Singular value decomposition of a matrix-valued impulse response

The singular value decomposition (SVD), a standard tool for theoretical and computational matrix analysis, represents a matrix as an ordered product of a unitary matrix, a nonnegative, real diagonal matrix, and a second unitary matrix. Our contribution is to extend the SVD representation to a matrix-valued impulse response. The representation involves a countably infinite number of vector-valued eigenfunctions and scalar singular values, and it provides the most concise and elegant description of the action of a matrix-valued filter (for example, a space-time communication channel) when driven by a finite-duration input signal.