Rotation by Householder matrices I have two vectors, let's say $u, v \in \mathbb{R}^n$ with the same norm $\|u\| = \|v\| = 1$. I want to map first vector to second using 2 Householder reflections $(I - 2pp^T), \|p\| = 1$. Is it always possible and what is the formula for such reflections?
For me it seems that it is true, because we need the rotation, so every rotation can be represented as a composition of 2 reflections. But I can't find exactly formula. Thanks for the help!

It is indeed always possible. One formula that works is this: take
$$
p=\frac{u-v}{\|u-v\|}
$$
To see that this works, note that
$$
p^Tu= \frac {u^Tu - v^Tu}{\|u-v\|}=
\frac 12 \frac{u^Tu +v^Tv - u^Tv - v^Tu}{\|u-v\|}=\\
\frac 12 \frac{(u-v)^T(u-v)}{\|u-v\|}= \frac 12 \|u-v\|
$$
