We recently derived a matrix fluctuation-dissipation (MFD) theorem, which directly relates the spectral intensities to the eigenvalue fluctuations of a quantum system. Here additional properties of the MFD theorem are presented. MFD is a microcanonical version of the fluctuation-dissipation theorem for a single high-energy state embedded in a dissipative quantum-mechanical bath. This is useful in applications to vibrational relaxation, which can be exactly described by the MFD formula if a single initial state carries all the oscillator strength. MFD versions of formulas for the dilution factor (sigma) and fraction of occupied phase space (F) are derived, which can be used to compute these quantities exactly without knowledge of the eigenfunctions. Computational applications are given in a companion paper.