Bilinear Forms

Bilinear Forms

Arun Ram
Department of Mathematics and Statistics
University of Melbourne
Parkville, VIC 3010 Australia
aram@unimelb.edu.au

Last updates: 21 September 2010

The groups Un, On, Sp2n

The unitary group is U(n) = g∈GLn(ℂ)∣ g g ‾ t = 1 where g ‾ = ( g ‾ ij ) if g=(gij). The orthogonal group is On(ℂ) = g∈GLn(ℂ)∣ g g t = 1 . The symplectic group is Sp2n(ℂ) = g∈GL2n(ℂ)∣ g J g t = 1 where J= 1 ⋱ 1 -1 ⋱ -1 or J= 1 ⋰ 1 -1 ⋰ -1

Let V be an 𝔽-vector space. A bilinear form on V is a map ⟨,⟩:V×V→𝔽 such that ⟨ c 1 v 1 + c 2 v 2 , v3 ⟩ = c1 ⟨ v1 , v3 ⟩ + c2 ⟨ v2 , v3 ⟩, and ⟨ v 1 , c 1 v 2 + c2 v3 ⟩ = c1 ⟨ v1 , v2 ⟩ + c2 ⟨ v2 , v3 ⟩, for v 1 , v 2 , v 3 ∈ V , c 1 , c 2 , c 3 ∈ 𝔽 . A bilinear form ⟨,⟩:V×V→𝔽 is symmetric if ⟨ v1 , v2 ⟩ = ⟨ v2 , v1 ⟩ for v1 , v2 ∈ V , and skew-symmetric if ⟨ v1 , v2 ⟩ = - ⟨ v2 , v1 ⟩ for v1 , v2 ∈ V . The orthogonal group is On(𝔽) = O(V,⟨⟩) = g∈GL(V)∣ ⟨ gv1,gv2 ⟩ = ⟨ v1v2 ⟩ for v1,v2∈V . The symplectic group is Spn(𝔽) = O(V,⟨⟩) = g∈GL(V)∣ ⟨ gv1,gv2 ⟩ = ⟨ v1v2 ⟩ for v1,v2∈V . Let 𝔽 be a field with an involution ‾ : 𝔽 → 𝔽, z→ z ‾ . Let V be a vector space over 𝔽.
A sesquilinear form is a map ⟨,⟩:V×V→𝔽 such that ⟨ c 1 v 1 + c 2 v 2 , v ⟩ = c1 ⟨ v1 , v3 ⟩ + c2 c2 ⟨ v2 , v ⟩, and ⟨ w , a 1 w 1 + a2 w2 ⟩ = a‾1 ⟨ w , w1 ⟩ + a‾1 ⟨ w , w2 ⟩, for v , v 2 , v 3 , w , w 1 , w 2 ∈ V and a 1 , a 2 , c 1 c 2 ∈ 𝔽 . A Hermitian form on V is a sesquilinear form ⟨,⟩:V×V→𝔽 such that ⟨ v1 , v2 ⟩= ⟨ v1 , v2 ⟩ ‾ for v1 , v2 ∈ V . A positive Hermitian form on V is a sesquilinear form ⟨,⟩:V×V→𝔽 such that ⟨ v1 , v2 ⟩ ∈ℝ≥0 for v1 , v2 ∈ V . The unitary group is Un = g∈GL(V)∣ ⟨ v1 , v2 ⟩ = ⟨ gv1 , gv2 ⟩ for v1 , v2 ∈ V . Let (V⊗V)* = bilinear forms on V S2 (V) * = symmetric bilinear forms on V ⋀2(V)* = skew-symmetric bilinear forms on V . The symmetric group S2 = 1,s acts on (V⊗V)* by ( s⋅ ⟨ ⟩ ) ( v1 , v2 ) = ⟨ v1 , v2 ⟩ for v2, v1 ∈ V . Then S2 (V) * = ⟨⟩∈ (V⊗V)* ∣ s⋅ ⟨⟩ = ⟨⟩ ⋀2(V)* = ⟨⟩∈ (V⊗V)* ∣ s⋅ ⟨⟩ = -⟨⟩ . The group GL(V) acts on (V⊗V)* by ( g⋅ ⟨ ⟩ ) ( v1 , v2 ) = ⟨ g-1 v1 , g-1 v2 ⟩ , for g∈GL(V), ⟨⟩∈ (V⊗V)* , v1, v2 ∈ V . The GL(V) action on (V⊗V)* commutes with the S2 action on (V⊗V)* , gs⋅⟨⟩ = sg⋅⟨⟩ for g∈GL(V) and ⟨⟩∈ (V⊗V)* and (V⊗V)* = S2 (V)* ⊕ ⋀2 (V)* as a (GL(V),S2) -bimodule. A choice of basis b1…bn provides a bijection (V⊗V)* → Mn(𝔽) ⟨,⟩ → A = ( ⟨ bi , bj ⟩ )1≤i,j≤n and this bijection identifies S2(V)* with Symn(𝔽) = A∈Mn(𝔽) ∣ A=At ⋀2(V)* with Skewn(𝔽) = A∈Mn(𝔽) ∣ A=-At The S2-action and the GL(V)-action on Mn(𝔽) are given by s⋅A=At and g⋅A = g-1 A (g-1)t for A∈Mn(𝔽) and g∈GLn(𝔽).

