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1.
Clément de Seguins Pazzis 《Linear algebra and its applications》2011,435(11):2708-2721
Let (K) be a field. Given an arbitrary linear subspace V of Mn(K) of codimension less than n-1, a classical result states that V generates the (K)-algebra Mn(K). Here, we strengthen this statement in three ways: we show that Mn(K) is spanned by the products of the form AB with (A,B)∈V2; we prove that every matrix in Mn(K) can be decomposed into a product of matrices of V; finally, when V is a linear perplane of Mn(K) and n>2, we show that every matrix in Mn(K) is a product of two elements of V. 相似文献
2.
Let K be a field and let Mm×n(K) denote the space of m×n matrices over K. We investigate properties of a subspace M of Mm×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 Mm×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. 相似文献
3.
Let Mn(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 Mn(R)×Mn(R) to V satisfies the condition that {u,u}={e,u} whenever u2=u, then there exists a linear map f from Mn(R) to V such that . Applying the main result we prove that an invertible linear transformation θ on Mn(R) preserves idempotent matrices if and only if it is a Jordan automorphism, and a linear transformation δ on Mn(R) is a Jordan derivation if and only if it is Jordan derivable at all idempotent points. 相似文献
4.
Let V be a 6-dimensional vector space over a field F, let f be a nondegenerate alternating bilinear form on V and let Sp(V,f)≅Sp6(F) denote the symplectic group associated with (V,f). The group GL(V) has a natural action on the third exterior power ?3V of V and this action defines five families of nonzero trivectors of V (four of whose are orbits for any choice of F). In this paper, we divide three of these five families into orbits for the action of Sp(V,f)⊆GL(V) on ?3V. 相似文献
5.
Changjiang Bu 《Journal of Applied Mathematics and Computing》2006,22(3):255-265
Let R be a PID,chR = 2,n > 1, Mn(R) be then xn full matrix algebra over R.f denotes any invertible linear map preserving {1}-inverses from Mn(R) to itself. In this paper, we have proven thatf is an invertible linear map on Mn(R) preserving {1}-inverses if and only iff satisfies any one of the following two conditions: (i) there exists a matrixP ? GL n(R) such thatf(A) =PAP ?1 for allA ? M n(R), (ii) there exists a matrixP ? GL n(R) such thatf(A) =PA t P?1 forA ? M n(R). 相似文献
6.
Clément de Seguins Pazzis 《Linear algebra and its applications》2010,433(3):625-866
Given an arbitrary field K and non-zero scalars α and β, we give necessary and sufficient conditions for a matrix A∈Mn(K) to be a linear combination of two idempotents with coefficients α and β. This extends results previously obtained by Hartwig and Putcha in two ways: the field K considered here is arbitrary (possibly of characteristic 2), and the case α≠±β is taken into account. 相似文献
7.
Let F be a field and let m and n be integers with m,n?3. Let Mn denote the algebra of n×n matrices over F. In this note, we characterize mappings ψ:Mn→Mm 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∈Mn and α∈F with ψ(In)≠0.
- 2.
- ψ is surjective and ψ(adj(A-B))=adj(ψ(A)-ψ(B)) for every A,B∈Mn.
8.
Roland Hildebrand 《Linear algebra and its applications》2008,429(4):901-932
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 xl∈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. 相似文献
9.
10.
Let K be an algebraically closed field of characteristic 0 and let Mn(K), n?3, be the matrix ring over K . We will show that the image of any multilinear polynomial in four variables evaluated on Mn(K) contains all matrices of trace 0. 相似文献
11.
