  We study finitely cyclic self-adjoint operators in a Hilbert space, i.e.
self-adjoint operators that posses such a finite subset in the domain that the
orbits of all its elements with respect to the operator are linearly dense in
the space. One of the main goals here is to obtain the representation theorem
for such operators in a form analogous to the one well-known in the cyclic
self-adjoint operators case. To do this, we present here a detailed
introduction to matrix measures, to the matrix measure $L^2$ spaces, and to the
multiplication by scalar functions operators in such spaces. This allows us to
formulate and prove in all the details the less known representation result,
saying that the finitely cyclic self-adjoint operator is unitary equivalent to
the multiplication by the identity function on $\mathbb{R}$ in the appropriate
matrix measure $L^2$ space. We study also some detailed spectral problems for
finitely cyclic self-adjoint operators, like the absolute continuity.
