Idempotent elements determined matrix algebras

Let M_n(R) be the algebra of all n×n matrices over a unital commutative ring R with 2 invertible, V be an R-module. It is shown in this article that, if a symmetric bilinear map {·,·} from M_n(R)×M_n(R) to V satisfies the condition that {u,u}={e,u} whenever u²=u, then there exists a linear map f from M_n(R) to V such that . Applying the main result we prove that an invertible linear transformation θ on M_n(R) preserves idempotent matrices if and only if it is a Jordan automorphism, and a linear transformation δ on M_n(R) is a Jordan derivation if and only if it is Jordan derivable at all idempotent points.     

Congruence of Hermitian matrices by Hermitian matrices

Two Hermitian matrices A,B∈M_n(C) are said to be Hermitian-congruent if there exists a nonsingular Hermitian matrix C∈M_n(C) such that B=CAC. In this paper, we give necessary and sufficient conditions for two nonsingular simultaneously unitarily diagonalizable Hermitian matrices A and B to be Hermitian-congruent. Moreover, when A and B are Hermitian-congruent, we describe the possible inertias of the Hermitian matrices C that carry the congruence. We also give necessary and sufficient conditions for any 2-by-2 nonsingular Hermitian matrices to be Hermitian-congruent. In both of the studied cases, we show that if A and B are real and Hermitian-congruent, then they are congruent by a real symmetric matrix. Finally we note that if A and B are 2-by-2 nonsingular real symmetric matrices having the same sign pattern, then there is always a real symmetric matrix C satisfying B=CAC. Moreover, if both matrices are positive, then C can be picked with arbitrary inertia.     

To what extent is a large space of matrices not closed under product?

Let (K) be a field. Given an arbitrary linear subspace V of M_n(K) of codimension less than n-1, a classical result states that V generates the (K)-algebra M_n(K). Here, we strengthen this statement in three ways: we show that M_n(K) is spanned by the products of the form AB with (A,B)∈V²; we prove that every matrix in M_n(K) can be decomposed into a product of matrices of V; finally, when V is a linear perplane of M_n(K) and n>2, we show that every matrix in M_n(K) is a product of two elements of V.     

On the diameters of commuting graphs

The commuting graph of a ring R, denoted by Γ(R), is a graph whose vertices are all non-central elements of R and two distinct vertices x and y are adjacent if and only if xy = yx. Let D be a division ring and n ? 3. In this paper we investigate the diameters of Γ(M_n(D)) and determine the diameters of some induced subgraphs of Γ(M_n(D)), such as the induced subgraphs on the set of all non-scalar non-invertible, nilpotent, idempotent, and involution matrices in M_n(D). For every field F, it is shown that if Γ(M_n(F)) is a connected graph, then diam Γ(M_n(F)) ? 6. We conjecture that if Γ(M_n(F)) is a connected graph, then diam Γ(M_n(F)) ? 5. We show that if F is an algebraically closed field or n is a prime number and Γ(M_n(F)) is a connected graph, then diam Γ(M_n(F)) = 4. Finally, we present some applications to the structure of pairs of idempotents which may prove of independent interest.     

Bijective linear maps on semimodules spanned by Boolean matrices of fixed rank

Let M_m,n(B) be the semimodule of all m×n Boolean matrices where B is the Boolean algebra with two elements. Let k be a positive integer such that 2?k?min(m,n). Let B(m,n,k) denote the subsemimodule of M_m,n(B) spanned by the set of all rank k matrices. We show that if T is a bijective linear mapping on B(m,n,k), then there exist permutation matrices P and Q such that T(A)=PAQ for all A∈B(m,n,k) or m=n and T(A)=PA^tQ for all A∈B(m,n,k). This result follows from a more general theorem we prove concerning the structure of linear mappings on B(m,n,k) that preserve both the weight of each matrix and rank one matrices of weight k². Here the weight of a Boolean matrix is the number of its nonzero entries.     

Multiplicative maps preserving the higher rank numerical ranges and radii

Let M_n be the semigroup of n×n complex matrices under the usual multiplication, and let S be different subgroups or semigroups in M_n including the (special) unitary group, (special) general linear group, the semigroups of matrices with bounded ranks. Suppose Λ_k(A) is the rank-k numerical range and r_k(A) is the rank-k numerical radius of A∈M_n. Multiplicative maps ?:S→M_n satisfying r_k(?(A))=r_k(A) are characterized. From these results, one can deduce the structure of multiplicative preservers of Λ_k(A).     

On the matrices of given rank in a large subspace

Let V be a linear subspace of M_n,p(K) with codimension lesser than n, where K is an arbitrary field and n?p. In a recent work of the author, it was proven that V is always spanned by its rank p matrices unless n=p=2 and K?F₂. Here, we give a sufficient condition on codim V for V to be spanned by its rank r matrices for a given r∈?1,p-1?. This involves a generalization of the Gerstenhaber theorem on linear subspaces of nilpotent matrices.     

Subspaces of matrices with special rank properties

Let K be a field and let M_m×n(K) denote the space of m×n matrices over K. We investigate properties of a subspace M of M_m×n(K) of dimension n(m-r+1) in which each non-zero element of M has rank at least r and enumerate the number of elements of a given rank in M when K is finite. We also provide an upper bound for the dimension of a constant rank r subspace of M_m×n(K) when K is finite and give non-trivial examples to show that our bound is optimal in some cases. We include a similar a bound for the maximum dimension of a constant rank subspace of skew-symmetric matrices over a finite field.     

