has coefficients of alternating signs.