Prove that if the norms of vectors are equal, there is a unitary transformation such that $Tx=y$

Question

Let $V$ be unitary space with finite dimension, and let $x,y\in V$ such that $||x||=||y||$.

Prove that there is an unitary linear transformation $T:V\rightarrow V$ such that $T(x)=y$.

I tried taking two orthonormal basis, $\{v_1,...v_n\}$ and $\{u_1,...u_n\}$ of $V$, and if I define $T$ such that $T(v_i)=u_i$ I know that $T$ is unitary since it takes an orthonormal basis and transform it onto other orthonormal basis.

Then, I write $x$ with the first basis, and $y$ with the second basis. However, it does not seem to help to prove that $Tx=y$.

How can I continue from here?

Answer 1

You can find orthonormal bases $\{v_1,v_2,..,v_n\}$ and $\{u_1,u_2,..,u_n\}$ with $v_1=\frac {x} {\|x\|}$ and $u_1=\frac {y} {\|y\|}$. Then your definition $T(v_1)=u_1$ automatically gives $Tx=y$.

Answer 2

Since $\|x\|=\|y\|=r$, we can map $x$ onto $y$ by reflecting it along the angle bisector. The corresponding unitary linear transformation is known as a Householder reflection. It is given as

$$ T = I - 2vv^T \qquad v = \frac{y-x}{\|y-x\|}$$