Rank preserver of Boolean matrices. (English) Zbl 1082.15003

Song, Seok-Zun; Kang, Kyung-Tae; Jun, Young-Bae

Bull. Korean Math. Soc. 42, No. 3, 501-507 (2005).

An \(m\)-by-\(n\) Boolean matrix \(A\) has rank one if there exist a nonzero \(m\)-by-1 matrix \(B\) and a nonzero 1-by-\(n\) matrix \(C\) such that \(A=BC\). It can be shown that such Boolean matrices \(B,C\) are unique, and one can then define a perimeter of rank-one \(A\) to be the total number of nonzero entries in \(B\) and \(C\).
Suppose \(n\geq 4\), \(m\geq 2\), and suppose \(k\not\in\{2,3,n+1\}\) is a fixed perimeter. The authors give a complete classification of linear operators that simultaneously preserve perimeters 2,3, and \(k\) of rank-one Boolean matrices.
It is also shown that the result is no longer valid when \(k=n+1\).
Reviewer’s Remark: It seems that there is a misprint in (e) of Theorem 3.6 (the printed formulation contradicts with Example 3.5). It should read: “\(T\) preserves the rank and the perimeters 2 and 3, and a fixed perimeter \(k\not\in\{2,3,n+1\}\) of all Boolean rank-one matrices.”

Reviewer: Bojan Kuzma (Ljubljana)

MSC:

15A04	Linear transformations, semilinear transformations
15A23	Factorization of matrices
15A03	Vector spaces, linear dependence, rank, lineability
15B36	Matrices of integers

Keywords:

perimeter; linear operator; \((U,V)\)-operator; Boolean matrix