classical groups
1 Classical Groups
It is commonplace to express the classical groups^{} with explicit matrices; however, the theory and classification of classical groups can benefit from a basis free consideration.
Given a finite dimensional vector space^{} $V$ over any field $k$, the set of all linear transformations on $V$ is denoted $GL(V)$ and called the general linear group^{}. We now define several significant related groups of $GL(V)$.
Theorem 1 (Birkhoffvon Neumann).
Given a reflexive nondegenerate sesquilinear form $b\mathrm{:}V\mathrm{\times}V\mathrm{\to}k$, then up to a constant $b$ is one of the following:

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Alternating: that is $b(v,v)=0$.

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Hermitian: so $b(v,w)=b{(w,v)}^{\sigma}$ where $\sigma $ is a field automorphism of $k$ of order $2$.

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Symmetric^{}: so $b(v,w)=b(w,v)$.
(Refer to [3, Theorem^{} 7.1] and [1, Chapter V].) We prefer the definition $b(v,v)=0$ over $b(v,w)=b(w,v)$ so that we can accommodate the fields of characteristic^{} 2. In all other characteristics these two properties are equivalent^{}.
In keeping with tradition for group theory, we let $GL(V)$ act on the vector space $V$ on the right hand side. This means given $v\in V$ and $f\in GL(V)$, $vf$ corresponds the vector in $V$ which $f$ sends $v$ to. If one thinks of $f$ as a matrix this requires $v$ to be a row vector^{}. It is also common to consider $f$ as a function^{} and use the notation $f(v)$.
Definition 2.
Given a reflexive nondegenerate sesquilinear form $b\mathrm{:}V\mathrm{\times}V\mathrm{\to}k$ we define
$$Isom(b)=\{f\in GL(V):b(vf,wf)=b(v,w),v,w\in V\}.$$ 
This is called the isometry group of $b$ in $G\mathit{}L\mathit{}\mathrm{(}V\mathrm{)}$.
Proposition 3.
$Isom(b)$ is a subgroup^{} of $G\mathit{}L\mathit{}\mathrm{(}V\mathrm{)}$.
Proof.
Given $f,g\in Isom(b)$, $v,w\in V$ then
$$b(v(fg),w(fg))=b((vf)g,(wf)g)=b(vf,wf)=b(v,w).$$ 
Hence $fg\in Isom(b)$. Clearly $1\in Isom(b)$ as well. Finally,
$$b(v{f}^{1},w{f}^{1})=b((v{f}^{1})f,(w{f}^{1})f)=b(v,w).$$ 
So ${f}^{1}\in Isom(b)$ and $Isom(b)$ is a subgroup of $GL(V)$. ∎
Now if we return to Theorem 1 we find that there are only three isometry group types, as there are only three types of reflexive nondegenerate sesquilinear forms. These receive the wellknown names:

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Symplectic group $Sp(V,b)$ if $b$ is alternating.

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Unitary group^{} $U(V,b)$ if $b$ is hermitian.

