Invertible Linear Map Research Articles

Polynomial maps with invertible sums of Jacobian matrices and directional derivatives

Let F:Cn→Cm be a polynomial map with degF=d≥2. We prove that F is invertible if m=n and ∑i=1d−1(JF)|αi is invertible for all αi∈Cn, which is trivially the case for invertible quadratic maps.More generally, we prove that for affine lines L={β+μγ∣μ∈C}⊆Cn (γ≠0), F∣L is linearly rectifiable, if and only if ∑i=1d−1(JF)|αi⋅γ≠0 for all αi∈L. This appears to be the case for all affine lines L when F is injective and d≤3.We also prove that if m=n and ∑i=1n(JF)|αi is invertible for all αi∈Cn, then F is a composition of an invertible linear map and an invertible polynomial map X+H with linear part X, such that the subspace generated by {(JH)|α∣α∈Cn} consists of nilpotent matrices.

Triple automorphisms of simple Lie algebras

An invertible linear map φ on a Lie algebra L is called a triple automorphism of it if φ([x, [y, z]]) = [φ(x), [φ(y), φ(z)]] for ∀ x, y, z ∈ L. Let g be a finite-dimensional simple Lie algebra of rank l defined over an algebraically closed field F of characteristic zero, p an arbitrary parabolic subalgebra of g. It is shown in this paper that an invertible linear map φ on p is a triple automorphism if and only if either φ itself is an automorphism of p or it is the composition of an automorphism of p and an extremal map of order 2.

Square-zero determined matrix algebras

Let N n (R) be the algebra of all n × n strictly upper triangular matrices over a commutative unital ring R. It is shown in this article that N n (R) is square-zero determined. More definitely, if a symmetric bilinear map φ from N n (R) × N n (R) to an R-module V satisfies the condition that φ(u, u) = 0 whenever u 2 = 0, then there exists a linear map ϕ from to V such that φ(x, y) = ϕ(xy + yx) for all x, y in N n (R). As applications of this result, we show that (i) a linear map on N n (R) is a square-zero derivation if and only if it is a quasi Jordan derivation; (ii) an invertible linear map on N n (R) is a square-zero preserving map if and only if it is a quasi Jordan automorphism.

Maps determined by action on identity-product elements

Let M n ( R ) be the algebra of all n × n matrices over a unital commutative ring R with 6 invertible. For a given element z ∈ M n ( R ) , a map δ on M n ( R ) is called preserving z -product if δ ( x ) δ ( y ) = δ ( z ) whenever xy = z . A map σ on M n ( R ) is called derivable at the given point z if σ ( x ) y + x σ ( y ) = σ ( z ) whenever xy = z . Using elementary matrix technique we show that if an invertible linear map δ on M n ( R ) preserves identity-product, then it is a Jordan automorphism; and a linear map σ on M n ( R ) is derivable at the identity matrix if and only if it is an inner derivation.

On bilinear maps determined by rank one matrices with some applications

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 ϕ from Mn×Mn to Mn satisfies the condition that ϕ(u,v)=ϕ(I,A) whenever u·v=A, then there exists a linear map φ from Mn to Mn such that ϕ(x,y)=φ(x·y),∀x,y∈Mn. If ϕ is further assumed to be symmetric then there exists a matrix B such that ϕ(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.

On the existence of an invariant non-degenerate bilinear form under a linear map

Let V be a vector space over a field F. Assume that the characteristic of F is large, i.e. char(F)>dimV. Let T:V→V be an invertible linear map. We answer the following question in this paper. When doesVadmit a T-invariant non-degenerate symmetric (resp. skew-symmetric) bilinear form? We also answer the infinitesimal version of this question.Following Feit and Zuckerman 2, an element g in a group G is called real if it is conjugate in G to its own inverse. So it is important to characterize real elements in GL(V,F). As a consequence of the answers to the above question, we offer a characterization of the real elements in GL(V,F).Suppose V is equipped with a non-degenerate symmetric (resp. skew-symmetric) bilinear form B. Let S be an element in the isometry group I(V,B). A non-degenerate S-invariant subspace W of (V,B) is called orthogonally indecomposable with respect to S if it is not an orthogonal sum of proper S-invariant subspaces. We classify the orthogonally indecomposable subspaces. This problem is non-trivial for the unipotent elements in I(V,B). The level of a unipotent T is the least integer k such that (T-I)k=0. We also classify the levels of unipotents in I(V,B).

Features analysis of a parametric PWL chaotic map and its utilization for secure transmissions

In this paper a complete analysis of a chaotic function is provided. The chaotic map is a parametric one-dimensional non invertible piecewise linear (PWL) map which is used to generate binary sequences. Implementation, key management and security associated to this chaotic generator are discussed. Then, we propose to use this chaotic map and its sequences in a communication system. The proposed communication system relies on the CSK principle and allows to transmit a binary information by switching between orthogonal binary sequences. The system is then analyzed by providing the bit-error rates derivation both theoretically and by computer simulations.

