Related papers: Two linear transformations each tridiagonal with r…

In this paper we further develop the connection between tridiagonal pairs and the q-tetrahedron algebra $\boxtimes_q$. Let V denote a finite dimensional vector space over an algebraically closed field and let A, A^* denote a tridiagonal…

A commutative associative algebra A with an identity over the field of real numbers which has a basis, where all elements are invertible, is considered in the work. Moreover, among matrixes consisting of the structure constants of A, there…

Let $\mathfrak{d}$ be the Lie superalgebra of superderivations of the sheaf of sections of the exterior algebra of the homogeneous vector bundle $E$ over the flag variety $G/P$, where $G$ is a simple finite-dimensional complex Lie group and…

Let $V$ be a finite dimensional vector space over a field $\mathrm{k}$ of characteristic $0$. Let $A$ be a linear mapping of $V$ into itself. This paper gives a normal form for $A$, which gives a better description of the structure of $A$…

We study the structure of certain modules $V$ over linear spaces $W$ with restrictions neither on the dimensions nor on the base field $\mathbb F$. A basis $\mathfrak B = \{v_i\}_{i\in I}$ of $V$ is called multiplicative respect to the…

We study orthogonal decompositions of symmetric and ordinary tensors using methods from linear algebra. For the field of real numbers we show that the sets of decomposable tensors can be defined be equations of degree 2. This gives a new…

Complex Lie point transformations are used to linearize a class of systems of second order ordinary differential equations (ODEs) which have Lie algebras of maximum dimension $d$, with $d\leq 4$. We identify such a class by employing…

In this paper we obtain the LU-decomposition of a noncommutative linear system of equations that, in the rank one case, characterizes the image of the Lepowsky homomorphism $U(\lieg)^{K}\to U(\liek)^{M}\otimes U(\liea)$. This…

In the context of operator valued W*-free probability theory, we study Haar unitaries, R-diagonal elements and circular elements. Several classes of Haar unitaries are differentiated from each other. The term bipolar decomposition is used…

Let $V$ be a vector space over a field $F$, $V^*$ its dual space and $L(V)$ the algebra of all linear operators on $V$. For an operator $a\in L(V)$ let $a*$ be its adjoint acting on $V*$, and for a subset $R$ of $L(V)$ let $R"$ be its…

A linear relation, i.e., a multivalued operator $T$ from a Hilbert space ${\mathfrak H}$ to a Hilbert space ${\mathfrak K}$ has Lebesgue type decompositions $T=T_{1}+T_{2}$, where $T_{1}$ is a closable operator and $T_{2}$ is an operator or…

It is well known that a commuting family of diagonalizable linear operators on a finite dimensional vector space is simultaneously diagonalizable. In this paper, we consider a family A of anti-commuting (complex) linear operators on a…

We examine the reduction process of a system of second-order ordinary differential equations which is invariant under a Lie group action. With the aid of connection theory, we explain why the associated vector field decomposes in three…

We say that a linear space is harmonious if it is resolvable and admits an automorphism group acting sharply transitively on the points and transitively on the parallel classes. Generalizing old results by the first author et al. we present…

Given an arbitrary field K and non-zero scalars a and b, we give necessary and sufficient conditions for a matrix A in M_n(K) to be a linear combination of two idempotents with coefficients a and b. This extends results previously obtained…

An alternative proof of Lie's approach for linearization of scalar second order ODEs is derived using the relationship between $\lambda$-symmetries and first integrals. This relation further leads to a new $\lambda$-symmetry linearization…

We give examples of stable rank 2 vector bundles on principally polarized abelian threefolds, and study their deformations. The starting point is the Serre construction, which gives a source of examples, and which we rephrase in terms of…

A square matrix is called {\it Hessenberg} whenever each entry below the subdiagonal is zero and each entry on the subdiagonal is nonzero. Let $V$ denote a nonzero finite-dimensional vector space over a field $\fld$. We consider an ordered…

V.I. Arnold [Russian Math. Surveys 26(2) (1971) 29-43] constructed a miniversal deformation of a square complex matrix under similarity; that is, a simple normal form to which not only a given square matrix A but all matrices B close to it…

Let $U$ and $V$ be finite-dimensional vector spaces over a (commutative) field $\mathbb{K}$, and $\mathcal{S}$ be a linear subspace of the space $\mathcal{L}(U,V)$ of all linear operators from $U$ to $V$. A map $F : \mathcal{S} \rightarrow…