Linear rank preservers of tensor products of rank one matrices

TL;DR

This paper characterizes linear maps on tensor product matrices that preserve rank-one tensors, extending recent results on rank-preserving maps for special matrix classes, with implications for understanding tensor structure.

Contribution

It provides a complete characterization of linear rank-preserving maps on tensor products of rank-one matrices, extending existing results to more general tensor settings.

Findings

Characterization of linear maps preserving rank-one tensor products

Extension of recent rank-preserving map results to tensor products

Implications for tensor and matrix rank structure analysis

Abstract

Let $n_1,\ldots,n_k $ be integers larger than or equal to 2. We characterize linear maps $\phi: M_{n_1\cdots n_k}\rightarrow M_{n_1\cdots n_k}$ such that $${\mathrm rank}\,(\phi(A_1\otimes \cdots \otimes A_k))=1\quad\hbox{whenever}\quad{\mathrm rank}\, (A_1\otimes \cdots \otimes A_k)=1 \quad \hbox{for all}\quad A_i \in M_{n_i},\, i = 1,\dots,k.$$ Applying this result, we extend two recent results on linear maps that preserving the rank of special classes of matrices.