Dual Spaces, Bilinear and Quadratic Forms, and Sylvester's Law of Inertia

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✓ 23 results · all verified · 8 also independently AI-judged

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1 · Prerequisites

Abstract Algebra

Combinatorics

Finite Counting, Factorials and Binomial Coefficients

Coxeter Groups

Symmetric Groups, Cycle Decomposition and the Sign Homomorphism

Linear Algebra

Real Analysis

Set Theory

2 · Logical flowchart

3 · Items

Definition

Alternating forms are skew-symmetric; the converse holds when charF≠2, while in characteristic 2 alternating forms are symmetric

Theorem

For the linear-first convention, a basis change by P sends a sesquilinear matrix A to P^TA σ(P); Hermitian forms satisfy A=σ(A)^T

Definition

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

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

Theorem

Sylvester's law of inertia: every real symmetric form is congruent to diag(Iₚ,-I_q,0ᵣ), and (p,q,r) is unique

Corollary

Two real symmetric bilinear forms are congruent if and only if they have the same inertia

Lemma

For symmetric M=beginpmatrixA&BB^T&C endpmatrix with A invertible, a block-unitriangular congruence gives A⊕(C-B^TA⁻¹B) and factors det M

Theorem

Sylvester's criterion: a real symmetric n× n matrix with n≥1 is positive definite if and only if all leading principal minors are positive

Sources

Dual Spaces, Bilinear and Quadratic Forms, and Sylvester's Law of Inertia

3 · Items

Linear functionals and the algebraic dual V^*=L(V,F)

The dual family (b^)_b∈ B associated to a Hamel basis B, defined by b^(c)=δ_bc

The dual family of a finite basis is a basis of the dual space, with the same dimension

For an infinite Hamel basis, its dual family is linearly independent but does not span the algebraic dual

Assuming choice, if v∉ U≤ V, some f∈ V^* vanishes on U and satisfies f(v)=1

The canonical evaluation map J_V:V→ V^** given by J_V(v)(f)=f(v)

Assuming choice, the canonical map J_V:V→ V^** is linear and injective

Assuming choice, J_V:V→ V^** is surjective if and only if V is finite-dimensional

The annihilator U^∘≤ V^* of U≤ V and the preannihilator ^∘ S≤ V of S≤ V^*

Assuming choice, ^∘(U^∘)=U; in finite dimension, dim U^∘=dim V-dim U

The transpose or algebraic adjoint T^:W^→ V^, T^(g)=g∘ T, of a linear map T:V→ W

Transpose is linear, sends identities to identities, and reverses composition: (S∘ T)^=T^∘ S^*

Assuming choice, ker T^=(imT)^∘ and imT^=(ker T)^∘; in finite dimensions rankT^*=rankT

In dual bases, the matrix of T^* is the transpose of the matrix of T

Bilinear forms, and symmetric, skew-symmetric, and alternating bilinear forms

Alternating forms are skew-symmetric; the converse holds when charF≠2, while in characteristic 2 alternating forms are symmetric

Bilinear forms on V correspond linearly and bijectively to linear maps V→ V^*

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

A basis change by P changes the matrix of a bilinear form from A to P^TAP

Congruent matrices have the same rank; hence rank and nondegeneracy of a bilinear form are basis-independent

Sesquilinear and Hermitian forms over a field with an involution, using the convention linear in the first variable

For the linear-first convention, a basis change by P sends a sesquilinear matrix A to P^TA σ(P); Hermitian forms satisfy A=σ(A)^T

A quadratic form q in arbitrary characteristic and its polar form b_q(u,v)=q(u+v)-q(u)-q(v)

If charF≠2, quadratic forms and symmetric bilinear forms correspond by q(v)=B(v,v) and B(u,v)=1/2 b_q(u,v)

Every symmetric bilinear form on a finite-dimensional space over a field of characteristic not 2 has an orthogonal basis

Over a field of characteristic not 2, every symmetric matrix is congruent to a diagonal matrix

Over a field of characteristic not 2, every quadratic form has diagonal coordinates q(x)=a₁x₁²+⋯+aₙxₙ²

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

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

Sylvester's law of inertia: every real symmetric form is congruent to diag(Iₚ,-I_q,0ᵣ), and (p,q,r) is unique

