About two trigonometric matrices (Q1826814)

The author studies the four matrices \[ M_{r,s} : =\sqrt{2}\left[ \sin \left(\pi {rmn}/{s}\right)\right] _{0<m,n<s},M_{2r,s}:=\sqrt{2}\left[ \sin \left( {2}\pi {rmn}/{s} \right) \right] _{0<m,n<s}, \] \[ M_{r,s}^{\prime } : =\sqrt{2}\left[ \sin \left( \pi{rmn}/{s} \right) \right] _{\substack{ 0<m,n<s\\ ( mn,s) =1}},M_{2r,s}^{\prime }:=\sqrt{2}\left[ \sin \left( {2}\pi{rmn}/{s} \right) \right] _{\substack{ 0<m,n<s\\ ( mn,s) =1}}, \] where \(r,s\) are odd integers with \(( r,s) =1\) and \(s>1\). More specifically the paper determines the eigenvalues of the above matrices and their multiplicities, as well as their characteristic polynomials.