Linear Rank Nonincreasing and Preserving Maps between Matrix Spaces
Abstract
We classify linear maps ϕ between possibly different rectangular matrix spaces over a field 𝔽 with |𝔽|≥5 such that rank ϕ(A)≤2 whenever rank A≤2. Every such map has one of six forms. If its image contains a matrix of rank greater than two, then ϕ(A)=MAN or ϕ(A)=MA⊤N for fixed matrices M,N. The remaining forms have images consisting entirely of matrices of rank at most two, including an exceptional form arising from 3×3 alternating matrices. As a consequence, we characterize linear maps preserving rank exactly two. We also prove that rank-k preservation implies rank-k nonincrease when |𝔽|>k+2, and that rank-k nonincrease implies rank-ℓ nonincrease for every admissible ℓ≥ k when |𝔽|>k+1. The classification uses the fact that the Chow rank of the generic 3×3 determinant is five, connecting the rank preserver problem with product decompositions of polynomials.
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BibTeX
@misc{huang2026linear,
title = {{Linear Rank Nonincreasing and Preserving Maps between Matrix Spaces}},
author = {Huajun Huang and Ming-Cheng Tsai},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.40995},
eprint = {2610.40995},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.40995},
note = {CC BY-NC-SA 4.0},
}
RIS
TY - UNPB
TI - Linear Rank Nonincreasing and Preserving Maps between Matrix Spaces
AU - Huajun Huang
AU - Ming-Cheng Tsai
PY - 2026
DA - 2026/10/29
PB - arXiv
DO - 10.48550/arxiv.2610.40995
UR - https://arxiv.org/abs/2610.40995
AB - We classify linear maps ϕ between possibly different rectangular matrix spaces over a field 𝔽 with |𝔽|≥5 such that rank ϕ(A)≤2 whenever rank A≤2. Every such map has one of six forms. If its image contains a matrix of rank greater than two, then ϕ(A)=MAN or ϕ(A)=MA^⊤N for fixed matrices M,N. The remaining forms have images consisting entirely of matrices of rank at most two, including an exceptional form arising from 3×3 alternating matrices. As a consequence, we characterize linear maps preserving rank exactly two. We also prove that rank-k preservation implies rank-k nonincrease when |𝔽|>k+2, and that rank-k nonincrease implies rank-ℓ nonincrease for every admissible ℓ≥ k when |𝔽|>k+1. The classification uses the fact that the Chow rank of the generic 3×3 determinant is five, connecting the rank preserver problem with product decompositions of polynomials.
ER -