## Bilinear, sesquilinear and quadratic forms

Bilinear forms. Let $A$ and $B$ be rings, $V$ a left $A$-module, $W$ a right $B$-module, and $F$ an $\left(A,B\right)$-bimodule.
A bilinear form is a function $⟨,⟩:V×W\to F$ such that

(a)   If ${v}_{1},{v}_{2}\in V$ and $w\in W$ then $⟨{v}_{1}+{v}_{2},w⟩=⟨{v}_{1},w⟩+⟨{v}_{2},w⟩$,
(b)   If $v\in V$ and ${w}_{1},{w}_{2}\in W$ then $⟨v,{w}_{1}+{w}_{2}⟩=⟨v,{w}_{1}⟩+⟨v,{w}_{2}⟩$,
(c)   If $a\in A$, $v\in V$ and $w\in W$ then $⟨av,w⟩=a⟨v,w⟩$,
(d)   If $b\in B$, $v\in V$ and $w\in W$ then $⟨v,wb⟩=⟨v,w⟩b$.

Sesquilinear forms. Let $A$ and $B$ be rings and let $\stackrel{‾}{\phantom{P}}:B\to B$ be an antiautomorphism so that

 if   ${b}_{1},{b}_{2}\in B$      then    $\stackrel{‾}{{b}_{1}{b}_{2}}=\stackrel{‾}{{b}_{2}}\phantom{\rule{0.2em}{0ex}}\stackrel{‾}{{b}_{1}}$.
Let $V$ be a left $A$-module, $W$ a left $B$-module, and $F$ an $\left(A,B\right)$-bimodule.
A sesquilinear form is a function $⟨,⟩:V×W\to F$ such that
(a)   If ${v}_{1},{v}_{2}\in V$ and $w\in W$ then $⟨{v}_{1}+{v}_{2},w⟩=⟨{v}_{1},w⟩+⟨{v}_{2},w⟩$,
(b)   If $v\in V$ and ${w}_{1},{w}_{2}\in W$ then $⟨v,{w}_{1}+{w}_{2}⟩=⟨v,{w}_{1}⟩+⟨v,{w}_{2}⟩$,
(c)   If $a\in A$, $v\in V$ and $w\in W$ then $⟨av,w⟩=a⟨v,w⟩$,
(d)   If $b\in B$, $v\in V$ and $w\in W$ then $⟨v,bw⟩=⟨v,w⟩\stackrel{‾}{b}$.

Quadratic forms. Let $A$ be a commutative ring and let $V$ be an $A$-module.
A quadratic form is a function ${‖\phantom{x}‖}^{2}:V\to A$ such that

(a)   If $a\in A$ and $v\in V$ then ${‖av‖}^{2}={a}^{2}{‖v‖}^{2}$,
(b)   The map $⟨,⟩:V×V\to A$ given by
 $⟨{v}_{1},{v}_{2}⟩={‖{v}_{1}+{v}_{2}‖}^{2}-{‖{v}_{1}‖}^{2}-{‖{v}_{2}‖}^{2}$
is a bilinear form.

Sesquilinear to bilinear. If $B$ is commutative then a bilinear form is a sesquilinear form with $\stackrel{‾}{\phantom{P}}=\mathrm{id}$.
Let $⟨,⟩:V×W\to F$ be a sesquilinear form. Define a right $B$-module by setting

 ${W}^{\sigma }$   to be   $W$      with     $wb={\sigma }^{-1}\left(b\right)w,\phantom{\rule{2em}{0ex}}\text{for}\phantom{\rule{2em}{0ex}}b\in B,w\in W$.
Then
 $⟨,⟩:V×{W}^{\sigma }\to F\phantom{\rule{2em}{0ex}}\text{given by}\phantom{\rule{2em}{0ex}}⟨v,w⟩=⟨v,w⟩$
is a bilinear form since $⟨v,wb⟩=⟨v,{\sigma }^{-1}\left(b\right)w⟩=⟨v,w⟩\sigma {\sigma }^{-1}b=⟨v,w⟩b$. This construction converts a sesquilinear form to a bilinear form.

