Orthogonal matrix whose first column is given

Question

What would be an orthogonal matrix whose first column is $\underline{x} = \begin{vmatrix} -1\\ \underline{y}\\ \end{vmatrix}$, where $\underline{y} \in {\rm I\!R}^{n-1} $, $\underline{x} \in {\rm I\!R}^{n}$?

I'm not sure where to begin.

Answer 1

HINTS:

An orthogonal matrix ${\bf M}$ has the property that ${\bf M}{\bf M}^{\top} = {\bf I}$, where $\top$ denotes the transpose, and ${\bf I}$ is the $n$-by-$n$ identity matrix. A corollary of this is that each colomn - thought of as a vector - must be unit length, and any two columns - thought of as vectors - must be mutually perpendicular.

Recall that a vector ${\bf v} = (v_1,\ldots,v_n)$ has length $\| {\bf v} \| = \sqrt{v_1^2+\cdots+v_n^2}$, and two non-zero vectors ${\bf v}$ and ${\bf w}$ are perpendicular if, and only if, the scalar product ${\bf v \cdot w}=0$, i.e. $v_1w_1+\cdots+v_nw_n=0$.