On Some Invariants of a Bilinear Form

1961 ◽  
Vol 4 (3) ◽  
pp. 261-264
Author(s):  
Jonathan Wild
Keyword(s):  
Vector Space ◽  
Bilinear Form ◽  
Arbitrary Field ◽  
Dimensional Vector ◽  
Dimensional Vector Space ◽  
Finite Dimensional Vector Space ◽  
Finite Dimensional ◽  
Special Cases ◽  
Orthogonal Subspace

Let E be a finite dimensional vector space over an arbitrary field. In E a bilinear form is given. It associates with every sub s pa ce V its right orthogonal sub space V* and its left orthogonal subspace *V. In general we cannot expect that dim V* = dim *V. However this relation will hold in some interesting special cases.

scholarly journals On the Index of a Symmetric Form

1960 ◽  
Vol 3 (3) ◽  
pp. 293-295
Author(s):  
Jonathan Wild
Keyword(s):  
Finite Field ◽  
Vector Space ◽  
Bilinear Form ◽  
Symmetric Bilinear Form ◽  
Dimensional Vector ◽  
Symmetric Form ◽  
Characteristic P ◽  
Dimensional Vector Space ◽  
Finite Dimensional Vector Space ◽  
Finite Dimensional

Let E be a finite dimensional vector space over a finite field of characteristic p > 0; dim E = n. Let (x,y) be a symmetric bilinear form in E. The radical Eo of this form is the subspace consisting of all the vectors x which satisfy (x,y) = 0 for every y ϵ E. The rank r of our form is the codimension of the radical.

The Invariant Subspace Lattice of a Linear Transformation

1967 ◽  
Vol 19 ◽  
pp. 810-822 ◽  
Author(s):  
L. Brickman ◽  
P. A. Fillmore
Keyword(s):  
Vector Space ◽  
Linear Transformation ◽  
Invariant Subspaces ◽  
Arbitrary Field ◽  
Dimensional Vector ◽  
Subspace Lattice ◽  
Dimensional Vector Space ◽  
Finite Dimensional Vector Space ◽  
Finite Dimensional ◽  
Algebraic Properties

The purpose of this paper is to study the lattice of invariant subspaces of a linear transformation on a finite-dimensional vector space over an arbitrary field. Among the topics discussed are structure theorems for such lattices, implications between linear-algebraic properties and lattice-theoretic properties, nilpotent transformations, and the conditions for the isomorphism of two such lattices. These topics correspond roughly to §§2, 3, 4, and 5 respectively.

scholarly journals Sets of idempotents that generate the semigroup of singular endomorphisms of a finite-dimensional vector space

1982 ◽  
Vol 25 (2) ◽  
pp. 133-139 ◽  
Author(s):  
R. J. H. Dawlings
Keyword(s):  
Vector Space ◽  
Dimensional Vector ◽  
Dimensional Vector Space ◽  
Finite Dimensional Vector Space ◽  
Finite Dimensional ◽  
Mathematical System

IfMis a mathematical system and EndMis the set of singular endomorphisms ofM, then EndMforms a semigroup under composition of mappings. A number of papers have been written to determine the subsemigroupSMof EndMgenerated by the idempotentsEMof EndMfor different systemsM. The first of these was by J. M. Howie [4]; here the case ofMbeing an unstructured setXwas considered. Howie showed that ifXis finite, then EndX=Sx.

scholarly journals Modular representations of abelian groups with regular rings of invariants

1982 ◽  
Vol 86 ◽  
pp. 229-248 ◽  
Author(s):  
Haruhisa Nakajima
Keyword(s):  
Vector Space ◽  
Polynomial Ring ◽  
Finite Subgroup ◽  
Dimensional Vector ◽  
Modular Representations ◽  
Invariant Polynomials ◽  
Dimensional Vector Space ◽  
Finite Dimensional Vector Space ◽  
Finite Dimensional ◽  
Regular Rings

Let k be a field of characteristic p and G a finite subgroup of GL(V) where V is a finite dimensional vector space over k. Then G acts naturally on the symmetric algebra k[V] of V. We denote by k[V]G the subring of k[V] consisting of all invariant polynomials under this action of G. The following theorem is well known.Theorem 1.1 (Chevalley-Serre, cf. [1, 2, 3]). Assume that p = 0 or (|G|, p) = 1. Then k[V]G is a polynomial ring if and only if G is generated by pseudo-reflections in GL(V).

scholarly journals The ideal structure of idempotent-generated transformation semigroups

1985 ◽  
Vol 28 (3) ◽  
pp. 319-331 ◽  
Author(s):  
M. A. Reynolds ◽  
R. P. Sullivan
Keyword(s):  
Vector Space ◽  
Dimensional Vector ◽  
Boolean Ring ◽  
Ideal Structure ◽  
Transformation Semigroups ◽  
Linear Transformations ◽  
Dimensional Vector Space ◽  
Finite Dimensional Vector Space ◽  
Finite Dimensional ◽  
The Ideal

Let X be a set and the semigroup (under composition) of all total transformations from X into itself. In ([6], Theorem 3) Howie characterised those elements of that can be written as a product of idempotents in different from the identity. We gather from review articles that his work was later extended by Evseev and Podran [3, 4] (and independently for finite X by Sullivan [15]) to the semigroup of all partial transformations of X into itself. Howie's result was generalized in a different direction by Kim [8], and it has also been considered in both a topological and a totally ordered setting (see [11] and [14] for brief summaries of this latter work). In addition, Magill [10] investigated the corresponding idea for endomorphisms of a Boolean ring, while J. A. Erdos [2] resolved the analogous problem for linear transformations of a finite–dimensional vector space.

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