I've always believed that the Spearman rank correlation matrix doesn't need to be positive semidefinite because the correlations are estimated pairwise so there's always a chance that it may not be. Recently, however, I have been looking deeper into this issue and I am confused, particularly after reading this:
On page #4, Theorem 2 it says:
A Spearman rank correlation matrix is PSD
Proof: A Spearman correlation is a Pearson correlation applied to ranks. Therefore the Spearman rank correlation matrix is PSD
So... can the Spearman correlation matrix be not positive semidefinite? Is there a counter-example people know? Or is it a misunderstanding to think that the Spearman correlation matrix could be non-positive definite?