Invertible linear maps preserving {1}-inverses of matrices over PID

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 ∃ GLn(R) such thatf(A) =PAP−1 for allA ∃ Mn(R), (ii) there exists a matrixP ∃ GLn(R) such thatf(A) =PAt P−1 forA ∃ Mn(R).

The Jacobian Conjecture: Linear triangularization for homogeneous polynomial maps in dimension three

Let k be a field of characteristic zero and F : k 3 → k 3 a polynomial map of the form F = x + H , where H is homogeneous of degree d ⩾ 2 . We show that the Jacobian Conjecture is true for such mappings. More precisely, we show that if JH is nilpotent there exists an invertible linear map T such that T −1 H T = ( 0 , h 2 ( x 1 ) , h 3 ( x 1 , x 2 ) ) , where the h i are homogeneous of degree d. As a consequence of this result, we show that all generalized Drużkowski mappings F = x + H = ( x 1 + L 1 d , … , x n + L n d ) , where L i are linear forms for all i and d ⩾ 2 , are linearly triangularizable if JH is nilpotent and rk J H ⩽ 3 .

Riesz basis of Coifman and Meyer's local sine and cosine type

A Riesz basis is the image of an orthonormal basis under an invertible continuous linear mapping. In many interesting applications, perturbing an orthonormal basis in a controlled manner yields a Riesz basis. In this paper we find Riesz basis for L 2( R) of the form { b I k ( x)sin λ k x}, with b I k ( x) bell functions, by perturbing the local sine and cosine orthonormal bases of Coifman and Meyer.

Invertible preservers and algebraic groups III: preservers of unitary similarity (congruence) invariants and overgroups of some unitary subgroups*

Let V = . We consider two groups of unitary operators on V, namely acting on V by conjugation, and acting on V via U:A→UAU 1. We study the overgroups of these unitary groups in GL(V). The results are then applied to classify the invertible linear maps on V and various subspaces of V which preserve some unitary similarity invariant or unitary congruence invariant property. This provides a unified proof for various existing results, and answers a number of open questions on linear preserver problems.

Invertible Preservers and Algebraic Groups II: Preservers of Similarity Invariants and Overgroups of PSLn (F)

Let F be an algebraically closed field. Identify PSLn (F) as a subgroup of SLn2 (F) via the conjugation representation on Mn (F). We find all overgroups of PSLn (F). This was studied by Doković and Platonov in characteristic zero. The results are then applied to classify the invertible linear maps on Mn (F)or sln (F) which preserve some similarity invariant property. This provides a unified proof for various existing results and yields some new results. We consider similar questions for actions on symmetric and exterior spaces.

Spectral Integration of Marcinkiewicz Multipliers

AbstractLet X be a closed subspace of LP(μ), where μ is an arbitrary measure and 1 < p < ∞. By extending the scope of spectral integration, we show that every invertible power-bounded linear mapping of X into X has a functional calculus implemented by the algebra of complex-valued functions on the unit circle satisfying the hypotheses of the Strong Marcinkiewicz Multiplier Theorem. This result expands the framework of the Strong Marcinkiewicz Multiplier Theorem to the setting of abstract measure spaces.

The involutive representation of solutions of the coupled KdV equation

In the present paper, the Lax representations of the equations of the coupled KdV type are obtained. By making use of a constrained relation between the potential and the eigenfunctions of the spectral problem, the Lax representations are nonlinearized. If one introduces a suitable symplectic structure on a complex space by means of an invertible linear mapping from the real to the complex space, then the nonlinearized Lax pairs become exactly the commutable flows of a finite-dimensional completely iterable system in the Liouville sense. Moreover, the representations of solutions of the equations of the coupled KdV type are given.

Linear maps on algebraic groups

Let G be an irreducible simple classical algebraic subgroup of SL ( n , C ), the n × n matrices of determinant one. We discuss the problem of classifying all invertible linear maps T on M ( n , C ) such that T ( G )⊆ G . This has already been solved with two general exceptions. We show how to eliminate one of these two cases. In general, T must have the form T(X)=PXQ or T(X)=PX t Q for suitable invertible matrices P and Q .

Linear maps which preserve a balanced nonsingular inertia class

Let n be an even integer such that n ⩾ 4. Let T be an invertible linear map on the space of n × n real symmetric matrices which maps the set of matrices having inertia( n 2 , n 2 , 0) into itself. Then there exist a nonsingular matrix S and ϵ = ± 1 such that T( A) = ϵS t AS. This is an analogue of a result obtained for Hermitian matrices by Pierce and Rodman.