  The monograph contains a systematic treatment of a circle of problems in
analysis and integral geometry related to inversion of the Radon transform on
the space of real rectangular matrices. This transform assigns to a function
$f$ on the matrix space the integrals of $f$ over the so-called matrix planes,
the linear manifolds determined by the corresponding matrix equations.
Different inversion methods are discussed. They rely on close connection
between the Radon transform, the Fourier transform, the Garding-Gindikin
fractional integrals, and matrix modifications of the Riesz potentials. A
special emphasis is made on new higher rank phenomena, in particular, on
possibly minimal conditions under which the Radon transform is well defined and
can be explicitly inverted. Apart of the space of Schwartz functions, we also
employ $L^p$-spaces and the space of continuous functions. Many classical
results for the Radon transform on $R^n$ are generalized to the higher rank
case.
