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The matrix, left and right radicals, rank, and nondegeneracy of a bilinear form on a finite-dimensional space
Definition
Let be a bilinear form on an -dimensional vector space , and let be an ordered basis. The matrix of in is
For coordinate columns and , one has .
The left radical and right radical are
The rank of is the rank of the associated map , . The form is nondegenerate when both radicals are zero. In finite dimension this is equivalent to being an isomorphism, and also to being invertible.
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
- 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
- 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 standard bilinear form ⟨ x,y⟩=∑_i<nxᵢyᵢ on Fⁿ Definition
- The form on ℝ² with matrix beginpmatrix0&10&0 endpmatrix is neither symmetric nor alternating Example
- 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
- The trace form of a finite extension is nondegenerate exactly when the extension is separable Theorem
Dependency tree · two levels
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Sources
- K. Conrad, Bilinear Forms, §§1 and 4 (standard reference, not scraped)
- H. Pinkham, Linear Algebra, Chapter 7 (standard reference, not scraped)