The Ranks of $m \times n \times (mn - 2)$ Tensors

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Abstract

It is shown that there is essentially only one $m \times n \times (mn - 2)$ tensor of rank $mn - 1$. It is also proved that, except for this tensor, all $m \times n \times p$ tensors with $p \leqq mn - 2$ have rank at most $mn - 2$. The main tool is Kronecker’s theory of matrix pencils which has already been applied directly by Ja’Ja’ [SIAM J. Comput., 8 (1979), pp. 443–462] to study the ranks of $m \times n \times 2$ tensors. We show that each nondegenerate $m \times n \times (mn - 2)$ tensor is determined by a related $m \times n \times 2$ tensor and apply the Kronecker theory to this related tensor.

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