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

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