  The K-matrix, also known as the "Wigner reaction matrix" in nuclear
scattering or "impedance matrix" in the electromagnetic wave scattering, is
given essentially by an M x M diagonal block of the resolvent (E-H)^{-1} of a
Hamiltonian H. For chaotic quantum systems the Hamiltonian H can be modelled by
random Hermitian N x N matrices taken from invariant ensembles with the Dyson
symmetry index beta=1,2,4. For beta=2 we prove by explicit calculation a
universality conjecture by P. Brouwer which is equivalent to the claim that the
probability distribution of K, for a broad class of invariant ensembles of
random Hermitian matrices H, converges to a matrix Cauchy distribution with
density ${\cal P}(K)\propto
\left[\det{({\lambda}^2+(K-{\epsilon})^2)}\right]^{-M}$ in the limit $N\to
\infty$, provided the parameter M is fixed and the spectral parameter E is
taken within the support of the eigenvalue distribution of H. In particular, we
show that for a broad class of unitary invariant ensembles of random matrices
finite diagonal blocks of the resolvent are Cauchy distributed. The cases
beta=1 and beta=4 remain outstanding.
