  We consider the properties of an observable (such as a single spin component
that squares to the identity) when expressed as a matrix in the basis of energy
eigenstates, and then truncated to a microcanonical slice of energies of
varying width. For a quantum chaotic system, we model the unitary or orthogonal
matrix that relates the spin basis to the energy basis as a random matrix
selected from the appropriate Haar measure. We find that the spectrum of
eigenvalues is given by a centered Jacobi distribution that approaches the
Wigner semicircle of a random hermitian matrix for small slices. For slices
that contain more than half the states, there is a set of eigenvalues of
exactly $\pm 1$. The transition to this qualitatively different behavior at
half size is similar to that seen in other quantities such as entanglement
entropy. Our results serve as a benchmark model for numerical calculations in
realistic physical systems.
