For a real square matrix $S$ with all eigenvalues on the imaginary axis, what can be said about $Tr(S^2)$? Since all eigenvalues of $S$ are on imaginary axis, $Tr(S)=0$ but I was wondering if anything more I can find about $Tr(S^2)$.
I know the following property is associated with the trace of the product of two positive semi-definite matrices $X\geq{0},Y\geq{0}$,
$Tr(X^{T}Y)\leq{Tr(X)}\lambda_{max}({Y})$ where $\lambda_{max}(Y)$ denotes the maximum eigenvalue of $Y$. Since $S$ is not specified whether positive semi-definite or not, I cannot use the property. Is there any other property that I can use to say anything about $Tr(S^2)$?