Consider a real but not symmetric matrix $A$. To test if the matrix has positive eigenvalues, I've learnt from this forum that a symmetric matrix will be given by $B=A+A^T$. If all the eigenvalues of $B$ are positive, then it follows that A also has all the eigenvalues positive. So this is a sufficient condition.
For example, consider
It so happens that $B$ has one negative eigenvalue. Whereas $A$ has both positive eigenvalues. So what is a necessary (and sufficient) condition that $A$ has all positive eigenvalues?
Looking at Gershgorin's theorem , further rises the possibility of complex eigenvalues.
- Tests for positive definiteness for nonsymmetric matrices
- p.322,Linear Algebra and its Applications, Gilbert Strang.
- Necessary and sufficient condition for all the eigenvalues of a real matrix to be non-negative