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A rank $1$ matrix has a factorization as ${\bf u}{\bf v}^t$ for vectors ${\bf u}$ and ${\bf v}$ of some orders. The arctic rank of a rank $1$ matrix is the half number of nonzero entries in ${\bf u}$ and ${\bf v}$. A matrix of rank $k$ can be expressed as the sum of $k$ rank $1$ matrices, a rank $1$ decomposition. The arctic rank of a matrix $A$ of rank $k$ is the minimum of the sums of arctic ranks of the rank $1$ matrices over all rank $1$ decomposition of $A$. In this paper we obtain characterizations of the linear operators that strongly preserve the symmetric arctic ranks of symmetric matrices over nonnegative reals.
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참고문헌 (10)
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LeRoy B. Beasley / 2015 / The arctic rank of a Boolean matrix / Journal of Algebra 433:168~182
LeRoy B. Beasley / 1984 / Boolean-rank-preserving operators and Boolean-rank-1 spaces / Linear Algebra and its Applications 59:55~77
LeRoy B. Beasley / 1987 / Term-rank, permanent, and rook-polynomial preservers / Linear Algebra and its Applications 90:33~46
LeRoy B. Beasley / 2016 / Primitive symmetric matrices and their preservers / Linear and Multilinear Algebra 65(1):129~139
LeRoy B. Beasley / 2016 / Symmetric arctic ranks of nonnegative matrices and their linear preservers / Linear and Multilinear Algebra 65(10):2000~2010
LeRoy B. Beasley / 1984 / Boolean-rank-preserving operators and Boolean-rank-1 spaces / Linear Algebra and its Applications 59:55~77
LeRoy B. Beasley / 1987 / Term-rank, permanent, and rook-polynomial preservers / Linear Algebra and its Applications 90:33~46
LeRoy B. Beasley / 2016 / Primitive symmetric matrices and their preservers / Linear and Multilinear Algebra 65(1):129~139
LeRoy B. Beasley / 2016 / Symmetric arctic ranks of nonnegative matrices and their linear preservers / Linear and Multilinear Algebra 65(10):2000~2010
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