On the similarity of Tensors

Abstract

Let $\mathbb{P}_n$ be the set of all matrices which have the same zero patterns with some permutation matrix of order $n$. In this paper, we prove the following result: Let $\mathbb{I}$ be the unit tensor of order $m\ge3$ and dimension $n\ge2$. Suppose that $P$ and $Q$ are two matrices with $P\mathbb{I}Q=\mathbb{I}$, then $P,Q\in \mathbb{P}_n$. This gives a characterization for the similarities of tensors with order $m\ge3$.