A simple estimation of the maximal rank of tensors with two slices by row and column operations, symmetrization and induction

The determination of the maximal ranks of a set of a given type of tensors is a basic problem both in theory and application. In statistical applications, the maximal rank is related to the number of necessary parameters to be built in a tensor model. JaJa [JA] and Sumi et. al [SMS1] developed an optimal bound theory based on Kronecker canonical form of the pencil of two matrices. Theory of matrix pencil is explained in several text book, for example, of Gantmacher [G]. Atkinson and Lloyd[AL], Atkinson and Stephens[AS] and Sumi et. al [SMS2] treated the maximal rank of tensors with 3 slices of matrices. In contrast we use an old theorem, which states that any real matrix can be expressed as a product of two real symmetric matrices. Based on this classical theorem (Bosch [B]) we will show the tight bound by simple row and column operations and symmetrization and mathematical induction. As far as the authors know, the inductive proof of the tight bound [3n/2] for 2×n×n tensors, which has been given by several authors based on eigenvalue theories, is the first result in this filed. It should be note that the inductive proof is shown to have a great difficulty for odd n. We overcame this in this paper. In Section 2 we list up several proofs for some particular cases, which are very interesting in themselves and became stepstones of our general proof. In Section 3 we will give a proof by symmetrization and an inductive proof for the maximal rank of 2× n× n. Finally, in Section 4, we will generalize the proof for the case of 2×m× n tensors.