  Let $A$ be a $d\times d$ complex self-adjoint matrix,
$\mathcal{X},\mathcal{Y}\subset \mathbb{C}^d$ be $k$-dimensional subspaces and
let $X$ be a $d\times k$ complex matrix whose columns form an orthonormal basis
of $\mathcal{X}$. We construct a $d\times k$ complex matrix $Y_r$ whose columns
form an orthonormal basis of $\mathcal{Y}$ and obtain sharp upper bounds for
the singular values $s(X^*AX-Y_r^*\,A\,Y_r)$ in terms of submajorization
relations involving the principal angles between $\mathcal{X}$ and
$\mathcal{Y}$ and the spectral spread of $A$. We apply these results to obtain
sharp upper bounds for the absolute variation of the Ritz values of $A$
associated with the subspaces $\mathcal{X}$ and $\mathcal{Y}$, that partially
confirm conjectures by Knyazev and Argentati.
