This paper shows that a quadratic form in a multivariate sample has a certain rank and its nonzero eigenvalues are distinct with probability one under the assumption that the matrix defining the quadratic form satisfies a certain rank condition and that the underlying distribution of the sample is absolutely continuous with respect to Lebesgue measure.
"Distinctness of the Eigenvalues of a Quadratic form in a Multivariate Sample." Ann. Statist. 1 (4) 763 - 765, July, 1973. https://doi.org/10.1214/aos/1176342472