Conditions for a matrix to have non-repeated eigenvalues

Question

I am wondering if anybody knows any reference/idea that can be used to adress the following seemingly simple question

"Is there any set of conditions so that all the eigenvalues of a real positive definite matrix are different?"

Motivation Duality in principal component analysis

Answer 1

One sufficient condition is that the Gersghorin discs associated with a given matrix do not intersect. This is easy to check in practical applications, since one can just compute the discs from the entries of the matrix itself.

Answer 2

More generally, let $A\in M_n(K)$ where $K$ is a field with $char(K)=0$ (we can generalize this condition) . Then

$A$ has no multiple eigenvalues in $\bar{K}$, the algebraic closure of $K$, IFF

$discrim(det(A-xI),x)\not=0$ where $discrim(P,x)$ is the discriminant of the polynomial $P$ with respect to $x$.

cf. https://en.wikipedia.org/wiki/Discriminant

Example. Let $a\in \mathbb{R}$. The real matrix $A=\begin{pmatrix}-93& -32& 8& 44\\-76& -74& 69& 92\\-72& a& 99& -31\\-2& 27& 29& 67\end{pmatrix}$ has $4$ complex distinct eigenvalues IFF

$discrim(\det(A-xI),x)=104185835672256a^5+45369178665625008a^4+20868533919078853824a^3+4464564616463758516336a^2+553796534057255432000352a-56405327830695680670639360\not= 0$.

In particular, if $a\approx 61.403389758845937922$, then $A$ admits the double eigenvalue $\approx 99.417913129590713400$.