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.     

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.     

Maximum nullity of outerplanar graphs and the path cover number

Let G=(V,E) be a graph with V={1,2,…,n}. Define S(G) as the set of all n×n real-valued symmetric matrices A=[a_ij] with a_ij≠0,i≠j if and only if ij∈E. By M(G) we denote the largest possible nullity of any matrix A∈S(G). The path cover number of a graph G, denoted P(G), is the minimum number of vertex disjoint paths occurring as induced subgraphs of G which cover all the vertices of G.There has been some success with relating the path cover number of a graph to its maximum nullity. Johnson and Duarte [5], have shown that for a tree T,M(T)=P(T). Barioli et al. [2], show that for a unicyclic graph G,M(G)=P(G) or M(G)=P(G)-1. Notice that both families of graphs are outerplanar. We show that for any outerplanar graph G,M(G)?P(G). Further we show that for any partial 2-path G,M(G)=P(G).     

Jordan higher all-derivable points on nontrivial nest algebras

Let A be a Banach algebra with unity I and M be a unital Banach A-bimodule. A family of continuous additive mappings D=(δ_i)_i∈N from A into M is called a higher derivable mapping at X, if δ_n(AB)=∑_i+j=nδ_i(A)δ_j(B) for any A,B∈A with AB=X. In this paper, we show that D is a Jordan higher derivation if D is a higher derivable mapping at an invertible element X. As an application, we also get that every invertible operator in a nontrivial nest algebra is a higher all-derivable point.     

A generalized Cayley-Hamilton theorem for polynomials with matrix coefficients

A family F of square matrices of the same order is called a quasi-commuting family if (AB-BA)C=C(AB-BA) for all A,B,C∈F where A,B,C need not be distinct. Let f_k(x₁,x₂,…,x_p),(k=1,2,…,r), be polynomials in the indeterminates x₁,x₂,…,x_p with coefficients in the complex field C, and let M₁,M₂,…,M_r be n×n matrices over C which are not necessarily distinct. Let and let δ_F(x₁,x₂,…,x_p)=detF(x₁,x₂,…,x_p). In this paper, we prove that, for n×n matrices A₁,A₂,…,A_p over C, if {A₁,A₂,…,A_p,M₁,M₂,…,M_r} is a quasi-commuting family, then F(A₁,A₂,…,A_p)=O implies that δ_F(A₁,A₂,…,A_p)=O.     

The linear preservers of real diagonalizable matrices

Using a recent result of Bogdanov and Guterman on the linear preservers of pairs of simultaneously diagonalizable matrices, we determine all the automorphisms of the vector space M_n(R) which stabilize the set of diagonalizable matrices. To do so, we investigate the structure of linear subspaces of diagonalizable matrices of M_n(R) with maximal dimension.     

Locally strongly convex hypersurfaces with constant affine mean curvature

Let be a locally strongly convex hypersurface, given by a strictly convex function xn+1=f(x₁,…,xn) defined in a convex domain Ω⊂An. We consider the Riemannian metric G# on M, defined by . In this paper we prove that if M is a locally strongly convex surface with constant affine mean curvature and if M is complete with respect to the metric G#, then M must be an elliptic paraboloid.     

Singular value estimates of oblique projections

Let W and M be two finite dimensional subspaces of a Hilbert space H such that H=W⊕M^⊥, and let P_W‖M^⊥ denote the oblique projection with range W and nullspace M^⊥. In this article we get the following formula for the singular values of P_W‖M^⊥

     

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 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.     

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.     

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.     

