A ug 2 00 8 Rank of 3 - tensors with 2 slices and Kronecker canonical form

Tensor type data are becoming important recently in various application fields (for example see Miwakeichi et al. [MI], Vasilescu and Terzopoulos [VT] and Muti and Bourennane [MB]). The factorization of a tensor to a sum of rank 1 tensors means that the data is expressed by a sum of data with most simpler structure, and we may have better understanding of data. This is an essential attitude for data analysis and therefore the problem of tensor factorization is an essential one for applications. In this paper we consider the rank problem of 3-tensors with 2 slices. This was studied in the 1970’s and 1980’s by many authors. Ja’ Ja’ [JA1, JA2, JA3] gave on a general upper bound for the maximal rank for them by using Kronecker canonical forms of the pencil of two matrices, and it is known that it is also a lower bound by using a result by Brockett and Dobkin [BD1, BD2]. Ja’ Ja’ showed that the rank of a Kronecker’s canonical form without regular pencils is equal to the sum of the ranks of direct summand. However, the rank of a Kronecker’s canonical form is not equal to the sum of the ranks of direct summand in general (see Remark 4.8). This causes to be difficult to determine the rank of tensors. Our aim is to determine completely determine the rank of tensors of type m × n × 2, which yields we also obtain the typical rank. In this paper we explicitly determine the rank of 3-tensors with 2 small slices over the complex and real number field.