Rank of 2-Slice and 3-Slice Tensors

Abstract

It is easy to compute the rank of a matrix: one transforms it to echelon form for example via Gaussian elimination; the rank can then be read off this form. In the first part of this chapter we show that similar results also hold for a pair of matrices. Following the Weierstraß-Kronecker theory, we define certain invariants for pairs of matrices and give a formula, due to Grigoriev [207, 208] and Ja’Ja’ [265, 266] for the rank of such a pair in terms of these invariants. For triples of matrices, resp. 3-slice tensors, no such formula is known. In the second part of this chapter we present some results due to Strassen [505]. We prove a lower bound for the border rank of 3-slice tensors. Moreover, we show that for the format (n, n, 3), n ≥ 3 odd, the complement of the set of tensors of maximal border rank is a hypersurface and we explicitly determine its equation. For this we will also rely on a result which will be obtained in Chap. 20.