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What properties do a real matrices need to satisfy so that all its eigenvalues are real?

I know very well that real symmetric matrices have this property. But they do not form an exhaustive set. I have also checked this question (What properties should a matrix have if all its eigenvalues are real?) but the property is not practically checkable.

I need a characterization for real matrices having real eigenvalues.

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  • $\begingroup$ There are things such as "is similar to an upper (or lower) triangular matrix", but I doubt any of them are easier to test than "all eigenvalues are real". $\endgroup$ – Henning Makholm Nov 4 '17 at 4:12
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I know that this is an old post but hopefully somebody is still interested. There is an old result of Emilie Haynsworth and Michael Drazin that a necessary and sufficient condition for a real matrix $A$ to have all real eigenvalues with $n$ linearly independent eigenvectors is that there exists a positive semidefinite matrix $S$ such that $AS$ is a symmetric matrix.

In the case where $S$ is positive definite then $A$ is self-adjoint under a modified inner product: $\langle v,w\rangle=v^T S w$

The Drazin-Haynsworth paper is below. Haynsworth wrote a number of nice papers on questions like this in the 60's.

https://link.springer.com/article/10.1007%2FBF01195188

I don't know if you want the matrices to have no eigenvector deficiency. If one drops the requirement that the matrix have $n$ linearly independent eigenvectors then I don't know but I doubt that there is a very nice condition. I hope that you find this helpful.

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