Frieder Knüppel

Products of Involutions in O + (V)

Abstract Let V be a finite-dimensional vector space over a commutative field of characteristic distinct from 2. Let V carry a symmetric nondegenerate bilinear form. The special orthogonal group is O + ( V ):={ πϵO ( V ): det π = 1}. Main result: An element π ∈ O + ( V ) is a product of two involutions in O + ( V ) ifand only if dim V ≢ 2 mod 4 or an orthogonal decomposition of V into orthogonally indecomposable π-modules contains a π-module of odd dimension.

On products of two involutions in the orthogonal group of a vector space

Let V be a finite-dimentional vector space over a commutative field of characteristic distinct from 2. Let V carry a symmetric nondegenerate bilinear form. Results: (A) Let π = ρσ, where π, ρ, σ ∈ O(V) and ρ, σ are involutions. There exists an orthogonal decomposition of V into orthogonally indecomposable π-modules which are simultaneously invariant under ρ and σ. (B) Let π ∈ O(V).One can find involutions ρ, σ ∈ O(V) such that π = ρσ and B(π) = B(ρ) + B(σ) holds if and only if an orthogonal decomposition of V into orthogonally indecomposable π-modules does not contain a term whose minimum polynomial is (x−1)α where α is even.

Products of symmetries in unitary groups

Abstract Given a regular − -hermitian form on a finite-dimensional vector space V over a commutative field K of characteristic ≠2 such that the norm on K is surjective onto the fixed field of − (this is true whenever K is finite). Call an element σ of the unitary group a symmetry if σ 2 = 1 and the negative space of σ is 1-dimensional. If π is unitary and det π ∈ {1, − 1}, we prove that π is a product of symmetries (with a few exceptions when K = GF 9 and dim V = 2) and we find the minimal number of factors in such a product.

k-fold anti-invariant subspaces of a linear mapping

Abstract Let π be a linear bijection on a finite-dimensional vector space and k ⩾0 an integer. A subspace T is called a k -fold anti-invariant subspace if T + Tπ +⋯+ Tπ k = T ⊕ Tπ ⊕⋯⊕ Tπ k . We find the maximal dimension of such subspaces T .

The product of two quadratic matrices

Abstract Let p=(x−β)(x−β −1 )∈K[x] where β 2 ≠β −2 and let V be a finite-dimensional vector space over the field K . A linear mapping M:V→V is called quadratic if p(M)=0 . We characterize products of two quadratic linear mappings.

PRODUCTS OF SIMILAR MATRICES

Abstract Given a nontrivial conjugacy class Ω of GL(V) which is contained in SL(V), we want to find a minimal integer k such that every element of SL(V) is a product of at most k elements of Ω. The analogous question is studied when Ω is replaced by Ф · Ф −1 , where Ф is a nontrivial conjugacy class of GL(V).

Construction of Symmetric and Symplectic Bilinear Forms

Let V be a finite-dimensional vector-space. A linear mapping ϕ on V is called simple if V(ϕ - 1) is 1-dimensional. Let S be a set of simple bijections on V. We discuss conditions entraining that each element of S is orthogonal (respectively symplectic) under an appropriate symmetric (respectively symplectic) bilinear form on V.

Products of Quasi-Involutions in Unitary Groups

AbstractGiven a regular –-hermitian form on an n-dimensional vector space V over a commutative field K of characteristic ≠ 2 (

The extended covering number of SLn is n + 1

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