Proving that largest root (obtained via P.C.A.) is a symmetric function

Question

Suppose, we are given $\textbf{X} = (X_1, X_2, \ldots,X_m)$ and $\textbf{Y} = (Y_1, Y_2, \ldots, Y_n)$.

Also, we are given, S = pooled variance.

If we implement Principal Component Analysis (P.C.A.) using S, then how do I proceed in proving that $\lambda_1$ (= largest root obtained via P.C.A.) is a symmetric function of $\textbf{X}$ and $\textbf{Y}$ ?