Let S $\displaystyle \in$ M_{nxn}(R) and D $\displaystyle \in$ M_{nxn}(R) be n $\displaystyle \times$ n matrices such that D^{T}SD = S

where S is non degenerate. Let $\displaystyle \mu \neq 0$ be an eigenvalue of $\displaystyle D$ with eigenvector

x.

(i) Show that Sxis an eigenvector of D^{T}and find its eigenvalue.

(ii) Show that $\displaystyle \frac{1}{\mu}$ is an eigenvalue of D.

[Hint: det(A) = det(A^{T}) for any square matrix A.]

This question has been giving me some real trouble and all help is welcome