Singular Value Decomposition of Spatial Matrices

Abstract

Singular value decomposition is a basic building block which is used in solution of many different problems. In cases when dimensionality of a problem exceeds two, a generalization of a singular value decomposition – tensor decompositions – are used. There is only one problem: the usage of tensor decompositions is not a good option in some cases. That is the reason why we propose to consider a generalization of a singular value decomposition on spatial matrices. In this article, the singular value decomposition of a spatial matrix is considered. Examples of tasks which can be successfully solved using multidimensional matrix algebra and examples of algorithms which have a natural generalization to multidimensional matrix algebra are given. Definition of the singular value decomposition is given and its properties are listed. Next, definitions of necessary concepts of multidimensional matrix algebra which were defined in Sokolov's research work are given; also, there are a couple of new definitions which are first formulated in this article. Thereafter, requirements for the sought decomposition are formulated based on desired properties of it. A method of creation the singular value decomposition of a spatial matrix is given; the method uses an idea of slicing a spatial matrix. This approach makes it possible to reduce the problem of finding the singular value decomposition of a spatial matrix to finding such decompositions for flat matrices. The property retention of such a decomposition is proved, and an example of this decomposition is given. Finally, ideas of applying the decomposition to problem solving are given and ideas of future work are proposed.

Keywords

• multidimensional matrix algebra
• matrix decomposition
• singular value decomposition