  We calculate the eigenvalues of a class of random matrices, namely the
randomly segmented tridiagonal quasi-Toeplitz (rstq-T) matrix, in exact
closed-form. The contexts under which these matrices arise are ubiquitous in
physics. In our case, they arise when studying the dynamics of a Rouse polymer
embedded in random environments. Unlike in the case of Rouse polymers in
homogeneous environments, where the dynamics give rise to a circulant matrix
and the diagonalization is achieved easily via a Fourier transform, analytical
diagonalization of the rstq-T matrix has remained unsolved thus far. We
analytically calculate the spectral distribution of the rstq-T matrix, which is
able to capture the effect of disorder on the modes.
