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Let M n {{\bf{M}}}_{n} be the set of all n × n n\times n real matrices. A nonsingular matrix A ∈ M n A\in {{\bf{M}}}_{n} is called a G-matrix if there exist nonsingular diagonal matrices D 1 {D}_{1} and D 2 {D}_{2} such that A − T = D 1 A D 2 {A}^{-T}={D}_{1}A{D}_{2} . For fixed nonsingular diagonal matrices D 1 {D}_{1} and D 2 {D}_{2} , let G ( D 1 , D 2 ) = { A ∈ M n : A − T = D 1 A D 2 } , {\mathbb{G}}\left({D}_{1},{D}_{2})=\left\{A\in {{\bf{M}}}_{n}:{A}^{-T}={D}_{1}A{D}_{2}\right\}, which is called a G-class. The purpose of this short article is to answer the following open question in the affirmative: do there exist two n × n n\times n G-classes having finite intersection when n ≥ 3 n\ge 3 ?

An m × n matrix R with nonnegative entries is called row stochastic if the sum of entries on every row of R is 1. Let Mm,n be the set of all m × n real matrices. For A, B ∈ Mm,n, we say that A is row Hadamard majorized by B (denoted by A ≺ RHB) if there exists an m × n row stochastic matrix R such that A = R ο B, where X ο Y is the Hadamard product (entrywise product) of matrices X, Y ∈ Mm,n. In this paper, we consider the concept of row Hadamard majorization as a relation on Mm,n and characterize the structure of all linear operators T: Mm,n → Mm,n preserving (or strongly preserving) row Hadamard majorization. Also, we find a theoretic graph connection with linear preservers (or strong linear preservers) of row Hadamard majorization, and we give some equivalent conditions for these linear operators on Mn.

Let $${\mathbf {M}}_{n,m}$$ be the set of all n-by-m real matrices and let the relation $$\prec $$ on $${\mathbf {M}}_{n,m}$$ be the multivariate majorization. A linear mapping $$\Phi \!:{\mathbf {M}}_{n,m}\longrightarrow {\mathbf {M}}_{n,m}$$ is said to be a linear preserver of multivariate majorization if $$\Phi (X)\prec \Phi (Y)$$ whenever $$X\prec Y$$. Also, $$\Phi $$ is said to be a local linear preserver of multivariate majorization if for each $$X\in {\mathbf {M}}_{n,m}$$ there exists a linear mapping $$\Phi _X$$ on $${\mathbf {M}}_{n,m}$$ that preserves, multivariate majorization and $$\Phi (X)=\Phi _X(X)$$. In this paper we introduce two types of strong local linear preservers of multivariate majorization on $${\mathbf {M}}_{n,m}$$. Then we find a relation between these types of strong local linear preservers and finally we characterize the structure of these linear maps.

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

This paper is a continuation of the article ‘Topological properties of J-orthogonal matrices’, Linear and Multilinear Algebra 66(2018), 2524–2533, by the authors. Let Mn be the set of all n×n real ...

Let $$\mathbf M _{m,n}$$ be the set of all $$m\times n$$ real matrices. A matrix $$A\in \mathbf M _{m,n}$$ is said to be a dense matrix if there are no zeros between two nonzero entries for every line (row or column) of this matrix. In this paper, we find the structure of linear maps $$T:\mathbf M _{m,n} \rightarrow \mathbf M _{m,n}$$ that strongly preserve dense matrices, i.e., T(A) is a dense matrix if and only if A is a dense matrix.

In this paper the set of all J-orthogonal matrices is considered and some interesting properties of these matrices are obtained. The main topic is a straightforward proof of the known topol...

Let Mm,n be the set of all m × n real matrices. A matrix A ∈ Mm,n is said to be row-dense if there are no zeros between two nonzero entries for every row of this matrix. We find the structure of linear functions T: Mm,n → Mm,n that preserve or strongly preserve row-dense matrices, i.e., T(A) is row-dense whenever A is row-dense or T(A) is row-dense if and only if A is row-dense, respectively. Similarly, a matrix A ∈ Mn,m is called a column-dense matrix if every column of A is a column-dense vector. At the end, the structure of linear preservers (strong linear preservers) of column-dense matrices is found.

