scholarly journals On the Index of a Quadratic Form

1958 ◽  
Vol 1 (3) ◽  
pp. 180-180
Author(s):  
Jonathan Wild
Keyword(s):  
Quadratic Form ◽  
Vector Space ◽  
Bilinear Form ◽  
Direct Proof ◽  
Symmetric Bilinear Form ◽  
Arbitrary Field ◽  
Analytic Geometry ◽  
Fundamental Fact

Given a vector space V = {x, y, ...} over an arbitrary field. In V a symmetric bilinear form (x,y) i s given. A subspace W is called totally isotropic [t.i.] if (x,y) = 0 for every pair x W, y W.Let Vn and Vm be two t.i. subspaces of V; n < m. Lower indices always indicate dimensions. It is a well known and fundamental fact of analytic geometry that there exists a t.i. subspace Wm of V containing Vn [cf. Dieudonné: Les Groupes classiques , P. 18]. As no simple direct proof seems to be available, we propose to supply one.

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 Right and Left Orthogonality

1961 ◽  
Vol 4 (2) ◽  
pp. 182-184
Author(s):  
Jonathan Wild
Keyword(s):  
Vector Space ◽  
Bilinear Form ◽  
Simple Proof ◽  
Arbitrary Field ◽  
Image Position

Let V be a vector space over an arbitrary field F. In V a bilinear formis given. If f is symmetric [(x, y) ≡ (y, x)] or skew-symmetric [(x, y) + (y, x) ≡ 0], then1Thus right and left orthogonality coincide. It is well known that (1) implies conversely that f is either symmetric or skew-symmetric in V. We wish to give a simple proof of this result.

Clifford and Heisenberg algebras

2011 ◽  
Author(s):  
Jean-Michel Bismut
Keyword(s):  
Vector Space ◽  
Clifford Algebra ◽  
Bilinear Form ◽  
Scalar Product ◽  
Heisenberg Algebra ◽  
Symmetric Bilinear Form ◽  
Rham Complex ◽  
Heisenberg Algebras ◽  
De Rham ◽  
Symplectic Vector Space

This chapter recalls various results on Clifford algebras and Heisenberg algebras. It first introduces the Clifford algebra of a vector space V equipped with a symmetric bilinear form B and then specializes the construction of the Clifford algebra to the case of V ⊕ V*. Next, the chapter argues that, if (V,ω‎) is a symplectic vector space, then the associated Heisenberg algebra is constructed and then specialized to the case of V ⊕ V*. Hereafter, the chapter considers the combination of the Clifford and Heisenberg algebras for V ⊕ V*, and constructs the complex Λ‎· (V*) ⊗ S· (V*), ̄ƌ) which is the subcomplex of polynomial forms in the de Rham complex. Finally, when V is equipped with a scalar product, this complex is related to a Witten complex over V.

Cone Preserving Mappings for Quadratic Cones Over Arbitrary Fields

1977 ◽  
Vol 29 (6) ◽  
pp. 1247-1253 ◽  
Author(s):  
J. A. Lester
Keyword(s):  
Vector Space ◽  
Bilinear Form ◽  
Symmetric Bilinear Form ◽  
Characteristic Two ◽  
Image Position ◽  
Singular Metric

Let F be a non-singular metric vector space, that is, a vector space over a field F not of characteristic two, upon which is defined a non-singular symmetric bilinear form ( , ). For any a ϵ V, we define the cone with vertex a to be the set

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.

scholarly journals The nonexistence of rank4IP tensors in signature(1,3)

2002 ◽  
Vol 31 (5) ◽  
pp. 259-269
Author(s):  
Kelly Jeanne Pearson ◽  
Tan Zhang
Keyword(s):  
Vector Space ◽  
Curvature Tensor ◽  
Bilinear Form ◽  
Symmetric Operator ◽  
Symmetric Bilinear Form ◽  
Real Vector Space ◽  
Real Vector ◽  
Algebraic Curvature Tensor ◽  
Algebraic Curvature

LetVbe a real vector space of dimension4with a nondegenerate symmetric bilinear form of signature(1,3). We show that there exists no algebraic curvature tensorRonVso that its associated skew-symmetric operatorR(⋅)has rank4and constant eigenvalues on the Grassmannian of nondegenerate2-planes inV.