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Orthogonal group^{} $O(V,b)$ if $b$ is symmetric.
A vector space $V$ equipped with a reflexive^{} nondegenerated sesquilinear form^{} $b$ is also given the designation symplectic, unitary^{}, and orthogonal^{} based on the classification of the form.
Because symplectic spaces have a standard hyperbolic basis it follows every symplectic group over a vector space of the same dimension^{} is isometric, meaning isomorphic^{} as vector spaces but with an isomorphism^{} which respects the forms. Thus we can write $Sp(V)$ instead of $Sp(V,b)$. For unitary and orthogonal groups more care is required.
Definition 4.
A classical group is any one of the family of groups derived from these three and the general linear group.
2 Classical Groups as Matrices
When expressing these groups with matrices it becomes necessary to establish the bilinear forms^{} with matrices. Given any $n\times n$matrix $B$ over some field $k$, and row vectors $v,w\in {k}^{n}$ we have a reflexive bilinear form defined by
$$b(v,w)=vB{w}^{t}.$$ 
Whence $b$ is nondegenerate if and only if $detB\ne 0$.
The most common example is the identity matrix^{} $B={I}_{n}$. For then
$$b(v,w)=v{w}^{t}$$ 
is the usual dot product^{}, only perhaps without the positive definite^{} axiom which makes sense only for ordered fields like the rationals $\mathbb{Q}$ and reals $\mathbb{R}$.
The isometry group of $I$ is nothing more than the invertible matrices $A$ where
$$AI{A}^{t}=I;A{A}^{t}=I.$$ 
Thus it is common for $O(n)$ to denote the orthogonal group over $\mathbb{R}$ and be given by
$$O(n)=\{A\in GL(n):A{A}^{t}=I\}.$$ 
For symplectic groups the form is the typical $J=\left[\begin{array}{cc}\hfill 0\hfill & \hfill I\hfill \\ \hfill I\hfill & \hfill 0\hfill \end{array}\right]$ matrix found in the definition of symplectic matrices. Hence the isometry condition for an alternating form
$$b(vA,wA)=vAJ{A}^{t}{w}^{t}=vJ{w}^{t}=b(v,w)$$ 
show that $AJ{A}^{t}=J$. Thus it is common to define
$$Sp(2m)=\{A\in GL(2m):AJ{A}^{t}=J\}.$$ 
Thus the symplectic matrices form a group of isometries.
3 Special subgroups
The commutator of $GL(V)$ is the special linear group^{} $SL(V)$ composed of all invertible linear transformations of determinant^{} 1. Given a reflexive nondegenerate sesquilinear form $b$ on $V$, we can create the groups
$$SIsom(V,b)=Isom(V,b)\cap SL(V).$$ 
These get the names
$$Sp(V),SU(V,b),\text{and}SO(V,b).$$ 
Notice that $Sp(V)\le SL(V)$ so we do not require a new name.
4 Projective groups
The projective geometry^{} of a vector space $V$, denoted $PG(V)$ is its lattice^{} of subspaces^{}. Clearly invertible linear maps^{} act on the projective geometry because they send points (1dimensional subspaces) to points, and lines (2dimensional subspaces) to lines, and in general $m$dimensional subspaces to other $m$dimensional subspaces.
However, the scalar transformations^{}, i.e.: those $f\in GL(V)$ such that $vf=v\lambda $ for some fixed $\lambda \in k$, do not move any of the subspaces of $V$ – they fix $PG(V)$. Therefore when we consider the action of $GL(V)$ on $PG(V)$ we factor out the kernel of the action – that is the scalar transforms (matrices.) We denote this group by $PGL(V)$. Because scalar matrices commute with all other, and not other matrices do, we notice this is the same as factoring by the center $Z(GL(V))$.
Immediately this gives rise the projective versions of each of the classical groups: Let $Z=Z(GL(V))$ – the set of scalar transformations (a group isomorphic to ${k}^{\times}$.)

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$PGL(V)=GL(V)/Z$, $PSL(V)=SL(V)/(Z\cap SL(V))$

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$PSp(V)=Sp(V)/(Z\cap Sp(V))$

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$PU(V)=U(V,b)/(Z\cap U(V,b))$, $PSU(V,b)=SU(V,b)/(Z\cap SU(V,b))$

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$PO(V)=O(V,b)/(Z\cap O(V,b))$, $PSO(V,b)=SO(V,b)/(Z\cap SO(V,b))$
Most of the time the projective special isometry groups are simple groups^{}. The exceptions arise for small dimensional vector spaces and/or small fields, or with the orthogonal groups.
References
 1 Gruenberg, K. W. and Weir, A.J. Linear Geometry^{} 2nd Ed. (English) [B] Graduate Texts in Mathematics. 49. New York  Heidelberg  Berlin: SpringerVerlag. (1977), pp. x198.
 2 Kantor, W. M. Lectures notes on Classical Groups.
 3 Taylor, Donald E. The geometry of the classical groups Sigma Series in Pure Mathematics. 9. Heldermann Verlag, Berlin, xii+229, (1992), ISBN 3885380099.
Title  classical groups 
Canonical name  ClassicalGroups 
Date of creation  20130322 15:50:13 
Last modified on  20130322 15:50:13 
Owner  Algeboy (12884) 
Last modified by  Algeboy (12884) 
Numerical id  23 
Author  Algeboy (12884) 
Entry type  Definition 
Classification  msc 11E57 
Synonym  linear algebraic groups 
Related topic  SemilinearTransformation 
Related topic  PolaritiesAndForms 
Related topic  SesquilinearFormsOverGeneralFields 
Defines  classical group 
Defines  isometry 