Matrix of a form. Let $⟨,⟩:V×W\to F$ be a sesquilinear form. Assume that $\left({e}_{1},{e}_{2},\dots ,{e}_{n}\right)$ is a basis of $V$ and $\left({f}_{1},{f}_{2},\dots ,{f}_{n}\right)$ is a basis of $W$. The matrix of $⟨,⟩$ with respect to the bases $\left({e}_{1},{e}_{2},\dots ,{e}_{n}\right)$ and $\left({f}_{1},{f}_{2},\dots ,{f}_{n}\right)$ is

 $\left(⟨{e}_{i},{f}_{j}⟩\right)$,     an element of    ${M}_{n×m}\left(F\right)$.

Orthogonals. Let $⟨,⟩:V×W\to F$ be a sesquilinear form. Let $N$ be a submodule of $W$. The orthogonal to $N$ is

 ${N}^{\perp }=\left\{v\in V\phantom{\rule{0.5em}{0ex}}|\phantom{\rule{0.5em}{0ex}}\text{if}\phantom{\rule{0.5em}{0ex}}n\in N\phantom{\rule{0.5em}{0ex}}\text{then}\phantom{\rule{0.5em}{0ex}}⟨v,n⟩=0\right\}$.

Rank and nondegeneracy. Let $⟨,⟩:V×W\to F$ be a bilinear form. Define

 $\begin{array}{rccl}s:& V& ⟶& {\mathrm{Hom}}_{B}\left(W,F\right)\\ & v& ⟼& \begin{array}{rccl}⟨v,\cdot ⟩:& W& \to & F\\ & w& ↦& ⟨v,w⟩\end{array}\\ \begin{array}{}\end{array}\end{array}\phantom{\rule{2em}{0ex}}\text{and}\phantom{\rule{2em}{0ex}}\begin{array}{rccl}d:& W& ⟶& {\mathrm{Hom}}_{A}\left(V,F\right)\\ & w& ⟼& \begin{array}{rccl}⟨\cdot ,w⟩:& V& \to & F\\ & v& ↦& ⟨v,w⟩\end{array}\\ \begin{array}{}\end{array}\end{array}$
satisfy
 $s\left(av\right)=as\left(v\right)\phantom{\rule{2em}{0ex}}\text{and}\phantom{\rule{2em}{0ex}}d\left(wb\right)=d\left(w\right)b$.
The form $⟨,⟩:V×W\to F$ is nondegenerate if both $s$ and $d$ are injective.

If $A=B=F$ is a field then the rank of $⟨,⟩:V×W\to F$ is

 $\mathrm{rk}s=\mathrm{dim}\left(\mathrm{im}s\right)=\mathrm{dim}\left(V/{W}^{\perp }\right)=\mathrm{dim}\left(W/{V}^{\perp }\right)=\mathrm{dim}\left(\mathrm{im}d\right)=\mathrm{rk}d$.

Adjoints. Let $A=B=F$ be a ring. Let ${⟨,⟩}_{1}:{V}_{1}×{W}_{1}\to F$ and ${⟨,⟩}_{2}:{V}_{2}×{W}_{2}\to F$ be nondegenerate sesquilinear forms. The left adjoint of $S:{V}_{1}\to {V}_{2}$ is ${S}^{*}:{W}_{2}\to {W}_{1}$ given by

 ${⟨S\left({v}_{1}\right),{w}_{2}⟩}_{2}={⟨{v}_{1},{S}^{*}\left({w}_{2}\right)⟩}_{1}$.
The right adjoint of $T:{W}_{1}\to {W}_{2}$ is ${T}^{*}:{V}_{2}\to {V}_{1}$ given by
 ${⟨{v}_{2},T\left({w}_{1}\right)⟩}_{2}={⟨{T}^{*}\left({v}_{2}\right),{w}_{1}⟩}_{1}$.

## Discriminants

Let $A=B=F$ be a commutative ring. Let ${⟨,⟩}_{1}:{V}_{1}×{W}_{1}\to F,\dots ,{⟨,⟩}_{m}:{V}_{m}×{W}_{m}\to F$ be sesquilinear forms. The product form is the sesquilinear form

 $⟨,⟩:\left({V}_{1}\otimes \cdots \otimes {V}_{m}\right)×\left({W}_{1}\otimes \cdots \otimes {W}_{m}\right)\to F$     given by     $⟨{v}_{1}\otimes \cdots \otimes {v}_{m},{w}_{1}\otimes \cdots \otimes {w}_{m}⟩={⟨{v}_{1},{w}_{1}⟩}_{1}\cdots {⟨{v}_{m},{w}_{m}⟩}_{m}$.

