A low-rank approximation of tensors and the topological group structure of invertible matrices

Abstract

By a tensor we mean an element of a tensor product of vector spaces over a field. Up to a choice of bases in factors of tensor products, every tensor may be coordinatized, that is, represented as an array consisting of numbers. This note is concerned with properties of the~tensor rank that is a natural generalization of the matrix rank. The topological group structure of invertible matrices is involved in this study. The multilinear matrix multiplication is discussed from a viewpoint of transformation groups. We treat a low-rank tensor approximation in finite-dimensional tensor products. It is shown that the problem on determining a best rank-$n$ approximation for a tensor of size $n\times n \times 2$ has no a solution.To this end, we make use of an~approximation by matrices with simple spectra.