Matrix constructs
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Matrices can be built and designed by applying procedures from lower order matrices. Matrix tensor products, direct sums of matrices and multiplication of matrices retain certain properties of the lower order matrices; matrices produced by these procedures are said to be separable. Here builders of matrices which retain properties of the lower order matrices or acquire new properties are described; the constructions are not separable. The builders may be combined to construct series of matrix types. A number of applications are given. The new systems allow the design of multidimensional non-separable systems. Methods with which to design multidimensional paraunitary matrices are derived; these have applications for wavelet and filter bank design. New entangled unitary matrices may be designed; these may be used in quantum information theory. Full diversity constellations of unitary matrices for space time applications are efficiently designed by the constructions.
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