Is there any quick way to conclude on the eigenvalues of $T = U S$ ? Can we express them with respect to the eigenvalues of $S$, $\lambda( S)$.
Where:
$$S = S^T $$ ($S$ is symmetric) $$ U = \begin{bmatrix} 1 & 0 & \ldots & & & & \ldots & 0 \\ 0 & 1 & 0 & 0 & 0 & 0 & \ldots & 0 \\ 0 & & \ddots & 0 & 0 & 0 & \ldots & 0 \\ 0 & & \ldots & 1 & 0 & 0 & \ldots & 0\\ 0 & & \ldots & 0 & -1 & 0 & \ldots & 0 \\ 0 & & \ldots & 0 & 0 & -1 & \ldots & 0 \\ 0 & & \ldots & 0 & 0 & 0 & \ddots & 0 \\ 0 & & \ldots & & & & \ldots & -1 \\ \end{bmatrix} $$ ($U$ is unitary and diagonal.Half top rows are negative the rows of an identity matrix, half bottom are exactly the rows of an identity matrix)