Let $\textbf{M}_{n }$ be the set of all $n \times n $ realmatrices. A matrix $D=[d_{ij}]\in\textbf{M}_{n } $ with nonnegative entries is called doubly stochastic if $\sum_{k=1}^{n} d_{ik}=\sum_{k=1}^{n} d_{kj}=1$ for all $1\leq i,j\leq n$. For $ X,Y \in \textbf{M}_{n}$ we say that $X$ is Hadamard-majorized by $Y$, denoted by $ X\prec_{H} Y$, if there exists an $n \times n$ doubly stochastic matrix $D$ such that $X=D\circ Y$.In this paper, some properties of$\prec_{H}$ on $\textbf{M}_{n}$ are first obtained, and then the (strong) linear preservers of$\prec_{H}$ on $\textbf{M}_{n }$ are characterized. For $n\geq3$, it is shown that the strong linear preservers of Hadamard majorization on $\textbf{M}_{n}$ are precisely the invertible linear maps on $\textbf{M}_{n}$ which preserve the set of matrices of term rank 1.An interesting graph theoretic connection to the linear preservers of Hadamard majorization is exhibited. A number of examples are also provided in the paper.

In this paper the concept of B-majorization is studied and some facts about this relation are found. Next, the linear preservers of B-majorization on R n and M n , m will be characterized. It is also shown that the nonzero linear preservers and strong linear preservers of B-majorization on R n are the same.

For vectors x , y ∈ R n , it is said that x is g-tridiagonal majorized by y (written as x ≺ g t y ) if there exists a tridiagonal g-doubly stochastic matrix D such that x = D y . In this paper, we continue the previous work (Armandnejad and Gashool (2012) [3] ) and give a complete characterization of linear operators preserving ≺ g t on R n .

Let M n , m be the set of all n × m real matrices. An n × n real matrix A is called circulant doubly stochastic if it is a convex combination of circulant permutation matrices. For x , y ∈ R n , x is said to be circulant majorized by y (written as x ≺ c y ), if there exists a circulant doubly stochastic matrix D such that x = D y . In this paper, the concept of circulant majorization is investigated and then the linear preservers and strong linear preservers of this concept are characterized on R n .

Let be the algebra of all real matrices. An real matrix is even doubly stochastic if it is a convex combination of even permutation matrices. For , is said to be even majorized by (written as ), if there exists an even doubly stochastic matrix such that . In this paper, the concept of even majorization is investigated and then the linear preservers and strong linear preservers of this concept are characterized on .

Let M n , m be the set of all n × m matrices with entries in R . For A , B ∈ M n , m , it is said that A is row majorized (respectively column-majorized) by B if every row (respectively column) of A is majorized by the corresponding row (respectively column) of B , i.e. for every i ( 1 ⩽ i ⩽ n ) there exists a doubly stochastic matrix D i such that A ( i ) = B ( i ) D i , where A ( i ) and B ( i ) are the i th rows of A and B respectively. In this paper the relations row and column-majorization on M n , m are studied and also all linear operators T : M n , m → M n , m preserving (or strongly preserving) row or column-majorization will be characterized.

Let M n be the set of all n × n matrices whose all entries are in , where is the field of real or complex numbers. In this paper we introduce the relation gw-majorization on M n . Also we characterize all linear operators that strongly preserve gw-majorization on M n .

Let \( {M_n} \) be the set of all \( n \times n \) real matrices. A matrix \( R \in {M_n} \) with nonnegative entries is called row stochastic if Re = e, where \( e = {(1, \ldots ,1)^t} \in {\mathbb{R}^n} \). For \( x,y \in {\mathbb{R}^n} \) we say that x is upper triangular majorized by y (written as \( x{ \prec _{ut}}y \) ) if there exists an upper triangular row stochastic matrix R such that x = Ry. In the present paper, some properties of ut-majorization are investigated. Also, the structure of all linear functions \( T:{\mathbb{R}^n} \to {\mathbb{R}^n} \) preserving (resp. strongly preserving) ut-majorization with additional condition Te1 ≠ 0 (resp. with no condition) will be characterized.

Dedicated to the memory of Batool Bagheri, the founder of mathematics house of Kerman. Abstract. Let Mn,m be the set of all nm real matrices. An nm matrix R = (rij) is g-row stochastic if Pm=1 rik is equal to 1 for all i (1 � in). Let X,Y 2 Mn,m. The matrix X is said to be gut-majorized by Y (denoted by Xgut Y ), if there exists an nn upper triangular g-row stochastic matrix R such that X = RY . Recall that a linear function T : Mn,m ! Mn,m preserves (or strongly preserves) a relation � , if TXTY whenever XY (or TXTY if and only if XY ). The present papers establishes some facts aboutgut on Mn,m, and characterizes the structure of all (strong) linear preservers ofgut on R n and strong linear preservers ofgut on Mn,m.