  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 {\em 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.
