Geodesic
Perimeter preserver of matrices over semifields
Czechoslovak Mathematical Journal, Tome 56 (2006) no. 2, pp. 515-524
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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.
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.
Classification : 15A03, 15A04, 15A23, 15A33
Keywords: linear operator; rank; dominate; perimeter; $(U, V)$-operator 
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     title = {Perimeter preserver of matrices over semifields},
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Song, Seok-Zun; Kang, Kyung-Tae; Jun, Young-Bae. Perimeter preserver of matrices over semifields. Czechoslovak Mathematical Journal, Tome 56 (2006) no. 2, pp. 515-524. http://geodesic.mathdoc.fr/item/CMJ_2006_56_2_a16/

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