Some Conditions for Matrices over an Incline To Be Invertible and General Linear Group on an Incline

Inclines are the additively idempotent semirings in which products are less than or equal to either factor. In this paper, some necessary and sufficient conditions for a matrix over L to be invertible are given, where L is an incline with 0 and 1. Also it is proved that L is an integral incline if and only if GLn（L） = PLn （L） for any n （n 〉 2）, in which GLn（L） is the group of all n × n invertible matrices over L and PLn（L） is the group of all n × n permutation matrices over L. These results should be regarded as the generalizations and developments of the previous results on the invertible matrices over a distributive lattice.     

The Semigroup of Circulant Matrices Over a Lattice

Let C_n(L) denote the set of all n × n circulant matrices over a distributive lattice L. Then C_n(L) forms a semigroup under the usual matrix product. In this paper, we shall characterize all idempotents in C_n(L), and also estabish the Euler-Fermat theorem for the semigroup C_n(L).AMS Subject Classification (2000): 20MSupported by the Educational Committee of Fujian, China.     

PARTIAL FERRERS MATRICES

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Nonlinear Lie derivations on upper triangular matrices

Let M _n (𝔸) and T _n (𝔸) be the algebra of all n?×?n matrices and the algebra of all n?×?n upper triangular matrices over a commutative unital algebra 𝔸, respectively. In this note we prove that every nonlinear Lie derivation from T _n (𝔸) into M _n (𝔸) is of the form A?→?AT???TA?+?A _??+?ξ(A)I _n , where T?∈?M _n (𝔸), ??:?𝔸?→?𝔸 is an additive derivation, ξ?:?T _n (𝔸)?→?𝔸 is a nonlinear map with ξ(AB???BA)?=?0 for all A,?B?∈?T _n (𝔸) and A _? is the image of A under???applied entrywise.     

On derivations arising in differential equations

Let L be a linear transformation on the set of all n×n matrices over an algebraically closed field of characteristic 0. It is shown that if AB=BA implies L(A)L(B)=L(B)L(A) and if either L is nonsingular or the implication in the hypothesis can also be reversed, then L is a sum of a scalar multiple of a similarity transformation and a linear functional times the identity transformation.     

Idempotent matrix lattices over distributive lattices

In this paper, the partially ordered set of idempotent matrices over distributive lattices with the partial order induced by a set of lattice matrices is studied. It is proved that this set is a lattice; the formulas for meet and join calculation are obtained. In the lattice of idempotent matrices over a finite distributive lattice, all atoms and coatoms are described. We prove that the lattice of quasi-orders over an n-element set Qord(n) is not graduated for n ≥ 3 and calculate the greatest and least lengths of maximal chains in this lattice. We also prove that the interval ([I, J]_≤, ≤) of idempotent (n × n)-matrices over {ie879-01}-lattices is isomorphic to the lattice of quasi-orders Qord(n). Using this isomorphism, we calculate the lattice height of idempotent {ie879-02}-matrices. We obtain a structural criterion of idempotent matrices over distributive lattices. __________ Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 13, No. 4, pp. 121–144, 2007.     

Lattice-Ordered Matrix Algebras Over Real GCD-Domains

Let R ? ? be a GCD-domain. In this article, Weinberg's conjecture on the n × n matrix algebra M _n (R) (n ≥ 2) is proved. Moreover, all the lattice orders (up to isomorphisms) on a full 2 × 2 matrix algebra over R are obtained.     

Similarity of matrices under SL(n,K)

Take a similarity class of n × n matrices over a field K Let p_i ,(λ) ^m(i) be the elementary divisors L_i , = K [λ]/(p_i ). Under conjugation by SL(n, K), the class splits into subclasses corresponding to the elements of K^×/Π(NL _i ^×) ^m(i).     

On the Multiplicative Monoid of n×n Matrices Over Artinian Chain Rings

Let R be an Artinian chain ring with a principal maximal ideal. We investigate properties of matrices over R and give matrix representations of R-submodules of R ⁿ first, then consider Green's relations, Green's relation equivalent classes, Schützenberger groups of 𝒟-classes, principal factors, and group ?-classes of the multiplicative monoid M _n (R) of n × n matrices over R. Furthermore, we show that M _n (R) is an eventually regular semigroup and derive basic numerical information of M _n (R) when R is finite.     

Pure Submodules of Finite Direct Sums of Co-Purely Indecomposable Modules

A torsion-free module M of finite rank over a discrete valuation ring R with prime p is co-purely indecomposable if M is indecomposable and rank M = 1 + dim _R/pR (M/pM). Co-purely indecomposable modules are duals of pure finite rank submodules of the p-adic completion of R. Pure submodules of cpi-decomposable modules (finite direct sums of co-purely indecomposable modules) are characterized. Included are various examples and properties of these modules.     

