TITLE: Parametrization of real-valued SU(N)
QUESTION [5 upvotes]: I want to construct a $SU(N)$ matrix $V$, with the following property:

All the elements of the first row are given, i.e. $V_{1,j}=a_i$ (with $\sum_i a_i^2=1$)
All matrix elements are real, i.e. $V_{i,j} \in \mathbb{R}$


How can I find a matrix $V$ that satifies the criteria? Specifically, how can I find the matrix elements as a function of $a_i$, i.e. $V_{i,j}(a_i)$?


Special case: SU(2)
$$
   V=
  \left[ {\begin{array}{cc}
   a_1 & a_2 \\
   V_{2,1} & V_{2,2} \\
  \end{array} } \right]
$$
We easily find $V_{2,1}=-a_2$ and $V_{2,2}=a_1$.

Special case: SU(3)
$$
   V=
  \left[ {\begin{array}{cc}
   a_1 & a_2 & a_3 \\
   V_{2,1} & V_{2,2} & V_{2,3} \\
   V_{3,1} & V_{3,2} & V_{3,3} \\
  \end{array} } \right]
$$
Here already I cannot find any feasible way to represent $V_{i,j}$ as a function of $a_1, a_2, a_3$. I have tried to use the generators of SU(3), the Gell-Mann matrices $\lambda_i$. In particular, $\lambda_2$, $\lambda_5$, $\lambda_7$ are the generators for real-valued SU(3) matrices. However, the resulting equation system involves multiple trigonometric functions for which I cannot solve $V_{i,j}(a_1, a_2, a_3)$.
The matrix $V$ is not unique, I just want any solution.

REPLY [3 votes]: In addition to the comments I made above about continuous solutions, I thought I'd point out a solution that works for all $n$ with only one point of discontinuity, namely
$$
(a_1\ a_2\ \ldots\ a_n) = (1\ 0\ \ldots\ 0).
$$
Away from this point, one can start with the following formulae:
$$
V_{i,1}= V_{1,i} = a_i\qquad\text{and}\qquad
V_{i,j} = \delta_{ij} - \frac{a_ia_j}{(1-a_1)}\quad\text{when}\ 1<i,j\le n
$$
Note that the resulting matrix is both orthgonal and symmetric.  However, it has determinant $-1$, so reversing a single row, say, the last one, will give a matrix in $\mathrm{SO}(n)$.