Linear preservers of row-dense matrices

Sara M. Motlaghian; Ali Armandnejad; Frank J. Hall

Displaying similar documents to “Linear preservers of row-dense matrices”

Linear preservers of rc-majorization on matrices

Mohammad Soleymani (2024)

Czechoslovak Mathematical Journal

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Let $A$ , $B$ be $n \times m$ matrices. The concept of matrix majorization means the $j$ th column of $A$ is majorized by the $j$ th column of $B$ and this is done for all $j$ by a doubly stochastic matrix $D$ . We define rc-majorization that extended matrix majorization to columns and rows of matrices. Also, the linear preservers of rc-majorization will be characterized.

G-matrices, $J$ -orthogonal matrices, and their sign patterns

Frank J. Hall, Miroslav Rozložník (2016)

Czechoslovak Mathematical Journal

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A real matrix $A$ is a G-matrix if $A$ is nonsingular and there exist nonsingular diagonal matrices $D_{1}$ and $D_{2}$ such that $A^{- T} = D_{1} A D_{2}$ , where $A^{- T}$ denotes the transpose of the inverse of $A$ . Denote by $J = diag (\pm 1)$ a diagonal (signature) matrix, each of whose diagonal entries is $+ 1$ or $- 1$ . A nonsingular real matrix $Q$ is called $J$ -orthogonal if $Q^{T} J Q = J$ . Many connections are established between these matrices. In particular, a matrix $A$ is a G-matrix if and only if $A$ is diagonally (with positive diagonals) equivalent to a column permutation...

Row Hadamard majorization on $𝐌_{m, n}$

Abbas Askarizadeh, Ali Armandnejad (2021)

Czechoslovak Mathematical Journal

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An $m \times n$ matrix $R$ with nonnegative entries is called row stochastic if the sum of entries on every row of $R$ is 1. Let $𝐌_{m, n}$ be the set of all $m \times n$ real matrices. For $A, B \in 𝐌_{m, n}$ , we say that $A$ is row Hadamard majorized by $B$ (denoted by $A ≺_{R H} B)$ if there exists an $m \times n$ row stochastic matrix $R$ such that $A = R \circ B$ , where $X \circ Y$ is the Hadamard product (entrywise product) of matrices $X, Y \in 𝐌_{m, n}$ . In this paper, we consider the concept of row Hadamard majorization as a relation on $𝐌_{m, n}$ and characterize the structure of all linear operators $T : 𝐌_{m, n} \to 𝐌_{m, n}$ preserving...

Controllable and tolerable generalized eigenvectors of interval max-plus matrices

Matej Gazda, Ján Plavka (2021)

Kybernetika

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By max-plus algebra we mean the set of reals $ℝ$ equipped with the operations $a \oplus b = max {a, b}$ and $a \otimes b = a + b$ for $a, b \in ℝ .$ A vector $x$ is said to be a generalized eigenvector of max-plus matrices $A, B \in ℝ (m, n)$ if $A \otimes x = λ \otimes B \otimes x$ for some $λ \in ℝ$ . The investigation of properties of generalized eigenvectors is important for the applications. The values of vector or matrix inputs in practice are usually not exact numbers and they can be rather considered as values in some intervals. In this paper the properties of matrices and vectors with inexact (interval)...

-simplicity of interval max-min matrices

Ján Plavka, Štefan Berežný (2018)

Kybernetika

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A matrix $A$ is said to have $𝐗$ -simple image eigenspace if any eigenvector $x$ belonging to the interval $𝐗 = {x : \underset{̲}{x} \leq x \leq \bar{x}}$ containing a constant vector is the unique solution of the system $A \otimes y = x$ in $𝐗$ . The main result of this paper is an extension of $𝐗$ -simplicity to interval max-min matrix $𝐀 = {A : \underset{̲}{A} \leq A \leq \bar{A}}$ distinguishing two possibilities, that at least one matrix or all matrices from a given interval have $𝐗$ -simple image eigenspace. $𝐗$ -simplicity of interval matrices in max-min algebra are studied and equivalent conditions for...

$(0, 1)$ -matrices, discrepancy and preservers

LeRoy B. Beasley (2019)

Czechoslovak Mathematical Journal

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

On $n$ -thin dense sets in powers of topological spaces

Adam Bartoš (2016)

Commentationes Mathematicae Universitatis Carolinae

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A subset of a product of topological spaces is called $n$ -thin if every its two distinct points differ in at least $n$ coordinates. We generalize a construction of Gruenhage, Natkaniec, and Piotrowski, and obtain, under CH, a countable $T_{3}$ space $X$ without isolated points such that $X^{n}$ contains an $n$ -thin dense subset, but $X^{n + 1}$ does not contain any $n$ -thin dense subset. We also observe that part of the construction can be carried out under MA.

Possible isolation number of a matrix over nonnegative integers

LeRoy B. Beasley, Young Bae Jun, Seok-Zun Song (2018)

Czechoslovak Mathematical Journal

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Let $ℤ_{+}$ be the semiring of all nonnegative integers and $A$ an $m \times n$ matrix over $ℤ_{+}$ . The rank of $A$ is the smallest $k$ such that $A$ can be factored as an $m \times k$ matrix times a $k \times n$ 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 $2 \times 2$ 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...

On row-sum majorization

Farzaneh Akbarzadeh, Ali Armandnejad (2019)

Czechoslovak Mathematical Journal

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Let $𝕄_{n, m}$ be the set of all $n \times m$ real or complex matrices. For $A, B \in 𝕄_{n, m}$ , we say that $A$ is row-sum majorized by $B$ (written as $A ≺^{rs} B$ ) if $R (A) ≺ R (B)$ , where $R (A)$ is the row sum vector of $A$ and $≺$ is the classical majorization on $ℝ^{n}$ . In the present paper, the structure of all linear operators $T : 𝕄_{n, m} \to 𝕄_{n, m}$ preserving or strongly preserving row-sum majorization is characterized. Also we consider the concepts of even and circulant majorization on $ℝ^{n}$ and then find the linear preservers of row-sum majorization of these relations on $𝕄_{n, m}$ . ...

On the combinatorial structure of $0 / 1$ -matrices representing nonobtuse simplices

Jan Brandts, Abdullah Cihangir (2019)

Applications of Mathematics

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A $0 / 1$ -simplex is the convex hull of $n + 1$ affinely independent vertices of the unit $n$ -cube $I^{n}$ . It is nonobtuse if none of its dihedral angles is obtuse, and acute if additionally none of them is right. Acute $0 / 1$ -simplices in $I^{n}$ can be represented by $0 / 1$ -matrices $P$ of size $n \times n$ whose Gramians $G = P^{⊤} P$ have an inverse that is strictly diagonally dominant, with negative off-diagonal entries. In this paper, we will prove that the positive part $D$ of the transposed inverse $P^{- ⊤}$ of $P$ is doubly stochastic and has the...

Circulant matrices with orthogonal rows and off-diagonal entries of absolute value $1$

Daniel Uzcátegui Contreras, Dardo Goyeneche, Ondřej Turek, Zuzana Václavíková (2021)

Communications in Mathematics

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It is known that a real symmetric circulant matrix with diagonal entries $d \geq 0$ , off-diagonal entries $\pm 1$ and orthogonal rows exists only of order $2 d + 2$ (and trivially of order $1$ ) [Turek and Goyeneche 2019]. In this paper we consider a complex Hermitian analogy of those matrices. That is, we study the existence and construction of Hermitian circulant matrices having orthogonal rows, diagonal entries $d \geq 0$ and any complex entries of absolute value $1$ off the diagonal. As a particular case, we consider...