The inequality says: Any matrices $X$ and $Y$ in $\mathcal{S}^n$ satisfy the inequality $$\left\| X - Y \right\| \geq \left\| \lambda(X) - \lambda(Y) \right\|,$$ where the equality holds if and only if $X$ and $Y$ have a simultaneous ordered spectral decomposition, i.e., there exists an orthogonal matrix $U$ such that $$Y = U \Lambda(Y) U^T, \quad X = U \Lambda(X)U^T.$$ I am confused about the only if part. How to prove this theorem (only if part)?
Norm is the frobenius norm and $\Lambda(X) = \mathrm{Diag} (\lambda(X))$ is the eigenvalues.