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For any $m\times n$ nonbinary Boolean matrix $A$, its spanning column rank is the minimum number of the columns of $A$ that spans its column space. We have a characterization of spanning column rank-$1$ nonbinary Boolean matrices. We investigate the linear operators that preserve the spanning column ranks of matrices over the nonbinary Boolean algebra. That is, for a linear operator $T$ on $m\times n$ nonbinary Boolean matrices, it preserves all spanning column ranks if and only if there exist an invertible nonbinary Boolean matrix $P$ of order $m$ and a permutation matrix $Q$ of order $n$ such that $T(A)=PAQ$ for all $m\times n$ nonbinary Boolean matrix $A$. We also obtain other characterizations of the (spanning) column rank preserver.
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참고문헌 (9)
참고문헌 신청
L. B. Beasley / 1984 / Boolean-rank-preserving operators and Boolean-rank-1spaces / Linear Algebra Appl 59:55~77
L. B. Beasley / 1988 / Semiring rank versus column rank / Linear Algebra Appl 101:33~48
L. B. Beasley / 2005 / Column ranks and their pre-servers over nonnegative real matrices / Linear Algebra Appl 399:3~16
K. H. Kim / 1982 / Boolean Matrix Theory and Applications / Marcel Dekker
S. Kirkland / 1993 / Linear operators preserving invariants of nonbinary Boolean matrices / Linear and Multilinear Algebra 33(3-4):295~300
L. B. Beasley / 1988 / Semiring rank versus column rank / Linear Algebra Appl 101:33~48
L. B. Beasley / 2005 / Column ranks and their pre-servers over nonnegative real matrices / Linear Algebra Appl 399:3~16
K. H. Kim / 1982 / Boolean Matrix Theory and Applications / Marcel Dekker
S. Kirkland / 1993 / Linear operators preserving invariants of nonbinary Boolean matrices / Linear and Multilinear Algebra 33(3-4):295~300
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