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It often happens that a signal processing application will involve a matrix operator that is invariant under specific preunitary and postunitary matrix multiplications. This invariance feature may be exploited to establish algebraic invariance properties possessed by the singular vectors associated with that matrice's SVD representation. This characterization can be of considerable theoretical as well as computational value. To illustrate the utility of this approach, the new class of bisymmetric matrices will be examined in detail. Interest in bisymmetric matrices arises from their frequent appearance in signal processing applications. For instance, it is shown that the forward-backward data matrix arising in linear prediction is bisymmetric. The more general class of exponential bisymmetric matrices, which includes among its members the discrete Fourier transform, is also briefly examined.