Linear Transformations Preserving the Real Orthogonal Group

1975 ◽  
Vol 27 (3) ◽  
pp. 561-572 ◽  
Author(s):  
Albert Wei
Keyword(s):  
Vector Space ◽  
General Problem ◽  
Orthogonal Group ◽  
Linear Transformations ◽  
The Real

Let K be a field and Mn﹛K) denote the vector space of n X n matrices over K. Marcus [4] posed the following general problem: Let W be a subspace of Mn(K) and S a subset of W. Describe the set L(S, W) of all linear transformations T on W such that T(S) is contained in S.

Vertex Subgroups of Irreducible Representations of Solvable Groups

1971 ◽  
Vol 23 (1) ◽  
pp. 12-21
Author(s):  
J. Malzan
Keyword(s):  
Vector Space ◽  
Unit Sphere ◽  
Convex Subset ◽  
Orthogonal Group ◽  
Solvable Groups ◽  
Fundamental Region ◽  
Irreducible Representations ◽  
Theoretical Problem ◽  
Real Vector ◽  
The Real

If ρ(G) is a finite, real, orthogonal group of matrices acting on the real vector space V, then there is defined [5], by the action of ρ(G), a convex subset of the unit sphere in V called a fundamental region. When the unit sphere is covered by the images under ρ(G) of a fundamental region, we obtain a semi-regular figure.The group-theoretical problem in this kind of geometry is to find when the fundamental region is unique. In this paper we examine the subgroups, ρ(H), of ρ(G) with a view of finding what subspace, W of V consists of vectors held fixed by all the matrices of ρ(H). Any such subspace lies between two copies of a fundamental region and so contributes to a boundary of both. If enough of these boundaries might be found, the fundamental region would be completely described.

scholarly journals THE INVARIANTS FIELD OF SOME FINITE PROJECTIVE LINEAR GROUP ACTIONS

2011 ◽  
Vol 85 (1) ◽  
pp. 19-25
Author(s):  
YIN CHEN
Keyword(s):  
Finite Field ◽  
Projective Space ◽  
Vector Space ◽  
Orthogonal Group ◽  
Homogeneous Polynomial ◽  
Classical Group ◽  
Projective Group ◽  
Dimensional Vector ◽  
Dimensional Vector Space ◽  
Projective Linear Group

AbstractLet Fq be a finite field with q elements, V an n-dimensional vector space over Fq and 𝒱 the projective space associated to V. Let G≤GLn(Fq) be a classical group and PG be the corresponding projective group. In this note we prove that if Fq (V )G is purely transcendental over Fq with homogeneous polynomial generators, then Fq (𝒱)PG is also purely transcendental over Fq. We compute explicitly the generators of Fq (𝒱)PG when G is the symplectic, unitary or orthogonal group.

scholarly journals Products of two idempotent transformations over arbitrary sets and vector spaces

1998 ◽  
Vol 57 (1) ◽  
pp. 59-71 ◽  
Author(s):  
Rachel Thomas
Keyword(s):  
Vector Space ◽  
Transformation Semigroup ◽  
Vector Spaces ◽  
Endomorphism Monoid ◽  
Linear Transformations ◽  
Arbitrary Vector ◽  
Full Transformation ◽  
Full Transformation Semigroup

In this paper we consider the characterisation of those elements of a transformation semigroup S which are a product of two proper idempotents. We give a characterisation where S is the endomorphism monoid of a strong independence algebra A, and apply this to the cases where A is an arbitrary set and where A is an arbitrary vector space. The results emphasise the analogy between the idempotent generated subsemigroups of the full transformation semigroup of a set and of the semigroup of linear transformations from a vector space to itself.

scholarly journals Cross-Ratio in Real Vector Space

2019 ◽  
Vol 27 (1) ◽  
pp. 47-60
Author(s):  
Roland Coghetto
Keyword(s):  
Vector Space ◽  
Real Line ◽  
Cross Ratio ◽  
Real Vector Space ◽  
Vector Spaces ◽  
Real Vector ◽  
Complex Vector ◽  
The Real ◽  
Mizar Mathematical Library ◽  
The Cross

Summary Using Mizar [1], in the context of a real vector space, we introduce the concept of affine ratio of three aligned points (see [5]). It is also equivalent to the notion of “Mesure algèbrique”1, to the opposite of the notion of Teilverhältnis2 or to the opposite of the ordered length-ratio [9]. In the second part, we introduce the classic notion of “cross-ratio” of 4 points aligned in a real vector space. Finally, we show that if the real vector space is the real line, the notion corresponds to the classical notion3 [9]: The cross-ratio of a quadruple of distinct points on the real line with coordinates x1, x2, x3, x4 is given by: $$({x_1},{x_2};{x_3},{x_4}) = {{{x_3} - {x_1}} \over {{x_3} - {x_2}}}.{{{x_4} - {x_2}} \over {{x_4} - {x_1}}}$$ In the Mizar Mathematical Library, the vector spaces were first defined by Kusak, Leonczuk and Muzalewski in the article [6], while the actual real vector space was defined by Trybulec [10] and the complex vector space was defined by Endou [4]. Nakasho and Shidama have developed a solution to explore the notions introduced by different authors4 [7]. The definitions can be directly linked in the HTMLized version of the Mizar library5. The study of the cross-ratio will continue within the framework of the Klein- Beltrami model [2], [3]. For a generalized cross-ratio, see Papadopoulos [8].

