Ben Kisley, Bryan Shader, Matrices whose permanent rank equals half their rank, Vol. 2025 (2025), Article ID 48, pp. 1-14

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DOI: 10.23952/cot.2025.49

Received July 24, 2024; Accepted March 28, 2025; Published online December 22, 2025

Abstract. The {permanent rank} of an $m\times n$ matrix $A$ over a field $\mathbb{F}$ generalizes the notion of the rank of $A$ and is the largest $k$ such that $A$ has a $k\times k$ submatrix whose permanent is nonzero. In 1999, Yu proved that the permanent rank of a matrix is always at least half the rank. This paper gives an explicit characterization the matrices for which equality holds; and demonstrates that, for characteristic different than 2, fixed $m$ , $n$ and even $r$ with $r\leq \min \{m,n\}$ there is essentially a unique $m \times n$ matrix over $\mathbb{F}$ with rank $r$ and permanent rank $r/2$ .

How to Cite this Article:
B. Kisley, B. Shader, Matrices whose permanent rank equals half their rank, Commun. Optim. Theory 2025 (2025) 49.