On commutativity of projectors

The purpose of this paper is to revisit two problems discussed previously in the literature, both related to the commutativity property P₁P₂ = P₂P₁, where P₁ and P₂ denote projectors (i.e., idempotent matrices). The first problem was considered by Baksalary et al. [J.K. Baksalary, O.M. Baksalary, T. Szulc, A property of orthogonal projectors, Linear Algebra Appl. 354 (2002) 35-39], who have shown that if P₁ and P₂ are orthogonal projectors (i.e., Hermitian idempotent matrices), then in all nontrivial cases a product of any length having P₁ and P₂ as its factors occurring alternately is equal to another such product if and only if P₁ and P₂ commute. In the present paper a generalization of this result is proposed and validity of the equivalence between commutativity property and any equality involving two linear combinations of two any length products having orthogonal projectors P₁ and P₂ as their factors occurring alternately is investigated. The second problem discussed in this paper concerns specific generalized inverses of the sum P₁ + P₂ and the difference P₁ − P₂ of (not necessary orthogonal) commuting projectors P₁ and P₂. The results obtained supplement those provided in Section 4 of Baksalary and Baksalary [J.K. Baksalary, O.M. Baksalary, Commutativity of projectors, Linear Algebra Appl. 341 (2002) 129-142].     

The nullity and rank of linear combinations of idempotent matrices

Baksalary and Baksalary [J.K. Baksalary, O.M. Baksalary, Nonsingularity of linear combinations of idempotent matrices, Linear Algebra Appl. 388 (2004) 25-29] proved that the nonsingularity of P₁ + P₂, where P₁ and P₂ are idempotent matrices, is equivalent to the nonsingularity of any linear combinations c₁P₁ + c₂P₂, where c₁, c₂ ≠ 0 and c₁ + c₂ ≠ 0. In the present note this result is strengthened by showing that the nullity and rank of c₁P₁ + c₂P₂ are constant. Furthermore, a simple proof of the rank formula of Groß and Trenkler [J. Groß, G. Trenkler, Nonsingularity of the difference of two oblique projectors, SIAM J. Matrix Anal. Appl. 21 (1999) 390-395] is obtained.     

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.     

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.     

Surjections on Grassmannians preserving pairs of elements with bounded distance

Let m and k be two fixed positive integers such that m>k?2. Let V be a left vector space over a division ring with dimension at least m+k+1. Let G_m(V) be the Grassmannian consisting of all m-dimensional subspaces of V. We characterize surjective mappings T from G_m(V) onto itself such that for any A,B in G_m(V), the distance between A and B is not greater than k if and only if the distance between T(A) and T(B) is not greater than k.     

Universality in complex Wishart ensembles for general covariance matrices with 2 distinct eigenvalues

We considered N×N Wishart ensembles in the class W_C(Σ_N,M) (complex Wishart matrices with M degrees of freedom and covariance matrix Σ_N) such that N₀ eigenvalues of Σ_N are 1 and N₁=N−N₀ of them are a. We studied the limit as M, N, N₀ and N₁ all go to infinity such that , and 0<c,β<1. In this case, the limiting eigenvalue density can either be supported on 1 or 2 disjoint intervals in R₊, and a phase transition occurs when the support changes from 1 interval to 2 intervals. By using the Riemann-Hilbert analysis, we have shown that when the phase transition occurs, the eigenvalue distribution is described by the Pearcey kernel near the critical point where the support splits.     

On orthogonal systems of matrix algebras

In this work it is shown that certain interesting types of orthogonal system of subalgebras (whose existence cannot be ruled out by the trivial necessary conditions) cannot exist. In particular, it is proved that there is no orthogonal decomposition of M_n(C)⊗M_n(C)≡M_n²(C) into a number of maximal abelian subalgebras and factors isomorphic to M_n(C) in which the number of factors would be 1 or 3.In addition, some new tools are introduced, too: for example, a quantity c(A,B), which measures “how close” the subalgebras A,B⊂M_n(C) are to being orthogonal. It is shown that in the main cases of interest, c(A^′,B^′) - where A^′ and B^′ are the commutants of A and B, respectively - can be determined by c(A,B) and the dimensions of A and B. The corresponding formula is used to find some further obstructions regarding orthogonal systems.