Let $A=B=F$ be a commutative ring. Let $⟨,⟩:V×W\to F$ be a sesquilinear form. The $m$th exterior product form is the sesquilinear form

 $⟨,⟩:{\Lambda }^{m}V×{\Lambda }^{m}W\to F$     given by     $⟨{v}_{1}\wedge \cdots \wedge {v}_{m},{w}_{1}\wedge \cdots \wedge {w}_{m}⟩=\mathrm{det}\left(⟨{v}_{i},{w}_{j}⟩\right)$.

Let $A=B=F$ be a commutative ring. Let $V$ be a free $𝔽$-module of dimension $n$ and let $⟨,⟩:V×V\to F$ be a sesquilinear form. The discriminant of $⟨,⟩$ is

 ${‖\phantom{P}‖}^{2}:{\Lambda }^{n}V\to F$     given by     ${‖{v}_{1}\wedge \cdots \wedge {v}_{n}‖}^{2}=⟨{v}_{1}\wedge \cdots \wedge {v}_{n},{v}_{1}\wedge \cdots \wedge {v}_{n}⟩=\mathrm{det}\left(⟨{v}_{i},{w}_{j}⟩\right)$.

Let $⟨,⟩:V×V\to F$ be a sesquilinear form. Let $\left\{{e}_{1},\dots ,{e}_{n}\right\}$ be a basis of $V$. The following are equivalent:

(a)   $s:V\to {V}^{*}$ is bijective.
(b)   $d:V\to {V}^{*}$ is bijective.
(c)   ${‖{e}_{1}\wedge \cdots \wedge {e}_{n}‖}^{2}$ is invertible in $F$.

## Hermitian, symmetric and skew-symmetric forms

Let $A=B=F$ be a ring and let $\stackrel{‾}{\phantom{P}}:F\to F$ be an involutive antiautomorphism so that

 if   $b,{b}_{1},{b}_{2}\in B$      then    $\stackrel{‾}{{b}_{1}{b}_{2}}=\stackrel{‾}{{b}_{2}}\phantom{\rule{0.2em}{0ex}}\stackrel{‾}{{b}_{1}},\phantom{\rule{2em}{0ex}}\text{and}\phantom{\rule{2em}{0ex}}\stackrel{‾}{\left(\stackrel{‾}{b}\right)}=b$.
Let $\epsilon \in Z\left(F\right)$. An $\epsilon$-Hermitian form is a sesquilinear form $⟨,⟩:V×V\to F$ such that
 if    ${v}_{1},{v}_{2}\in V$    then    $⟨{v}_{2},{v}_{1}⟩=\epsilon \stackrel{‾}{⟨{v}_{1},{v}_{2}⟩}$.
A Hermitian form is a sesquilinear form $⟨,⟩:V×V\to F$ such that $\epsilon =1$ so that
 if ${v}_{1},{v}_{2}\in V$ then $⟨{v}_{2},{v}_{1}⟩=\stackrel{‾}{⟨{v}_{1},{v}_{2}⟩}$.
A symmetric form is a sesquilinear form $⟨,⟩:V×V\to F$ such that $\stackrel{‾}{\phantom{P}}=\mathrm{id}$ and $\epsilon =1$ so that
 if ${v}_{1},{v}_{2}\in V$ then $⟨{v}_{2},{v}_{1}⟩=⟨{v}_{1},{v}_{2}⟩$.
A skew-symmetric form is a sesquilinear form $⟨,⟩:V×V\to F$ such that $\stackrel{‾}{\phantom{P}}=\mathrm{id}$ and $\epsilon =-1$ so that
 if ${v}_{1},{v}_{2}\in V$ then $⟨{v}_{2},{v}_{1}⟩=-⟨{v}_{1},{v}_{2}⟩$.

PUT Gram-Schmidt and principal minors here, following [Bou, Ch. 9 §6 Prop. 1].

