The response of an appropriate system to a noise input is studied. The systems studied are ``generalized lossless networks'' and may include, for example, electron beams and parametrically pumped components. The noise may be introduced through several ports in an uncorrelated or correlated way. The problem is conveniently handled by the matrix analog of an eigenvector analysis. The noise input is represented as a matrix of which the diagonal elements are the self‐power density spectra, and the off‐diagonal elements the cross‐power density spectra. This matrix is resolved into ``eigenmatrices'' which are then recombined at the output in accordance with the ``2‐index eigenvalues.'' The noise invariants found by Haus (Sand &pgr;) are appropriate to a system with two degrees of freedom. For a system withndegrees of freedom, there are at leastninvariants of the noise—i.e., components of the noise that are unchanged by the system. The theory is finally applied to a generalized coupler, in which a single circuit mode is coupled to a degenerate multiplicity of beam modes, such as would occur in a hollow beam. It is shown that a ``Kompfner‐null'' condition exists. The physical significance of this condition is studied and its effect on beam noise determined.