Writing the ortogonality condition for any element of O(n), I've arrived to:
If we take n=2, we know that $\Lambda\Lambda^{T}=\mathbb{I}$, so:
$$\begin{pmatrix} x & y \\ z & t \end{pmatrix} \begin{pmatrix} x & z \\ y & t \end{pmatrix} = \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix} $$
This yields:
$$\begin{cases} x^{2} + y^{2} = 1\\ xz + yt = 0 \\ z^{2} + t^{2}= 1 \end{cases}$$
Visually, these equations mean that we can find two orthonormal vectors.
We could generalise our reasoning to arbitrary dimension easily.
How could I prove rigorously, without plugging in any numbers nor functions, that this system of equations has solution (infinite, in fact)?
Is there any method to prove the existence of solution of non non linear equations (or systems)?
Best Answer
The first and third equations states that points (x, y) and (z, t) are on a unit circle. Let's write that as: $x = \cos\phi$
$y = \sin\phi$
$z = \cos\psi$
$t = \sin\psi$
Now we may write the second equation as:
$\cos\phi\cos\psi + \sin\phi\sin\psi = 0$
This may be rewritten as:
$\cos (\phi-\psi) = 0$
Which has a solution:
$\phi - \psi = \frac{\pi}{2} + k\pi$
We should restrict the angles to $(-\pi, \pi)$:
$\phi = \psi \pm \frac{\pi}{2}$
Which means that for each (x,y) on a unit circle there are two possible (z, t).