The Space Groups of Two Dimensional Minkowski Space

1978 ◽  
Vol 30 (5) ◽  
pp. 1103-1120
Author(s):  
George Maxwell
Keyword(s):  
Vector Space ◽  
Minkowski Space ◽  
Bilinear Form ◽  
Orthogonal Group ◽  
Affine Space ◽  
Symmetric Bilinear Form ◽  
Two Dimensional ◽  
Space Groups ◽  
Dimensional Minkowski Space ◽  
Real Affine

LetEbe an n-dimensional real affine space,Vits vector space of translations andA(E)the affine group ofE.Suppose that (. , .) is a nondegenerate symmetric bilinear form on F of signature(n —1, 1), O(V) its orthogonal group andS(V)its group of similarities.

scholarly journals Bounds on real finite linear groups that preserve a symmetric or skew symmetric bilinear form

2019 ◽  
Vol 22 (4) ◽  
pp. 545-554
Author(s):  
Michael J. Collins
Keyword(s):  
Quadratic Form ◽  
Bilinear Form ◽  
Symmetric Bilinear Form ◽  
Linear Groups ◽  
Compact Groups ◽  
Orthogonal Groups ◽  
Jordan Type ◽  
Finite Subgroups ◽  
Finite Linear ◽  
Real Quadratic

AbstractWe extend the results of our earlier work on Jordan-type bounds for finite subgroups of complex classical groups to real groups. The bounds that we obtain are related to our previous results by means of structural results for finite linear groups that we can generalise here to compact groups. For the real analogue of orthogonal groups, we take into account the signature of a real quadratic form to determine bounds in every case.

A Characterization of Spin Representations

1971 ◽  
Vol 23 (5) ◽  
pp. 896-906 ◽  
Author(s):  
Robert B. Brown
Keyword(s):  
Vector Space ◽  
Clifford Algebra ◽  
Bilinear Form ◽  
Automorphism Groups ◽  
Symmetric Bilinear Form ◽  
The Status ◽  
Spin Representations ◽  
Clifford Groups

Associated with a non-degenerate symmetric bilinear form on a vector space is a Clifford algebra and various Clifford groups, which have spin representations on minimal right ideals of the Clifford algebra. Several invariants for these representations have been known for some time. In this paper the forms are assumed to be “split”, and several relations between the invariants are derived and promoted to the status of axioms. Then it is shown that any system satisfying the axioms comes from a minimal right ideal in a Clifford algebra and that the automorphism groups associated with the system are the Clifford groups. Hence, the axioms characterize spin representations.A description of split forms and spin representations is in section two. In section three the invariants and their properties are described.

scholarly journals Matroid connectivity and singularities of configuration hypersurfaces

2021 ◽  
Vol 111 (1) ◽  
Author(s):  
Graham Denham ◽  
Mathias Schulze ◽  
Uli Walther
Keyword(s):  
Bilinear Form ◽  
Symmetric Bilinear Form ◽  
Smooth Locus

AbstractConsider a linear realization of a matroid over a field. One associates with it a configuration polynomial and a symmetric bilinear form with linear homogeneous coefficients. The corresponding configuration hypersurface and its non-smooth locus support the respective first and second degeneracy scheme of the bilinear form. We show that these schemes are reduced and describe the effect of matroid connectivity: for (2-)connected matroids, the configuration hypersurface is integral, and the second degeneracy scheme is reduced Cohen–Macaulay of codimension 3. If the matroid is 3-connected, then also the second degeneracy scheme is integral. In the process, we describe the behavior of configuration polynomials, forms and schemes with respect to various matroid constructions.

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