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...

Let Mm×n(F) be the vector space of all m×n matrices over a field F. In the case where m ≥ n, char(F) ≠ 2 and F has at least five elements, we give a complete characterization of linear maps Φ: Mm×n(F) → Mm×n(F) such that spark(Φ(A)) = spark(A) for any A ∈Mm×n(F).

Letr Σn(C) denote the space of all n χ n symmetric matrices over the complex field C. The main objective of this paper is to prove that the maps Φ : Σn(C) -> Σn (C) satisfying for any fixed irre- ducible characters X, X' -SC the condition dx(A +aB) = dχ·(Φ(Α ) + αΦ(Β)) for all matrices A,В ε Σ„(С) and all scalars a ε C are automatically linear and bijective. As a corollary of the above result we characterize all such maps Φ acting on ΣИ(С).

Let $\mathscr{H}$ be a complex Hilbert space, $A$ a positive operator with closed range in $ℬ\left(\mathscr{H}\right)$ and ${ℬ}_A\left(\mathscr{H}\right)$ the sub-algebra of $ℬ\left(\mathscr{H}\right)$ of all $A$-self-adjoint operators. Assume $\phi :{ℬ}_A\left(\mathscr{H}\right)$ onto itself is a linear continuous map. This paper shows that if $\phi$ preserves $A$-unitary operators such that $\phi \left(I\right)=P$ then $\psi$ defined by $\psi \left(T\right)=P\phi \left(PT\right)$ is a homomorphism or an anti-homomorphism and $\psi \left(T^{♯}\right)=\psi {\left(T\right)}^{♯}$ for all $T\in {ℬ}_A\left(\mathscr{H}\right)$, where $P=A^{+}A$ and $A^{+}$ is the Moore-Penrose inverse of $A$. A similar result is also true if $\phi$ preserves $A$-quasi-unitary operators in both directions such that there exists an...

Let X and Y be Banach spaces and ℬ(X) and ℬ(Y) the algebras of all bounded linear operators on X and Y, respectively. We say that A,B ∈ ℬ(X) quasi-commute if there exists a nonzero scalar ω such that AB = ωBA. We characterize bijective linear maps ϕ : ℬ(X) → ℬ(Y) preserving quasi-commutativity. In fact, such a characterization can be proved for much more general algebras. In the finite-dimensional case the same result can be obtained without the bijectivity assumption.

The set of all $m\times n$ Boolean matrices is denoted by ${𝕄}_{m,n}$. We call a matrix $A\in {𝕄}_{m,n}$ regular if there is a matrix $G\in {𝕄}_{n,m}$ such that $AGA=A$. 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 $min\{m,n\}\le 2$, then all operators on ${𝕄}_{m,n}$ strongly preserve regular matrices, and if $min\{m,n\}\ge 3$, 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 $T\left(X\right)=UXV$ for all $X\in {𝕄}_{m,n}$, or $m=n$ and $T\left(X\right)=U{X}^{T}V$ for all $X\in {𝕄}_n$.

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...

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...

Our purpose is to present a number of new facts about the structure of semipositive matrices, involving patterns, spectra and Jordon form, sums and products, and matrix equivalence, etc. Techniques used to obtain the results may be of independent interest. Examples include: any matrix with at least two columns is a sum, and any matrix with at least two rows, a product, of semipositive matrices. Any spectrum of a real matrix with at least $2$ elements is the spectrum of a square semipositive matrix,...

The objective of this manuscript is to investigate the structure of linear maps on the space of real symmetric matrices ${𝒮}^n$ that leave invariant the closed convex cones of copositive and completely positive matrices (${\mathrm{COP}}_n$ and ${\mathrm{CP}}_n$). A description of an invertible linear map on ${𝒮}^n$ such that $L\left({\mathrm{CP}}_n\right)\subset C{P}_n$ is obtained in terms of semipositive maps over the positive semidefinite cone ${𝒮}_{+}^n$ and the cone of symmetric nonnegative matrices ${𝒩}_{+}^n$ for $n\le 4$, with specific calculations for $n=2$. Preserver properties of the Lyapunov map $X\mapsto AX+X{A}^t$, the...

Let ${𝔹}_k$ be the general Boolean algebra and $T$ a linear operator on $M_{m,n}\left({𝔹}_k\right)$. If for any $A$ in $M_{m,n}\left({𝔹}_k\right)$ ($M_n\left({𝔹}_k\right)$, respectively), $A$ is regular (invertible, respectively) if and only if $T\left(A\right)$ is regular (invertible, respectively), then $T$ is said to strongly preserve regular (invertible, respectively) matrices. In this paper, we will give complete characterizations of the linear operators that strongly preserve regular (invertible, respectively) matrices over ${𝔹}_k$. Meanwhile, noting that a general Boolean algebra ${𝔹}_k$ is isomorphic...

We study continuous maps on alternate matrices over complex field which preserve zeros of Lie product.