Positive and negative definiteness, the inertia $(p,q,r)$, rank $p+q$, and signature $p-q$ of a real symmetric bilinear or quadratic form

Depends on

• A quadratic form $q$ in arbitrary characteristic and its polar form $b_q(u,v)=q(u+v)-q(u)-q(v)$
• The reals form a totally ordered field
• The matrix, left and right radicals, rank, and nondegeneracy of a bilinear form on a finite-dimensional space

Used by

• Negative definiteness of the primitive part of the Neron-Severi space Corollary
• Positive determinant does not imply positive definiteness Counterexample
• Same complexification with different killing form signatures Counterexample
• Finite large spherical complexes, their almost-negative matrices, the metric flag condition, and links Definition
• Irreducible affine Coxeter type: the corank-one form, the radical quotient, and the affine slice Definition
• Nondegenerate critical points, nullity, index, and coindex Definition
• Primitive subspaces, hard Lefschetz and the Hodge-Riemann signs: the dictionary of hypotheses S, hL and HR Definition
• The Coxeter nerve and its Moussong metric Definition
• The real Coxeter form, its radical, reflections, and form-preserving maps Definition
• The signature of a closed oriented manifold of dimension divisible by four Definition
• The trace, the invariant intersection form on Bott-Samelson bimodules and its reduction Definition
• A cycle and an overlong arm: explicit non-positive witnesses Example
• An indefinite Coxeter form with a faithful canonical reflection representation Example
• An indefinite Coxeter form: infinite, but not of affine type Example
• Bₙ and Cₙ define the same Coxeter diagram and the same Coxeter group Example
• Dihedral diagrams I₂(m): Gram determinants, the infinite case, and the low-rank coincidences Example
• Gram determinants and principal minors of the non-crystallographic types H₃ and H₄ Example
• Link angles in A2, affine A2 and the universal Coxeter nerve Example
• Path determinants dₖ=dₖ₋₁-cos²(π/m)dₖ₋₂ and the three-arm inequality 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 affine Ã₂ nerve: every edge exists, the Gram determinant vanishes, and the perimeter is exactly 2π Example
• The all-right triangle must be filled; the disconnected universal-Coxeter nerve is CAT(1) vacuously Example
• The Coxeter complex of A₃: a triangulation of the sphere and the residue of a proper parabolic Example
• The Coxeter complex of I₂(5): the circle triangulated by the ten chambers Example
• The Hodge index theorem on a blowup of the projective plane Example
• The Hodge index theorem on a product of projective lines Example
• A nondegenerate symmetric form with a half-dimensional isotropic subspace has zero signature Lemma
• Cartan-number products, allowed edge labels, tree scalings and reflection stability Lemma
• Crystallographic alcove diagrams: the affine list realized by Weyl types A–G Lemma
• Disconnected diagrams, direct products, and comparison of invariant forms Lemma
• Exclusions for positive definite diagrams: trees, valency, labels, chains and arms Lemma
• Face links of large metric flag complexes, and the inductive local CAT(1) criterion 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
• If v₁,…,vₘ∈ℝⁿ satisfy ⟨ vᵢ,vⱼ⟩=t≥0 for i≠ j and ⟨ vᵢ,vᵢ⟩>t, they are linearly independent 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
• Signature constancy in a Lefschetz family, the L-stable subspace lemma and the weak-Lefschetz substitute Lemma
• Sylvester inertia makes the Morse index intrinsic Lemma

…and 9 more results.

Dependency tree · two levels

17 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

• H. Pinkham, Linear Algebra, §7.7 (standard reference, not scraped)
• J. Kuan, Positive Definite Matrices (standard reference, not scraped)