Large spaces of symmetric matrices of bounded rank are decomposable ∗

Let k and n be positive integers such that kn. Let S_n (F) denote the space of all n×n symmetric matrices over the field F with char F≠2. A subspace L of S_n (F) is said to be a k-subspace if rank Ak for every A?L.

Now suppose that k is even, and write k=2r. We say a k∥-subspace of S_n (F) is decomposable if there exists in Fⁿ a subspace W of dimension n?r such that x^tAx=0 for every x?W A?L.

We show here, under some mild assumptions on k n and F, that every k∥-subspace of S_n (F) of sufficiently large dimension must be decomposable. This is an analogue of a result obtained by Atkinson and Lloyd for corresponding subspaces of F^m,n .     

Canonical form with respect to semiscalar equivalence for a matrix pencil with nonsingular first matrix

Polynomial n × n matrices A(x) and B(x) over a field \mathbbF \mathbb{F} are called semiscalar equivalent if there exist a nonsingular n × n matrix P over \mathbbF \mathbb{F} and an invertible n × n matrix Q(x) over \mathbbF \mathbb{F} [x] such that A(x) = PB(x)Q(x). We give a canonical form with respect to semiscalar equivalence for a matrix pencil A(x) = A _0x - A ₁, where A ₀ and A ₁ are n × n matrices over \mathbbF \mathbb{F} , and A ₀ is nonsingular.     

The group-theoretic complexity of subsemigroups of boolean matrices

We find the group-theoretic complexity of many subsemigroups of the semigroup B_n of n × n Boolean matrices, including Hall matrices, reflexive matrices, fully indecomposable matrices, upper triangular matrices, row-rank-n matrices, and others.     

Noncirculant Toeplitz matrices all of whose powers are Toeplitz

Let a, b and c be fixed complex numbers. Let M _n (a, b, c) be the n × n Toeplitz matrix all of whose entries above the diagonal are a, all of whose entries below the diagonal are b, and all of whose entries on the diagonal are c. For 1 ⩽ k ⩽ n, each k × k principal minor of M _n (a, b, c) has the same value. We find explicit and recursive formulae for the principal minors and the characteristic polynomial of M _n (a, b, c). We also show that all complex polynomials in M _n (a, b, c) are Toeplitz matrices. In particular, the inverse of M _n (a, b, c) is a Toeplitz matrix when it exists.     

A result concerning nilpotent values with generalized skew derivations on Lie ideals

ABSTRACT

Let n≥1 be a fixed integer, R a prime ring with its right Martindale quotient ring Q, C the extended centroid, and L a non-central Lie ideal of R. If F is a generalized skew derivation of R such that (F(x)F(y)?yx)ⁿ = 0 for all x,y∈L, then char(R) = 2 and R?M₂(C), the ring of 2×2 matrices over C.     

Linear operators preserving unitarily invariant norms of matrices

Let F _{m × n} be the set of all m × n matrices over the field F = C or R Denote by U_n (F) the group of all n × n unitary or orthogonal matrices according as F = C or F-R. A norm N() on F m ×n, is unitarily invariant if N(UAV) = N(A): for all A ∈ F _m×n U ∈ U _m (F). and V ∈ U_n (F). We characterize those linear operators T F _{m × n} → F _{m × n} which satisfy N (T(A)) = N(A)for all A ∈ F _{m × n}

for a given unitarily invariant norm N(). It is shown that the problem is equivalent to characterizing those operators which preserve certain subsets in F _{m × n} To develop the theory we prove some results concerning unitary operators on F _{m × n} which are of independent interest.     

Identities of the Semigroup of Upper Triangular Tropical Matrices

A new family of identities satisfied by the semigroups U _n (𝕋) of n × n upper triangular tropical matrices is constructed and an elementary proof is given.     

Linear spaces and preservers of symmetric matrices of bounded rank-two

Let 𝔽 be a field of characteristic two. Let S _n (𝔽) denote the vector space of all n?×?n symmetric matrices over 𝔽. We characterize i. subspaces of S _n (𝔽) all whose elements have rank at most two where n???3,

ii. linear maps from S _m (𝔽) to S _n (𝔽) that sends matrices of rank at most two into matrices of rank at most two where m, n???3 and |𝔽|?≠?2.

     

Adjacency preserving bijective maps on triangular matrices over any division ring

Let 𝕋 _n (D) be the set of n × n upper triangular matrices over a division ring D. We characterize the adjacency preserving bijective maps in both directions on 𝕋 _n (D) (n ≥ 3). As applications, we describe the ring semi-automorphisms and the Jordan automorphisms on upper triangular matrices over a simple Artinian ring.