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Keywords:
linear operator; rank; dominate; perimeter; $(U,V)$-operator
linear operator; rank; dominate; perimeter; $(U,V)$-operator
Summary:
For a rank-$1$ matrix $A= {\bold a \bold b}^t$, we define the perimeter of $A$ as the number of nonzero entries in both $\bold a$ and $\bold b$. We characterize the linear operators which preserve the rank and perimeter of rank-$1$ matrices over semifields. That is, a linear operator $T$ preserves the rank and perimeter of rank-$1$ matrices over semifields if and only if it has the form $T(A)=U A V$, or $T(A)=U A^t V$ with some invertible matrices U and V.
References:
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[3] S. Z. Song, S. R. Park: Maximal column rank preservers of fuzzy matrices. Discuss. Math. Gen. Algebra Appl. 21 (2001), 207–218. DOI 10.7151/dmgaa.1038 | MR 1894316
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