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Matrices are built and designed by applying procedures from lower ordermatrices. Matrix tensor products, direct sums or multiplication of matrices aresuch procedures and a matrix built from these is said to be a {\em separable}matrix. A {\em non-separable} matrix is a matrix which is not separable and isoften referred to as {\em an entangled matrix}. The matrices built may retainproperties of the lower order matrices or may also acquire new desiredproperties not inherent in the constituents.  Here design methods for non-separable matrices of required types are derived.These can retain properties of lower order matrices or have new desirableproperties. Infinite series of required non-separable matrices areconstructible by the general methods.  Non-separable matrices are required for applications and other uses; they cancapture the structure in a unique way and thus perform much better thanseparable matrices. General new methods are developed with which to construct{\em multidimensional entangled paraunitary matrices}; these have applicationsfor wavelet and filter bank design. The constructions are in addition used todesign new systems of non-separable unitary matrices; these have applicationsin quantum information theory. Some consequences include the design of fulldiversity constellations of unitary matrices, which are used in MIMO systems,and methods to design infinite series of special types of Hadamard matrices.

Non-separable matrix builders for signal processing, quantum information and MIMO applications