Two real symmetric bilinear forms are congruent if and only if they have the same inertia

For symmetric M=beginpmatrixA&BB^T&C endpmatrix with A invertible, a block-unitriangular congruence gives A⊕(C-B^TA⁻¹B) and factors det M

Sylvester's criterion: a real symmetric n× n matrix with n≥1 is positive definite if and only if all leading principal minors are positive

Dual Spaces, Bilinear and Quadratic Forms, and Sylvester's Law of Inertia

3 · Items

Linear functionals and the algebraic dual V^*=L(V,F)

The dual family (b^*)_b∈ B associated to a Hamel basis B, defined by b^*(c)=δ_bc

The dual family of a finite basis is a basis of the dual space, with the same dimension

For an infinite Hamel basis, its dual family is linearly independent but does not span the algebraic dual

Assuming choice, if v∉ U≤ V, some f∈ V^* vanishes on U and satisfies f(v)=1

The canonical evaluation map J_V:V→ V^** given by J_V(v)(f)=f(v)

Assuming choice, the canonical map J_V:V→ V^** is linear and injective

Assuming choice, J_V:V→ V^** is surjective if and only if V is finite-dimensional

The annihilator U^∘≤ V^* of U≤ V and the preannihilator ^∘ S≤ V of S≤ V^*

Assuming choice, ^∘(U^∘)=U; in finite dimension, dim U^∘=dim V-dim U

The transpose or algebraic adjoint T^*:W^*→ V^*, T^*(g)=g∘ T, of a linear map T:V→ W

Transpose is linear, sends identities to identities, and reverses composition: (S∘ T)^*=T^*∘ S^*

Assuming choice, ker T^*=(imT)^∘ and imT^*=(ker T)^∘; in finite dimensions rankT^*=rankT

In dual bases, the matrix of T^* is the transpose of the matrix of T

Bilinear forms, and symmetric, skew-symmetric, and alternating bilinear forms

Alternating forms are skew-symmetric; the converse holds when charF≠2, while in characteristic 2 alternating forms are symmetric

Bilinear forms on V correspond linearly and bijectively to linear maps V→ V^*

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

A basis change by P changes the matrix of a bilinear form from A to P^TAP

Congruent matrices have the same rank; hence rank and nondegeneracy of a bilinear form are basis-independent

Sesquilinear and Hermitian forms over a field with an involution, using the convention linear in the first variable

For the linear-first convention, a basis change by P sends a sesquilinear matrix A to P^TA σ(P); Hermitian forms satisfy A=σ(A)^T

A quadratic form q in arbitrary characteristic and its polar form b_q(u,v)=q(u+v)-q(u)-q(v)

If charF≠2, quadratic forms and symmetric bilinear forms correspond by q(v)=B(v,v) and B(u,v)=1/2 b_q(u,v)

Every symmetric bilinear form on a finite-dimensional space over a field of characteristic not 2 has an orthogonal basis

Over a field of characteristic not 2, every symmetric matrix is congruent to a diagonal matrix

Over a field of characteristic not 2, every quadratic form has diagonal coordinates q(x)=a₁x₁²+⋯+aₙxₙ²

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

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

Sylvester's law of inertia: every real symmetric form is congruent to diag(Iₚ,-I_q,0ᵣ), and (p,q,r) is unique

Two real symmetric bilinear forms are congruent if and only if they have the same inertia

For symmetric M=beginpmatrixA&BB^T&C endpmatrix with A invertible, a block-unitriangular congruence gives A⊕(C-B^TA⁻¹B) and factors det M

Sylvester's criterion: a real symmetric n× n matrix with n≥1 is positive definite if and only if all leading principal minors are positive

The dual family (b^)_b∈ B associated to a Hamel basis B, defined by b^(c)=δ_bc

The transpose or algebraic adjoint T^:W^→ V^, T^(g)=g∘ T, of a linear map T:V→ W

Transpose is linear, sends identities to identities, and reverses composition: (S∘ T)^=T^∘ S^*

Assuming choice, ker T^=(imT)^∘ and imT^=(ker T)^∘; in finite dimensions rankT^*=rankT