Matrices are built and designed by applying procedures from lower order matrices. Matrix tensor products, direct sums or multiplication of matrices are such 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 is often referred to as {\em an entangled matrix}. The matrices built may retain properties of the lower order matrices or may also acquire new desired properties 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 desirable properties. Infinite series of required non-separable matrices are constructible by the general methods. Non-separable matrices are required for applications and other uses; they can capture the structure in a unique way and thus perform much better than separable matrices. General new methods are developed with which to construct {\em multidimensional entangled paraunitary matrices}; these have applications for wavelet and filter bank design. The constructions are in addition used to design new systems of non-separable unitary matrices; these have applications in quantum information theory. Some consequences include the design of full diversity constellations of unitary matrices, which are used in MIMO systems, and methods to design infinite series of special types of Hadamard matrices.

翻译：矩阵可通过低阶矩阵的运算过程构建与设计。矩阵的张量积、直和或矩阵乘法即为此类过程，由此构建的矩阵称为可分离矩阵。非分离矩阵则指不可分离的矩阵，常被称为纠缠矩阵。所构建的矩阵可能保留低阶矩阵的性质，也可能获得低阶矩阵本身不具备的新期望性质。本文推导了所需类型非分离矩阵的设计方法，这些方法既能保留低阶矩阵的性质，也能赋予新期望特性。通过通用方法可构造所需非分离矩阵的无穷序列。非分离矩阵在应用与其他用途中具有必要性：它们能以独特方式捕捉结构特征，从而表现远优于可分离矩阵。我们发展了构建多维纠缠酋矩阵的全新通用方法，此类矩阵在小波与滤波器组设计中具有应用价值。此外，这些构造方法还可用于设计新型非分离么正矩阵系统，这在量子信息理论中有重要应用。相关成果包括设计用于MIMO系统的满分级酉矩阵星座，以及构建特殊类型阿达马矩阵无穷序列的方法。