The matrix, left and right radicals, rank, and nondegeneracy of a bilinear form on a finite-dimensional space

Depends on

• Bilinear forms on $V$ correspond linearly and bijectively to linear maps $V\to V^*$
• The dual family of a finite basis is a basis of the dual space, with the same dimension
• Coordinate columns $[v]_{\mathcal B}$ and matrices $[T]_{\mathcal B}^{\mathcal C}$ of linear maps relative to ordered bases
• Rank and nullity of a linear map with finite-dimensional domain
• Rank-nullity: $\dim_F V=\operatorname{nullity}T+\operatorname{rank}T$
• A square matrix is invertible exactly when its multiplication map is a linear isomorphism; matrices preserve inverses of linear isomorphisms

Used by

• Every nontrivial number field has a ramified finite prime Corollary
• A faithful canonical realization that is not reflection faithful: the affine rank-two system Counterexample
• A nondegenerate indefinite symmetric form can have W∩ W^⊥≠0 Counterexample
• In characteristic 2, a symmetric bilinear form need not have an orthogonal basis Counterexample
• Irreducible affine Coxeter type: the corank-one form, the radical quotient, and the affine slice Definition
• Positive and negative definiteness, the inertia (p,q,r), rank p+q, and signature p-q of a real symmetric bilinear or quadratic form Definition
• The real Coxeter form, its radical, reflections, and form-preserving maps Definition
• The standard bilinear form ⟨ x,y⟩=∑_i<nxᵢyᵢ on Fⁿ Definition
• The trace, the invariant intersection form on Bott-Samelson bimodules and its reduction Definition
• A null normal admits no reflection of the displayed form Example
• Reducible positive semidefinite forms: factorwise treatment and the square alcove Example
• Reflection matrices in a positive plane, a Lorentzian plane, and a plane with radical Example
• The form on ℝ² with matrix beginpmatrix0&10&0 endpmatrix is neither symmetric nor alternating Example
• A nondegenerate symmetric form with a half-dimensional isotropic subspace has zero signature Lemma
• Gram-map image, full-rank fibers, and rank-stratum dimensions Lemma
• Hard Lefschetz gives the primitive decomposition, orthogonality and the signature formula Lemma
• Hecke and Lie seam contract compatibility: the Artin-to-Hecke map, normalization conversions, root-length matching, and the reflection-faithfulness boundary Lemma
• Invariant forms on an indecomposable Soergel bimodule are unique up to scalar, non-degenerate, and agree with the restricted Bott-Samelson form Lemma
• Positive radical, corank one, positive definiteness of proper submatrices, and domination exclusions Lemma
• Reflections: involutivity, form invariance, fixed hyperplane, and exact rank-two order Lemma
• The nilpotent Lefschetz filtration is canonical and self-dual Lemma
• The tensor product of nondegenerate real symmetric forms has multiplicative signature Lemma
• A basis change by P changes the matrix of a bilinear form from A to P^TAP Theorem
• Every alternating form on a finite-dimensional space has a basis of symplectic pairs followed by a basis of its radical; in particular its rank is even Theorem
• Ramification is detected by the number-field discriminant Theorem
• The trace form of a finite extension is nondegenerate exactly when the extension is separable Theorem

Dependency tree · two levels

27 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

• K. Conrad, Bilinear Forms, §§1 and 4 (standard reference, not scraped)
• H. Pinkham, Linear Algebra, Chapter 7 (standard reference, not scraped)