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.     

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.     

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.     

Semidefinite descriptions of low-dimensional separable matrix cones

Let K⊂E, K^′⊂E^′ be convex cones residing in finite-dimensional real vector spaces. An element y in the tensor product E⊗E^′ is K⊗K^′-separable if it can be represented as finite sum , where x_l∈K and for all l. Let S(n), H(n), Q(n) be the spaces of n×n real symmetric, complex Hermitian and quaternionic Hermitian matrices, respectively. Let further S₊(n), H₊(n), Q₊(n) be the cones of positive semidefinite matrices in these spaces. If a matrix A∈H(mn)=H(m)⊗H(n) is H₊(m)⊗H₊(n)-separable, then it fulfills also the so-called PPT condition, i.e. it is positive semidefinite and has a positive semidefinite partial transpose. The same implication holds for matrices in the spaces S(m)⊗S(n), H(m)⊗S(n), and for m?2 in the space Q(m)⊗S(n). We provide a complete enumeration of all pairs (n,m) when the inverse implication is also true for each of the above spaces, i.e. the PPT condition is sufficient for separability. We also show that a matrix in Q(n)⊗S(2) is Q₊(n)⊗S₊(2)- separable if and only if it is positive semidefinite.     

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).     

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.     

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.     

An upper bound for the minimum rank of a graph

For a graph G of order n, the minimum rank of G is defined to be the smallest possible rank over all real symmetric n×n matrices A whose (i,j)th entry (for i≠j) is nonzero whenever {i,j} is an edge in G and is zero otherwise. We prove an upper bound for minimum rank in terms of minimum degree of a vertex is valid for many graphs, including all bipartite graphs, and conjecture this bound is true over for all graphs, and prove a related bound for all zero-nonzero patterns of (not necessarily symmetric) matrices. Most of the results are valid for matrices over any infinite field, but need not be true for matrices over finite fields.     

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 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)     

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.     

Strong linear preservers of rank reverse permutability on triangular matrices

Let F be a field with ∣F∣ > 2 and T_n(F) be the set of all n × n upper triangular matrices, where n ? 2. Let k ? 2 be a given integer. A k-tuple of matrices A₁, …, A_k ∈ T_n(F) is called rank reverse permutable if rank(A₁ A₂ ? A_k) = rank(A_k A_k−1 ? A₁). We characterize the linear maps on T_n(F) that strongly preserve the set of rank reverse permutable matrix k-tuples.     

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.     

A linear-algebra problem from algebraic coding theory

Given an m×n matrix M over E=GF(q^t) and an ordered basis A={z₁,…,z_t} for field E over K=GF(q), expand each entry of M into a t×1 vector of coordinates of this entry relative to A to obtain an mt×n matrix M¹ with entries from the field K. Let r=rank(M) and r¹=rank(M¹). We show that r?r¹?min{rt,n}, and we determine the number b(m,n,r,r¹,q,t) of m×n matrices M of rank r over GF(q^t) with associated mt×n matrix M¹ of rank r¹ over GF (q).     

Rings of matrix invariants in positive characteristic

Denote by R_n,m the ring of invariants of m-tuples of n×n matrices (m,n?2) over an infinite base field K under the simultaneous conjugation action of the general linear group. When char(K)=0, Razmyslov (Izv. Akad. Nauk SSSR Ser. Mat. 38 (1974) 723) and Procesi (Adv. Math. 19 (1976) 306) established a connection between the Nagata-Higman theorem and the degree bound for generators of R_n,m. We extend this relationship to the case when the base field has positive characteristic. In particular, we show that if 0<char(K))?n, then R_n,m is not generated by its elements whose degree is smaller than m. A minimal system of generators of R_2,m is determined for the case char(K)=2: it consists of 2^m+m−1 elements, and the maximum of their degrees is m. We deduce a consequence indicating that the theory of vector invariants of the special orthogonal group in characteristic 2 is not analogous to the case char(K)≠2. We prove that the characterization of the R_n,m that are complete intersections, known before when char(K)=0, is valid for any infinite K. We give a Cohen-Macaulay presentation of R_2,4, and analyze the difference between the cases char(K)=2 and char(K)≠2.     

Rank-one nonincreasing mappings on triangular matrices and some related preserver problems

Let T_n (F) be the algebra of all n×n upper triangular matrices over an arbitrary field F. We first characterize those rank-one nonincreasing mappings ψ: T_n (F)→T_m (F)n?m such that ψ(I_n ) is of rank n. We next deduce from this result certain types of singular rank-one r-potent preservers and nonzero r-potent preservers on T_n (F). Characterizations of certain classes of homomorphisms and semi-homomorphisms on T_n (F) are also given.     

Absolutely indecomposable symmetric matrices

Let A be a symmetric matrix of size n×n with entries in some (commutative) field K. We study the possibility of decomposing A into two blocks by conjugation by an orthogonal matrix T∈Mat_n(K). We say that A is absolutely indecomposable if it is indecomposable over every extension of the base field. If K is formally real then every symmetric matrix A diagonalizes orthogonally over the real closure of K. Assume that K is a not formally real and of level s. We prove that in Mat_n(K) there exist symmetric, absolutely indecomposable matrices iff n is congruent to 0, 1 or −1 modulo 2s.     

The maximum dimension of a subspace of nilpotent matrices of index 2

A matrix M is nilpotent of index 2 if M²=0. Let V be a space of nilpotent n×n matrices of index 2 over a field k where and suppose that r is the maximum rank of any matrix in V. The object of this paper is to give an elementary proof of the fact that . We show that the inequality is sharp and construct all such subspaces of maximum dimension. We use the result to find the maximum dimension of spaces of anti-commuting matrices and zero subalgebras of special Jordan Algebras.