On bilinear maps determined by rank one matrices with some applications

Let M_n be the algebra of all n×n matrix over a field F, A a rank one matrix in M_n. In this article it is shown that if a bilinear map ? from M_n×M_n to M_n satisfies the condition that ?(u,v)=?(I,A) whenever u·v=A, then there exists a linear map φ from M_n to M_n such that . If ? is further assumed to be symmetric then there exists a matrix B such that ?(x,y)=tr(xy)B for all x,y∈M_n. Applying the main result we prove that if a linear map on M_n is desirable at a rank one matrix then it is a derivation, and if an invertible linear map on M_n is automorphisable at a rank one matrix then it is an automorphism. In other words, each rank one matrix in M_n is an all-desirable point and an all-automorphisable point, respectively.     

Homomorphisms of matrix semigroups over division rings from dimension two to four

Let D be an arbitrary division ring and M_n(D) the multiplicative semigroup of all n×n matrices over D. We study non-degenerate, injective homomorphisms from M₂(D) to M₄(D). In particular, we present a structural result for the case when D is the ring of quaternions.     

On classical adjoint-commuting mappings between matrix algebras

Let F be a field and let m and n be integers with m,n?3. Let M_n denote the algebra of n×n matrices over F. In this note, we characterize mappings ψ:M_n→M_m that satisfy one of the following conditions:

1.

|F|=2 or |F|>n+1, and ψ(adj(A+αB))=adj(ψ(A)+αψ(B)) for all A,B∈M_n and α∈F with ψ(I_n)≠0.

2.

ψ is surjective and ψ(adj(A-B))=adj(ψ(A)-ψ(B)) for every A,B∈M_n.

Here, adjA denotes the classical adjoint of the matrix A, and I_n is the identity matrix of order n. We give examples showing the indispensability of the assumption ψ(I_n)≠0 in our results.     

Accretive operators and Cassels inequality

Let a,b>0 and let Z∈M_n(R) such that Z lies into the operator ball of diameter [aI,bI]. Then for all positive definite A∈M_n(R),

     

Similarity preserving linear maps on upper triangular matrix algebras

In this note we consider similarity preserving linear maps on the algebra of all n × n complex upper triangular matrices T_n. We give characterizations of similarity invariant subspaces in T_n and similarity preserving linear injections on T_n. Furthermore, we considered linear injections on T_n preserving similarity in M_n as well.     

Multiplicative mappings at some points on matrix algebras

Let M_n be the algebra of all n×n matrices, and let φ:M_n→M_n be a linear mapping. We say that φ is a multiplicative mapping at G if φ(ST)=φ(S)φ(T) for any S,T∈M_n with ST=G. Fix G∈M_n, we say that G is an all-multiplicative point if every multiplicative linear bijection φ at G with φ(I_n)=I_n is a multiplicative mapping in M_n, where I_n is the unit matrix in M_n. We mainly show in this paper the following two results: (1) If G∈M_n with detG=0, then G is an all-multiplicative point in M_n; (2) If φ is an multiplicative mapping at I_n, then there exists an invertible matrix P∈M_n such that either φ(S)=PSP^-1 for any S∈M_n or φ(T)=PT^trP^-1 for any T∈M_n.     

The singular linear preservers of non-singular matrices

Given an arbitrary field K, we reduce the determination of the singular endomorphisms f of M_n(K) such that f(GL_n(K))⊂GL_n(K) to the classification of n-dimensional division algebras over K. Our method, which is based upon Dieudonné’s theorem on singular subspaces of M_n(K), also yields a proof for the classical non-singular case.     

Maximum order-index of matrices over commutative inclines: an answer to an open problem

This paper proves that the maximum order-index of n × n matrices over an arbitrary commutative incline equals (n − 1)² + 1. This is an answer to an open problem “Compute the maximum order-index of a member of M_n(L)”, proposed by Cao, Kim and Roush in a monograph Incline Algebra and Applications, 1984, where M_n(L) is the set of all n × n matrices over an incline L.     

On injective Jordan semi-triple maps of matrix algebras

We show that every injective Jordan semi-triple map on the algebra M_n(F) of all n × n matrices with entries in a field F (i.e. a map Φ:M_n(F)→M_n(F) satisfying

Φ(ABA)=Φ(A)Φ(B)Φ(A)     

Invertible and nilpotent matrices over antirings

In this paper, we characterize invertible matrices over an arbitrary commutative antiring S with 1 and find the structure of GL_n(S). We find the number of nilpotent matrices over an entire commutative finite antiring. We prove that every nilpotent n×n matrix over an entire antiring can be written as a sum of ⌈log₂n⌉ square-zero matrices and also find the necessary number of square-zero summands for an arbitrary trace-zero matrix to be expressible as their sum.     

Number theory of matrix semigroups

Unlike factorization theory of commutative semigroups which are well-studied, very little literature exists describing factorization properties in noncommutative semigroups. Perhaps the most ubiquitous noncommutative semigroups are semigroups of square matrices and this article investigates the factorization properties within certain subsemigroups of M_n(Z), the semigroup of n×n matrices with integer entries. Certain important invariants are calculated to give a sense of how unique or non-unique factorization is in each of these semigroups.