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Linear operators that preserve column rank of Boolean matrices. (English) Zbl 0802.15006

Let \(T\) be a linear transformation of the matrix algebra (of order \(\geq 4)\) over a Boolean algebra. The main result shows that \(T\) preserves the column rank if and only if there are invertible matrices \(U\) and \(V\) such that \(T(A) = UAV\) for all \(A\).

MSC:

15A30 Algebraic systems of matrices
15A03 Vector spaces, linear dependence, rank, lineability
15A04 Linear transformations, semilinear transformations
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References:

[1] LeRoy B. Beasley and Seok-Zun Song, A comparison of nonnegative real ranks and their preservers, Linear and Multilinear Algebra 31 (1992), no. 1-4, 37 – 46. · Zbl 0781.15006 · doi:10.1080/03081089208818120
[2] LeRoy B. Beasley and Norman J. Pullman, Semiring rank versus column rank, Linear Algebra Appl. 101 (1988), 33 – 48. · Zbl 0642.15002 · doi:10.1016/0024-3795(88)90141-3
[3] LeRoy B. Beasley and Norman J. Pullman, Boolean-rank-preserving operators and Boolean-rank-1 spaces, Linear Algebra Appl. 59 (1984), 55 – 77. · Zbl 0536.20044 · doi:10.1016/0024-3795(84)90158-7
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