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morphism
#4
Feb25-12, 01:31 PM
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P: 2,020
Casting 2x2 matrix with unit determinant in another form

For the Iwasawa decomp of SL(2,R), these notes of Keith Conrad are nice.

I should mention that what's going on here is part of a bigger picture: the Iwasawa decomposition is really a statement about decomposing certain types of groups (e.g. semisimple Lie groups) of which SL(2,R) is an example.

For an example of another decomposition, you can refine the polar decomposition that you learned in linear algebra to obtain the so-called KAK decomposition, which also applies to a broad class of Lie groups. For SL(2,R), this states that any A can be written as
[tex]A = \begin{pmatrix} \cos\theta & -\sin\theta \\ \sin\theta & \cos\theta \end{pmatrix} \begin{pmatrix} a & 0 \\ 0 & a^{-1}\end{pmatrix} \begin{pmatrix} \cos\psi & -\sin\psi \\ \sin\psi & \cos\psi \end{pmatrix},[/tex]
where [itex]\theta,\psi \in [0,2\pi)[/itex] and [itex]a>0[/itex] are uniquely determined by A.

Yet another example is the Bruhat decomposition, which states that [itex]A \in \text{SL}(2,\mathbb R)[/itex] is either upper-triangular, or else can be decomposed as
[tex]\begin{pmatrix} a & b \\ c & d \end{pmatrix} = \begin{pmatrix} 1 & ac^{-1} \\ 0 & 1 \end{pmatrix} \begin{pmatrix} 0 & -1 \\ 1 & 0 \end{pmatrix} \begin{pmatrix} c & d \\ 0 & c^{-1} \end{pmatrix}.[/tex] Note that c is nonzero since we're assuming A is not upper-triangular, so it makes sense to invert it. Also note that b appears to have vanished, but of course it's still there as [itex]b = (ad-1)c^{-1}[/itex].