Let K be a (algebraically closed ) field. A morphism A ⟼ g
−1
Ag, where A ∈ M(n) and g ∈ GL(n), defines an action of a general linear group GL(n) on an n × n-matrix space M(n), referred to as an adjoint action. In correspondence with the adjoint action is the coaction α: K[M(n)] → K[M(n)] ⊗ K[GL(n)] of a Hopf algebra K[GL(n)] on a coordinate algebra K[M(n)] of an n × n-matrix space, dual to the conjugation morphism. Such is called an adjoint coaction. We give coinvariants of an adjoint coaction
for the case where K is a field of arbitrary characteristic and one of the following conditions is satisfied: (1) q is not a root of unity; (2) char K = 0 and q = ±1; (3) q is a primitive root of unity of odd degree. Also it is shown that under the conditions specified, the category of rational
GL
q
× GL
q
-modules is a highest weight category. 相似文献
12.
Toma? Kosem 《Linear algebra and its applications》2006,418(1):153-160
The conjecture posed by Aujla and Silva [J.S. Aujla, F.C. Silva, Weak majorization inequalities and convex functions, Linear Algebra Appl. 369 (2003) 217-233] is proved. It is shown that for any m-tuple of positive-semidefinite n × n complex matrices Aj and for any non-negative convex function f on [0, ∞) with f(0) = 0 the inequality ?f(A1) + f(A2) + ? + f(Am)? ? ? f(A1 + A2 + ? + Am)? holds for any unitarily invariant norm ? · ?. It is also proved that ?f(A1) + f(A2) + ? + f(Am)? ? f(?A1 + A2 + ? + Am?), where f is a non-negative concave function on [0, ∞) and ? · ? is normalized. 相似文献
13.
Risong Li 《Communications in Nonlinear Science & Numerical Simulation》2012,17(7):2815-2823
Let f : X → X be a continuous map of a compact metric space X. The map f induces in a natural way a map fM on the space M(X) of probability measures on X, and a transformation fK on the space K(X) of closed subsets of X. In this paper, we show that if (X, f) is a chain transitive system with shadowing property, then exactly one of the following two statements holds:
- (a)
- fn and (fK)n are syndetically sensitive for all n ? 1.
- (b)
- fn and (fK)n are equicontinuous for all n ? 1.
14.
Let Mn be the algebra of all n×n matrix over a field F, A a rank one matrix in Mn. In this article it is shown that if a bilinear map ? from Mn×Mn to Mn satisfies the condition that ?(u,v)=?(I,A) whenever u·v=A, then there exists a linear map φ from Mn to Mn 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∈Mn. Applying the main result we prove that if a linear map on Mn is desirable at a rank one matrix then it is a derivation, and if an invertible linear map on Mn is automorphisable at a rank one matrix then it is an automorphism. In other words, each rank one matrix in Mn is an all-desirable point and an all-automorphisable point, respectively. 相似文献
15.
16.
Let Mn be the semigroup of n×n complex matrices under the usual multiplication, and let S be different subgroups or semigroups in Mn including the (special) unitary group, (special) general linear group, the semigroups of matrices with bounded ranks. Suppose Λk(A) is the rank-k numerical range and rk(A) is the rank-k numerical radius of A∈Mn. Multiplicative maps ?:S→Mn satisfying rk(?(A))=rk(A) are characterized. From these results, one can deduce the structure of multiplicative preservers of Λk(A). 相似文献
17.
We show that every injective Jordan semi-triple map on the algebra Mn(F) of all n × n matrices with entries in a field F (i.e. a map Φ:Mn(F)→Mn(F) satisfying
Φ(ABA)=Φ(A)Φ(B)Φ(A) 相似文献
18.
Mihály Weiner 《Linear algebra and its applications》2010,433(3):520-533
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 Mn(C)⊗Mn(C)≡Mn2(C) into a number of maximal abelian subalgebras and factors isomorphic to Mn(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⊂Mn(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. 相似文献
19.
Clément de Seguins Pazzis 《Linear algebra and its applications》2011,435(1):147-2271
Let V be a linear subspace of Mn,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. 相似文献
20.
Janko Marovt 《Linear algebra and its applications》2010,432(6):1595-1607
Let D be an arbitrary division ring and Mn(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. 相似文献