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I would like to prove the Iwasawa decomposition for ${\rm SL}_2(\mathbf R)$ , the $2 \times 2$ special linear group over the real numbers. That is, if we let $U$ be all matrices of the form $\begin{bmatrix}1 & x \\ 0 & 1 \end{bmatrix}$ where $x$ is real, $A$ be all matrices of the form $\begin{bmatrix}a & 0 \\ 0 & a^{-1} \end{bmatrix}$ where $a > 0$, and $K$ be all matrices of the form $\begin{bmatrix}\cos x & \sin x \\ -\sin x & \cos x \end{bmatrix}$ where $x$ is real, it is clear that $U,A,$ and $K$ are all subgroups of ${\rm SL}_2(\mathbf R)$ with $K$ being a subgroup of the real unitary matrices and $U$ being a subgroup of unipotent matrices. How do I show that ${\rm SL}_2(\mathbf R)=UAK$, that is, all matrices in ${\rm SL}_2(\mathbf R)$ can be written as a product of a unipotent, positive diagonal, and unitary matrix in that order? Thank you a lot.

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    Typing `$\begin{bmatrix}a & b \\ c & d \end{bmatrix}$` creates this: $\begin{bmatrix}a & b \\ c & d \end{bmatrix}$. Generalize and make your question readable. :) (To do so, click on "edit" just below the list of tags that are below your question.) – 2017-02-13
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    Have you looked in books that discuss the Iwasawa decomposition? – 2017-02-13
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    @JohnHughes thank you very much – 2017-02-13
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    Have a look at the book... named $SL_2(\mathbb{R})$ by the great Serge Lang, freely loadable at (https://archive.org/details/springer_10.1007-978-1-4612-5142-2) – 2017-02-13

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