I want to construct a $SU(N)$ matrix $V$, with the following property:


1. All the elements of the first row are given, i.e. $V\_{1,j}=a\_i$ (with $\sum\_i a\_i^2=1$)
2. 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](https://en.wikipedia.org/wiki/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.



Parametrization of real-valued SU(N)
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 $\mathrm{SO}(n)$.


