1 - 7 of 7
Number of results to display per page
Search Results
2. Invertible commutativity preservers of matrices over max algebra
- Creator:
- Song, Seok-Zun, Kang, Kyung-Tae, and Jun, Young-Bae
- Format:
- bez média and svazek
- Type:
- model:article and TEXT
- Subject:
- max algebra, linear operator, and pair of commuting matrices
- Language:
- English
- Description:
- The max algebra consists of the nonnegative real numbers equipped with two binary operations, maximization and multiplication. We characterize the invertible linear operators that preserve the set of commuting pairs of matrices over a subalgebra of max algebra.
- Rights:
- http://creativecommons.org/publicdomain/mark/1.0/ and policy:public
3. Linear maps that strongly preserve regular matrices over the Boolean algebra
- Creator:
- Kang, Kyung-Tae and Song, Seok-Zun
- Format:
- bez média and svazek
- Type:
- model:article and TEXT
- Subject:
- Boolean algebra, regular matrix, and $(U,V)$-operator
- Language:
- English
- Description:
- The set of all $m\times n$ Boolean matrices is denoted by ${\mathbb M}_{m,n}$. We call a matrix $A\in {\mathbb M}_{m,n}$ regular if there is a matrix $G\in {\mathbb M}_{n,m}$ such that $AGA=A$. In this paper, we study the problem of characterizing linear operators on ${\mathbb M}_{m,n}$ that strongly preserve regular matrices. Consequently, we obtain that if $\min \{m,n\}\le 2$, then all operators on ${\mathbb M}_{m,n}$ strongly preserve regular matrices, and if $\min \{m,n\}\ge 3$, then an operator $T$ on ${\mathbb M}_{m,n}$ strongly preserves regular matrices if and only if there are invertible matrices $U$ and $V$ such that $T(X)=UXV$ for all $X\in {\mathbb M}_{m,n}$, or $m=n$ and $T(X)=UX^TV$ for all $X\in {\mathbb M}_{n}$.
- Rights:
- http://creativecommons.org/publicdomain/mark/1.0/ and policy:public
4. Linear operators that preserve Boolean rank of Boolean matrices
- Creator:
- Beasley, Leroy B. and Song, Seok-Zun
- Format:
- bez média and svazek
- Type:
- model:article and TEXT
- Subject:
- Boolean matrix, Boolean rank, and Boolean linear operator
- Language:
- English
- Description:
- The Boolean rank of a nonzero $m\times n$ Boolean matrix $A$ is the minimum number $k$ such that there exist an $m\times k$ Boolean matrix $B$ and a $k\times n$ Boolean matrix $C$ such that $A=BC$. In the previous research L. B. Beasley and N. J. Pullman obtained that a linear operator preserves Boolean rank if and only if it preserves Boolean ranks $1$ and $2$. In this paper we extend this characterizations of linear operators that preserve the Boolean ranks of Boolean matrices. That is, we obtain that a linear operator preserves Boolean rank if and only if it preserves Boolean ranks $1$ and $k$ for some $1<k\leq m$.
- Rights:
- http://creativecommons.org/publicdomain/mark/1.0/ and policy:public
5. Linear operators that preserve graphical properties of matrices: isolation numbers
- Creator:
- Beasley, LeRoy B., Song, Seok-Zun, and Jun, Yizng Bae
- Format:
- bez média and svazek
- Type:
- model:article and TEXT
- Subject:
- Boolean matrix, Boolean rank, Boolean linear operator, and isolation number
- Language:
- English
- Description:
- Let $A$ be a Boolean $\{0,1\}$ matrix. The isolation number of $A$ is the maximum number of ones in $A$ such that no two are in any row or any column (that is they are independent), and no two are in a $2\times 2$ submatrix of all ones. The isolation number of $A$ is a lower bound on the Boolean rank of $A$. A linear operator on the set of $m\times n$ Boolean matrices is a mapping which is additive and maps the zero matrix, $O$, to itself. A mapping strongly preserves a set, $S$, if it maps the set $S$ into the set $S$ and the complement of the set $S$ into the complement of the set $S$. We investigate linear operators that preserve the isolation number of Boolean matrices. Specifically, we show that $T$ is a Boolean linear operator that strongly preserves isolation number $k$ for any $1\leq k\leq \min \{m,n\}$ if and only if there are fixed permutation matrices $P$ and $Q$ such that for $X\in {\mathcal M}_{m,n}(\mathbb B)$ $T(X)=PXQ$ or, $m=n$ and $T(X)=PX^tQ$ where $X^t$ is the transpose of $X$.
- Rights:
- http://creativecommons.org/publicdomain/mark/1.0/ and policy:public
6. Perimeter preserver of matrices over semifields
- Creator:
- Song, Seok-Zun, Kang, Kyung-Tae, and Jun, Young-Bae
- Format:
- bez média and svazek
- Type:
- model:article and TEXT
- Subject:
- linear operator, rank, dominate, perimeter, and $(U,V)$-operator
- Language:
- English
- Description:
- 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.
- Rights:
- http://creativecommons.org/publicdomain/mark/1.0/ and policy:public
7. Zero-term ranks of real matrices and their preservers
- Creator:
- Beasley, LeRoy B. , Jun, Young-Bae , and Song, Seok-Zun
- Format:
- bez média and svazek
- Type:
- model:article and TEXT
- Subject:
- linear operator, zero-term rank, and $P,Q,B$-operator
- Language:
- English
- Description:
- Zero-term rank of a matrix is the minimum number of lines (rows or columns) needed to cover all the zero entries of the given matrix. We characterize the linear operators that preserve zero-term rank of the $m \times n$ real matrices. We also obtain combinatorial equivalent condition for the zero-term rank of a real matrix.
- Rights:
- http://creativecommons.org/publicdomain/mark/1.0/ and policy:public