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Rank and perimeter preserver of rank-1 matrices over max algebra

Seok-Zun SongKyung-Tae Kang — 2003

Discussiones Mathematicae - General Algebra and Applications

For a rank-1 matrix A = a b t over max algebra, we define the perimeter of A as the number of nonzero entries in both a and b. We characterize the linear operators which preserve the rank and perimeter of rank-1 matrices over max algebra. That is, a linear operator T preserves the rank and perimeter of rank-1 matrices if and only if it has the form T(A) = U ⊗ A ⊗ V, or T ( A ) = U A t V with some monomial matrices U and V.

Linear maps that strongly preserve regular matrices over the Boolean algebra

Kyung-Tae KangSeok-Zun Song — 2011

Czechoslovak Mathematical Journal

The set of all m × n Boolean matrices is denoted by 𝕄 m , n . We call a matrix A 𝕄 m , n regular if there is a matrix G 𝕄 n , m such that A G A = A . In this paper, we study the problem of characterizing linear operators on 𝕄 m , n that strongly preserve regular matrices. Consequently, we obtain that if min { m , n } 2 , then all operators on 𝕄 m , n strongly preserve regular matrices, and if min { m , n } 3 , then an operator T on 𝕄 m , n strongly preserves regular matrices if and only if there are invertible matrices U and V such that T ( X ) = U X V for all X 𝕄 m , n , or m = n and T ( X ) = U X T V for all X 𝕄 n .

Perimeter preserver of matrices over semifields

Seok-Zun SongKyung-Tae KangYoung Bae Jun — 2006

Czechoslovak Mathematical Journal

For a rank- 1 matrix A = 𝐚 𝐛 t , we define the perimeter of A as the number of nonzero entries in both 𝐚 and 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.

Perimeter preservers of nonnegative integer matrices

Seok-Zun SongKyung-Tae KangSucheol Yi — 2004

Commentationes Mathematicae Universitatis Carolinae

We investigate the perimeter of nonnegative integer matrices. We also characterize the linear operators which preserve the rank and perimeter of nonnegative integer matrices. That is, a linear operator T preserves the rank and perimeter of rank- 1 matrices if and only if it has the form T ( A ) = P ( A B ) Q , or T ( A ) = P ( A t B ) Q with appropriate permutation matrices P and Q and positive integer matrix B , where denotes Hadamard product.

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