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### $\left(0,1\right)$-matrices, discrepancy and preservers

Czechoslovak Mathematical Journal

Let $m$ and $n$ be positive integers, and let $R=\left({r}_{1},...,{r}_{m}\right)$ and $S=\left({s}_{1},...,{s}_{n}\right)$ be nonnegative integral vectors. Let $A\left(R,S\right)$ be the set of all $m×n$ $\left(0,1\right)$-matrices with row sum vector $R$ and column vector $S$. Let $R$ and $S$ be nonincreasing, and let $F\left(R\right)$ be the $m×n$ $\left(0,1\right)$-matrix, where for each $i$, the $i$th row of $F\left(R,S\right)$ consists of ${r}_{i}$ 1’s followed by $\left(n-{r}_{i}\right)$ 0’s. Let $A\in A\left(R,S\right)$. The discrepancy of A, $\mathrm{disc}\left(A\right)$, is the number of positions in which $F\left(R\right)$ has a 1 and $A$ has a 0. In this paper we investigate linear operators mapping $m×n$ matrices over the binary...

### Linear operators that preserve Boolean rank of Boolean matrices

Czechoslovak Mathematical Journal

The Boolean rank of a nonzero $m×n$ Boolean matrix $A$ is the minimum number $k$ such that there exist an $m×k$ Boolean matrix $B$ and a $k×n$ 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...

### Zero-term ranks of real matrices and their preservers

Czechoslovak Mathematical Journal

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 $m×n$ real matrices. We also obtain combinatorial equivalent condition for the zero-term rank of a real matrix.

### Linear operators that preserve graphical properties of matrices: isolation numbers

Czechoslovak Mathematical Journal

Let $A$ be a Boolean $\left\{0,1\right\}$ 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 $2×2$ 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 $m×n$ 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...

### Possible isolation number of a matrix over nonnegative integers

Czechoslovak Mathematical Journal

Let ${ℤ}_{+}$ be the semiring of all nonnegative integers and $A$ an $m×n$ matrix over ${ℤ}_{+}$. The rank of $A$ is the smallest $k$ such that $A$ can be factored as an $m×k$ matrix times a $k×n$ matrix. The isolation number of $A$ is the maximum number of nonzero entries in $A$ such that no two are in any row or any column, and no two are in a $2×2$ submatrix of all nonzero entries. We have that the isolation number of $A$ is a lower bound of the rank of $A$. For $A$ with isolation number $k$, we investigate the possible values of the rank of $A$...

### Spaces of rank-2 matrices over GF(2).

ELA. The Electronic Journal of Linear Algebra [electronic only]

### Properties of a covariance matrix with an application to $D$-optimal design.

ELA. The Electronic Journal of Linear Algebra [electronic only]

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