Suppose we have a matrix $A \in \mathbb{R}^{n \times n}$ with $\textrm{eig}(A)=\{ \lambda_1, \lambda_2, \ldots, \lambda_n\}$ such that $\lambda_i \in \mathbb{R}$.
Does the realness of the eigenvalues imply some structure in $A$?
I know of that $A=A^{\sf T}$ (real symmetric) implies all the eigenvalues will be real, but I'm wondering about implications in the other direction.
Is there some additional condition + realness of eigenvalues which would imply the matrix is symmetric?
Context: I'm trying to classify special types of matrices according to the properties of their eigenvalues. e.g. $\lambda_i \geq 0$, then $A$ is positive semidefinite, etc. If anyone can point me to a list of such correspondences between eigenvalue constraints and matrix properties it would be much appreciated.
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