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)∣gg‾t=1 where g‾=(g‾ij)ifg=(gij). The orthogonal group is On(ℂ)=g∈GLn(ℂ)∣ggt=1. The symplectic group is Sp2n(ℂ)=g∈GL2n(ℂ)∣gJgt=1 where J=1⋱1-1⋱-1orJ=1⋰1-1⋰-1

Let V be an 𝔽-vector space. A bilinear form on V is a map ⟨,⟩:V×V→𝔽 such that ⟨c1v1+c2v2,v3⟩=c1⟨v1,v3⟩+c2⟨v2,v3⟩,and⟨v1,c1v2+c2v3⟩=c1⟨v1,v2⟩+c2⟨v2,v3⟩, for v1,v2,v3∈V, c1,c2,c3∈𝔽. A bilinear form ⟨,⟩:V×V→𝔽 is symmetric if ⟨v1,v2⟩=⟨v2,v1⟩forv1,v2∈V, and skew-symmetric if ⟨v1,v2⟩=-⟨v2,v1⟩forv1,v2∈V. The orthogonal group is On(𝔽)=O(V,⟨⟩)=g∈GL(V)∣⟨gv1,gv2⟩=⟨v1v2⟩forv1,v2∈V. The symplectic group is Spn(𝔽)=O(V,⟨⟩)=g∈GL(V)∣⟨gv1,gv2⟩=⟨v1v2⟩forv1,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 ⟨c1v1+c2v2,v⟩=c1⟨v1,v3⟩+ c2⟨v2,v⟩,and⟨w,a1w1+a2w2⟩=a‾1⟨w,w1⟩+a‾1⟨w,w2⟩, for v,v2,v3,w,w1,w2∈V and a1,a2,c1c2∈𝔽. A Hermitian form on V is a sesquilinear form ⟨,⟩:V×V→𝔽 such that ⟨v1,v2⟩=⟨v1,v2⟩‾forv1,v2∈V. A positive Hermitian form on V is a sesquilinear form ⟨,⟩:V×V→𝔽 such that ⟨v1,v2⟩∈ℝ≥0forv1,v2∈V. The unitary group is Un=g∈GL(V)∣⟨v1,v2⟩=⟨gv1,gv2⟩forv1,v2∈V. Let (V⊗V)*=bilinear forms onVS2(V)*=symmetric bilinear forms onV⋀2(V)*=skew-symmetric bilinear forms onV. The symmetric group S2=1,s acts on (V⊗V)* by (s⋅⟨⟩)(v1,v2)=⟨v1,v2⟩forv2,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-1v1,g-1v2⟩, 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⋅⟨⟩forg∈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)*withSymn(𝔽)=A∈Mn(𝔽)∣A=At⋀2(V)*withSkewn(𝔽)=A∈Mn(𝔽)∣A=-At The S2-action and the GL(V)-action on Mn(𝔽) are given by s⋅A=Atandg⋅A=g-1A(g-1)t for A∈Mn(𝔽) and g∈GLn(𝔽).

Under the action of GLn𝔽 on the set Skewn𝔽=A∈Mn𝔽∣A=At the orbits have representatives Jr=010 ⋱⋱⋱ 010 -100 ⋱⋱⋱ -100 000 ⋱⋱⋱ 000 for0≤r≤⌊n/2⌋. Under the action of Under the action of GLn𝔽 on the set Hermn𝔽=A∈Mn𝔽∣A=A‾t, the orbits have representatives (α10⋯00α2⋱⋮αr⋮0⋱0⋯0)withαi=α‾i≠0. Note that if ‾:𝔽→𝔽 is the identity, then Hermn𝔽=Skewn𝔽.

		Proof.
	Choose a basis of V, b1,…,bn, such that for each l the form ⟨,⟩ is nondegenerate on Spanb1,…,bl. Let Dij=cof⟨bi,bj⟩1≤i,j≤ljl and el=Dll-1D1lb1+D2lb2+⋯+Dllbl. Then ⟨ei,ej⟩=0ifi≠jand,⟨el,el⟩=Dll-1Dl+1,l+1. If P=(Dll-1Dil ),thenPtAP=(D11-1D220⋯0D22-1D330⋮0⋱). Homework: Do an example. Note: This is the same as Gram-Schmidt. □

Two more G-actions

(1) GL(V)×GL(W) acts on V⊗W*. This corresponds so the action of GLn𝔽×GLm𝔽 on Mn(𝔽) by P,Q⋅A=PAQ-1.
(2) GL(V) acts on V. This corresponds so the action of GLn𝔽 on Mn(𝔽) by P⋅A=PAP-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]→Mn(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∈GLm(R)andQ∈GLn(R) such that PAQ=a1a2⋱ar0⋱0whereai≠0anda1∣a2∣⋯∣ar. This is accomplished with elementary matrices hi(c)=1⋱1c1⋱1Xij(c)=1⋱1c⋱1⋱1 sij=1⋱1011⋱1101⋱1andyi,j(u,v,s,t)=1⋱1usvt1⋱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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