## Positive Hermitian forms

Let $𝔽$ be a maximal ordered field and let
(a)   $A=𝔽$ and $\stackrel{‾}{\phantom{P}}=\mathrm{id}$,
(b)   $A=𝔽\left[i\right]$ with ${i}^{2}=-1$ and $\stackrel{‾}{\phantom{P}}$ the conjugation antiautomorphism,
(c)   $A=𝔽\left[i,j,k\right]$ a quaternion algebra over $𝔽$ and $\stackrel{‾}{\phantom{P}}$ the conjugation antiautomorphism,
A positive Hermitian form is a Hermitian form $⟨,⟩:V×V\to A$ such that
 if $v\in V$ then $⟨v,v⟩\in {F}_{\ge 0}$.
(Note that $⟨v,v⟩\in F$ since $⟨v,v⟩=\stackrel{‾}{⟨v,v⟩}$.)

Assume $A=𝔽$ or $A=𝔽\left[i\right]$, $E$ is finite dimensional and $⟨,⟩:V×V\to A$ is a positive nondegenerate Hermitian form. Let $k\in {ℤ}_{>0}$. Then

(a)
 $⟨,⟩:{V}^{\otimes k}×{V}^{\otimes k}\to A\phantom{\rule{2em}{0ex}}\text{and}\phantom{\rule{2em}{0ex}}⟨,⟩:{\Lambda }^{k}V×{\Lambda }^{k}W\to A$
are positive nondegenerate.
(b)   If $x,y\in V$ then
 $⟨x,y⟩\stackrel{‾}{⟨x,y⟩}\le ⟨x,x⟩⟨y,y⟩$.
(c)   If $A$ is commutative then the principal minors of the matrix of $⟨,⟩$ are positive.

## Notes and References

These notes follow Bourbaki [Bou]. In particular the definitions of adjoints follows [Bou, Ch. 9 §1 no. 8], the definitions of product forms follows [Bou, Ch. 9. §1 no. 9], the definition of discriminants follows [Bou, Ch. 9 §2], the definitions of Hermitian forms follows [Bou, Ch. 9 §3] the definitions of positive Hemitian forms follows [Bou, Ch. 9 §7 no. 1]. WARNING: These statements have not been checked carefully, and a careful check, probably a lecture presentation needs to be done to check everything thoroughly. The proof of the Cauchy-Schwartz identity appears in [Bourbaki, Topological Vector Spaces Ch. V §2 Prop. 2] with the same proof as in [Ru]. The proof here comes from [Bou], where it is pointed out that the restriction of a positive form, to the two dimensional space spanned by $x$ and $y$ is positive. A third proof is via Lagrange's identity, found in [Bou, ???], and in the complex case, in [Ah, §1.4 Ex. 5]. In the real case Lagrange's identity and the Schwarz inequality are

 $0\le \frac{1}{2}\sum _{i,j=1}^{n}{\left({x}_{i}{y}_{j}-{x}_{j}{y}_{i}\right)}^{2}={‖x‖}^{2}{‖y‖}^{2}-{⟨x,y⟩}^{2}$
and, in the complex case,
 ${|\sum _{i=1}^{n}{a}_{i}{b}_{i}|}^{2}=\sum _{i=1}^{n}{|{a}_{i}|}^{2}\sum _{i=1}^{n}{|{b}_{i}|}^{2}-\sum _{1\le i.
SHOULDN'T THERE BE A bar ON THE ${b}_{i}$ on the LHS? CHECK THIS. NO, there isn't a bar in Ahlfors, or in any of the identities in [Ah, § 1.5].

## References

[Ah] Lars V. Ahlfors, Complex analysis. An introduction to the theory of analytic functions of one complex variable, Third edition, International Series in Pure and Applied Mathematics. McGraw-Hill Book Co., New York, 1978. xi+331 pp. ISBN: 0-07-000657-1, 30-01 MR0510197.

[Bou] N. Bourbaki, Algèbre, Chapitre 9: Formes sesquilinéaires et formes quadratiques, Actualités Sci. Ind. no. 1272 Hermann, Paris, 1959, 211 pp. MR0107661.

[Ru] W. Rudin, Real and complex analysis, Third edition, McGraw-Hill, 1987. MR0924157.