TITLE: Describing $SU(n,C)$
QUESTION [1 upvotes]: For $A \in SU(2,C)$, it is clear that $A$ is completly determined by its first row (well any row or column, but let's say first column). In the general $SU(n,C)$-case this is no longer true. In fact, it seems that for every complex $n$-vector of unit norm, there exists a family of matrices for which the vector is the first column. By working with messy simultaneous equations I seem to have shown that every two elements of this family differ by a multiple of $SU(n-1)$ embedded in the bottom right hand corner with 1 in the first entry of the first column and zeros everywhere else. I suspect that there is a bijective correspondence between elements of this family and $SU(n-1)$ but can't prove it.
Can anyone give a simple slick reworking of all of this?

REPLY [3 votes]: Use the fact that matrices act on vectors. $SU(n)$ acts transitively on the space of unit-length vectors; the stabilizer of a point  is $SU(n-1)$ by Thorny's argument. For example, for the vector $(1,0,...,0)$ the stabilizer is the subgroup $\left(\begin{matrix}1&0\\\\ 0&A\end{matrix}\right)\approx SU(n-1)$. Now by the orbit-stabilizer theorem, the space of unit-length vectors is identified with $SU(n)/SU(n-1)$. Fixing one vector $(1,0,...,0)$ fixes this identification, and then each other vector corresponds to a coset $gSU(n-1)$ which is the family you describe.
This isn't really using much about matrices or geometry; I referred to this as the "orbit-stabilizer theorem" above, but it is really just the basic structural feature of group actions. It's certainly something you can understand by yourself; if it's not immediately obvious, you can think about some simpler examples. In the group of permutations $S_n$, consider the family of permutations that map $1\mapsto 3$ -- how do elements of this family differ from each other? You could also try extending your argument to understand the first column of matrices in $GL(n,\mathbb{R})$; you will of course find a similar answer, but the details are interestingly different. Another fun example is to consider linear functions from $\mathbb{R}\to\mathbb{R}$, and look at the family of linear functions taking $2\mapsto 7$.
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