Article
Full entry |
PDF
(0.1 MB)
Feedback
PDF
(0.1 MB)
Feedback
Keywords:
linear operator; rank; perimeter; $(P, Q, B)$-operator
linear operator; rank; perimeter; $(P, Q, B)$-operator
Summary:
We investigate the perimeter of nonnegative integer matrices. We also characterize the linear operators which preserve the rank and perimeter of nonnegative integer matrices. That is, a linear operator $T$ preserves the rank and perimeter of rank-$1$ matrices if and only if it has the form $T(A)=P(A\circ B)Q$, or $T(A)=P(A^t \circ B)Q $ with appropriate permutation matrices $P$ and $Q$ and positive integer matrix $B$, where $\circ$ denotes Hadamard product.
References:
[1] Beasley L.B., Pullman N.J.: Boolean rank-preserving operators and Boolean rank-$1$ spaces. Linear Algebra Appl. 59 (1984), 55-77. MR 0743045 | Zbl 0536.20044
[2] Beasley L.B., Gregory D.A., Pullman N.J.: Nonnegative rank-preserving operators. Linear Algebra Appl. 65 (1985), 207-223. MR 0774353 | Zbl 0561.15002
[3] Beasley L.B., Pullman N.J.: Term-rank, permanent and rook-polynomial preservers. Linear Algebra Appl. 90 (1987), 33-46. MR 0884107 | Zbl 0617.15001
[4] Beasley L.B., Song S.Z., Lee S.G.: Zero term rank preservers. Linear and Multilinear Algebra 48 (2001), 313-318. MR 1928400 | Zbl 0987.15002
[5] Song S.Z.: Linear operators that preserve maximal column ranks of nonnegative integer matrices. Proc. Amer. Math. Soc. 126 (1998), 2205-2211. MR 1443409 | Zbl 0896.15009
Search
Partner of