scholarly journals Promotion and Evacuation

10.37236/75 ◽  
2009 ◽  
Vol 16 (2) ◽  
Author(s):  
Richard P. Stanley
Keyword(s):  
Vector Space ◽  
Hecke Algebra ◽  
Linear Extensions ◽  
Linear Transformations ◽  
Basic Properties ◽  
Finite Poset ◽  
Order Ideals ◽  
Maximal Chains

Promotion and evacuation are bijections on the set of linear extensions of a finite poset first defined by Schützenberger. This paper surveys the basic properties of these two operations and discusses some generalizations. Linear extensions of a finite poset $P$ may be regarded as maximal chains in the lattice $J(P)$ of order ideals of $P$. The generalizations concern permutations of the maximal chains of a wider class of posets, or more generally bijective linear transformations on the vector space with basis consisting of the maximal chains of any poset. When the poset is the lattice of subspaces of ${\Bbb F}_q^n$, then the results can be stated in terms of the expansion of certain Hecke algebra products.

Linear Transformations On Matrices: The Invariance of Generalized Permutation Matrices, I

1976 ◽  
Vol 28 (3) ◽  
pp. 455-472 ◽  
Author(s):  
Hock Ong ◽  
E. P. Botta
Keyword(s):  
Vector Space ◽  
Unitary Group ◽  
Linear Transformations ◽  
Permutation Matrices ◽  
Linear Maps

Let F be a field, Mn(F) be the vector space of all w-square matrices with entries in F and a subset of Mn(F). It is of interest to determine the structure of linear maps T : Mn(F) →Mn(F) such that . For example: Let be GL(n, C), the group of all nonsingular n X n matrices over C [5]; the subset of all rank 1 matrices in MmXn(F) [4] (MmXn(F) is the vector space of all m X n matrices over F) ; the unitary group [2] ; or the set of all matrices X in Mn(F) such that det(X) = 0 [1]. Other results in this direction can be found in [3].

THE REGULAR PART OF A SEMIGROUP OF LINEAR TRANSFORMATIONS WITH RESTRICTED RANGE

2017 ◽  
Vol 103 (3) ◽  
pp. 402-419 ◽  
Author(s):  
WORACHEAD SOMMANEE ◽  
KRITSADA SANGKHANAN
Keyword(s):  
Finite Field ◽  
Vector Space ◽  
Dimensional Subspace ◽  
Finite Dimensional Subspace ◽  
Restricted Range ◽  
Linear Transformations ◽  
Dimensional Vector Space ◽  
Finite Dimensional ◽  
Idempotent Rank ◽  
Image Position

Let$V$be a vector space and let$T(V)$denote the semigroup (under composition) of all linear transformations from$V$into$V$. For a fixed subspace$W$of$V$, let$T(V,W)$be the semigroup consisting of all linear transformations from$V$into$W$. In 2008, Sullivan [‘Semigroups of linear transformations with restricted range’,Bull. Aust. Math. Soc.77(3) (2008), 441–453] proved that$$\begin{eqnarray}\displaystyle Q=\{\unicode[STIX]{x1D6FC}\in T(V,W):V\unicode[STIX]{x1D6FC}\subseteq W\unicode[STIX]{x1D6FC}\} & & \displaystyle \nonumber\end{eqnarray}$$is the largest regular subsemigroup of$T(V,W)$and characterized Green’s relations on$T(V,W)$. In this paper, we determine all the maximal regular subsemigroups of$Q$when$W$is a finite-dimensional subspace of$V$over a finite field. Moreover, we compute the rank and idempotent rank of$Q$when$W$is an$n$-dimensional subspace of an$m$-dimensional vector space$V$over a finite field$F$.

scholarly journals A Norm Inequality for Linear Transformations

1961 ◽  
Vol 4 (3) ◽  
pp. 239-242
Author(s):  
B.N. Moyls ◽  
N.A. Khan
Keyword(s):  
Positive Integer ◽  
Vector Space ◽  
Hermitian Operator ◽  
Norm Inequality ◽  
Dimensional Vector ◽  
Linear Transformations ◽  
Dimensional Vector Space ◽  
Image Position ◽  
Ky Fan

In 1949 Ky Fan [1] proved the following result: Let λ1…λn be the eigenvalues of an Hermitian operator H on an n-dimensional vector space Vn. If x1, …, xq is an orthonormal set in V1, and q is a positive integer such n that 1 ≤ q ≤ n, then1

Hypersurfaces Fixed by Equiaffinities

1973 ◽  
Vol 16 (1) ◽  
pp. 129-131
Author(s):  
J. C. Fisher
Keyword(s):  
Vector Space ◽  
Canonical Form ◽  
Parameter Family ◽  
Present Approach ◽  
Real Plane ◽  
Dimensional Vector ◽  
Dimensional Vector Space ◽  
The Real ◽  
Geometric Argument

In this note we state and prove the followingAny equiaffinity acting on the points of an n-dimensional vector space (n ≥2) leaves invariant the members of a one parameter family of hypersurfaces defined by polynomials p(xl…,xn)=c of degree m ≤n.The theorem, restricted to the real plane, appears to have been discovered almost simultaneously by Coxeter [4] and Komissaruk [5]. The former paper presents an elegant geometric argument, showing that the result follows from the converse of Pascal's theorem. The present approach is more closely related to that of [5], in which the transformations are reduced to a canonical form.

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