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A Boolean rank one matrix can be factored as ${\bf u}{\bf v}^t$ for vectors ${\bf u}$ and ${\bf v}$ of appropriate orders. The perimeter of this Boolean rank one matrix is the number of nonzero entries in ${\bf u}$ plus the number of nonzero entries in ${\bf v}$. A Boolean matrix of Boolean rank $k$ is the sum of $k$ Boolean rank one matrices, a rank one decomposition. The perimeter of a Boolean matrix $A$ of Boolean rank $k$ is the minimum over all Boolean rank one decompositions of $A$ of the sums of perimeters of the Boolean rank one matrices. The arctic rank of a Boolean matrix is one half the perimeter. In this article we characterize the linear operators that preserve the symmetric arctic rank of symmetric Boolean matrices.
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참고문헌 (10)
참고문헌 신청
L. B. Beasley / 2016 / Preservers of term ranks and star cover numbers / Electronic J. Linear Algebra 31:549~564
L. B. Beasley / Clique covers of graphs and ranks of Boolean matrices / Journal of Combinatorial Mathematics and Combinatorial Computing
L. B. Beasley / 2015 / The arctic rank of a Boolean matrix / J. Algebra 433:168~182
L. B. Beasley / 1984 / Boolean-rank-preserving operators and Boolean-rank-1spaces / Linear Algebra Appl 59:55~77
L. B. Beasley / 1987 / Term-rank, permanent and rook-polynomial preservers / Linear Algebra Appl 90:33~46
L. B. Beasley / Clique covers of graphs and ranks of Boolean matrices / Journal of Combinatorial Mathematics and Combinatorial Computing
L. B. Beasley / 2015 / The arctic rank of a Boolean matrix / J. Algebra 433:168~182
L. B. Beasley / 1984 / Boolean-rank-preserving operators and Boolean-rank-1spaces / Linear Algebra Appl 59:55~77
L. B. Beasley / 1987 / Term-rank, permanent and rook-polynomial preservers / Linear Algebra Appl 90:33~46
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