  Let $\bbK=\mathbb R, \mathbb C, \mathbb H$ be the field of real, complex or
quaternionic numbers and $M_{p, q}(\bbK)$ the vector space of all $p\times
q$-matrices. Let $X$ be the matrix unit ball in $M_{n-r, r}(\bbK)$ consisting
of contractive matrices. As a symmetric space, $X=G/K=O(n-r, r)/O(n-r)\times
O(r)$, $U(n-r, r)/U(n-r)\times U(r)$ and respectively $Sp(n-r, r)/Sp(n-r)\times
Sp(r)$. The matrix unit ball $y_0$ in $M_{r^\prime-r, r}$ with $r^\prime \le
n-1$ is a totally geodesic submanifold of $X$ and let $Y$ be the set of all
$G$-translations of the submanifold $y_0$. The set $Y$ is then a manifold and
an affine symmetric space. We consider the Radon transform $\mathcal Rf(y)$ for
functions $f\in C_0^\infty(X)$ defined by integration of $f$ over the subset
$y$, and the dual transform $\mathcal R^t F(x), x\in X$ for functions $F(y)$ on
$Y$. We find inversion formulas by constructing explicit certain invariant
differential operators.
