  Let $Sp(n)$ be the symplectic group of quaternionic $(n\times n)$-matrices.
For any $1\leq k\leq n$, an element $A$ of $Sp(n)$ can be decomposed in $A=
\begin{bmatrix} \alpha&T\cr \beta&P \end{bmatrix}$ with $P$ a $(k\times
k)$-matrix. In this work, starting from a singular value decomposition of $P$,
we obtain what we call a relative singular value decomposition of $A$. This
feature is well adapted for the study of the quaternionic Stiefel manifold
$X_{n,k}$, and we apply it to the determination of the Lusternik-Schnirelmann
category of $Sp(k)$ in $X_{2k-j,k}$, for $j= 0,\,1,\,2$