Under the action of G L n 𝔽 on the set Skew n 𝔽 = A ∈ M n 𝔽 ∣ A = A t the orbits have representatives Jr = 0 1 0 ⋱ ⋱ ⋱ 0 1 0 -1 0 0 ⋱ ⋱ ⋱ -1 0 0 0 0 0 ⋱ ⋱ ⋱ 0 0 0 for 0≤r≤⌊n/2⌋. Under the action of Under the action of G L n 𝔽 on the set Herm n 𝔽 = A ∈ M n 𝔽 ∣ A = A ‾ t , the orbits have representatives ( α 1 0 ⋯ 0 0 α 2 ⋱ ⋮ α r ⋮ 0 ⋱ 0 ⋯ 0 ) with α i = α ‾ i ≠ 0. Note that if ‾ : 𝔽 → 𝔽 is the identity, then Herm n 𝔽 = Skew n 𝔽 .
Proof.
  1. Choose a basis of V, b 1 , … , b n , such that for each l the form ⟨ , ⟩ is nondegenerate on Span b 1 , … , b l . Let D i j = cof ⟨ b i , b j ⟩ 1 ≤ i , j ≤ l jl and e l = D l l - 1 D 1l b 1 + D 2l b 2 + ⋯ + D ll b l . Then ⟨ e i , e j ⟩ = 0 if i ≠ j and, ⟨ e l , e l ⟩ = D l l - 1 D l + 1 , l + 1 . If P = ( D l l - 1 D i l ) , then P t A P = ( D 1 1 - 1 D 2 2 0 ⋯ 0 D 2 2 - 1 D 3 3 0 ⋮ 0 ⋱ ) . Homework: Do an example.
    Note: This is the same as Gram-Schmidt.
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Two more G-actions

(1) GL (V) × GL (W) acts on V⊗ W* . This corresponds so the action of G L n 𝔽 ×G L m 𝔽 on M n ( 𝔽 ) by P , Q ⋅ A = P A Q - 1 .
(2) GL (V) acts on V . This corresponds so the action of G L n 𝔽 on M n ( 𝔽 ) by P ⋅ A = P A P - 1 . In this case the orbits are indexed by Jordan forms.
The answer coincides with the clasification of finitely generated 𝔽[x]-modules: :𝔽[x] → M n (𝔽) = End(V) x → A i.e. representations V of 𝔽[x] of dimension n.

Smith normal form

If A is an m×n matrix over a principal ideal domain R, then there are P ∈ G L m ( R ) and Q ∈ G L n ( R ) such that P A Q = a 1 a 2 ⋱ a r 0 ⋱ 0 where a i ≠ 0 and a 1 ∣ a 2 ∣ ⋯ ∣ a r . This is accomplished with elementary matrices h i (c) = 1 ⋱ 1 c 1 ⋱ 1 X i j (c) = 1 ⋱ 1 c ⋱ 1 ⋱ 1 s i j = 1 ⋱ 1 0 1 1 ⋱ 1 1 0 1 ⋱ 1 and y i , j (u,v,s,t) = 1 ⋱ 1 u s v t 1 ⋱ 1 .

Notes and References

In [BJN] the classification of the GL(V)×GL(W) orbits is termed Smith Normal Form and is proved, over a PID, by row reduction, in Theorem 3.2 of Chaper 20. Invariant factors are defined on p. 399.
In [Ar] this appeas in Theorem 4.3 of Chapter 12. Again the proof is by row reduction. Artin Chapter 7 is on bilinear forms. The classification of GLn(ℝ) orbits on Symn(ℝ) is in Theorem 2.9 of Chapter 7 and is termed Sylvester's law. The proof is the Gram-Schmidt induction. The classification of Un(ℂ) orbits on Hermn×n(ℂ) is done in Theorem 5.4, finding a basis of eigenvectors. This is termed, the Spectral Theorem.
In [Bo], Chapter 9, Forme sesquilinéaires et formes quadratiques, the GLn(𝔽) orbits of Hermn×n(𝔽) are classified in Corollary 2 to Theorem 1 of §6 no. 1 and the explicit version of the Gram-Schmidt process is in Proposition 1 of §6 no. 1, as noted in the sentence immediately beore Proposition 2 on p. 93. As noted there, the Gram-Schmidt process works perfectly well over commutative rings. In [Ar], the Jordan normal form (GLn(𝔽)) orbits on GLn(𝔽) is derived from the smith normal form applied to the ring 𝔽[t]. This is Theorem 7.13. The rational canonical form appears in Theorem 7.9. In [BJN] the rational canonical form is §4 of Chapter 21 and the Jordan Canonical form is in §5 of Chapter 21. In [Bo], Chapter VII the Jordan Canonical form is Proposition 4 of §5 no. 3 and Proposition 8 of §5 no. 4. The rational canonical form is in §5 no. 1 and the smith normal form is Corollary 1 of §4 no. 6. There is a thorough discussion of invariant factors and elementary divisors.

Bibliography

[Ar] M. Artin, Algebra, Prentice Hall, 1991.

[BJN] P.B Bhattacharya, S.K. Jain and S.R. Nagpaul, Basic Abstract Algebra, 2nd Edition, Cambridge University Press, 1994.

[Bo] N. Bourbaki, Algebrè,

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