Maximal Ranks

Abstract

In this chapter, we consider the maximal rank of tensors with format (m, n, 2) or (m, n, 3). The rank of a tensor with format (m, n, 2) is computable because of the list of representatives of $\mathrm {GL}(m,\mathbb {F})\times \mathrm {GL}(n,\mathbb {F})$-equivalent classes of tensors with format (m, n, 2), which is known as the Kronecker–Weierstrass canonical form. We consider the rank of tensors of the Kronecker–Weierstrass canonical form. The rank of tensors with format $(m,n,mn-1)$ or $(m,n,mn-2)$ is related to ‘dual’ tensors with format (m, n, 1) or (m, n, 2), respectively. It is difficult to compute the rank of tensors with format (m, n, 3), even the maximal rank is not obtained fully. A 3-tensor can be considered as a collection of matrices; then, we obtain results from the viewpoint of the theory of matrices. We introduce known upper bounds and lower bounds of the ranks of tensors with format (m, n, p), where $p\ge 3$.