Find a Givens rotation matrix such that $y=Gx$

Question

Assume that $x,y \in \mathbb{R}^2$ with $||x||_2=||y||_2=1$.

Find a Givens rotation matrix $G=\begin{bmatrix}c & s \\ -s & c \end{bmatrix}$ (i.e., find $c$ and $d$ with $c^2+d^2=1$) such that $y=Gx$.

Answer:

Let $x=(x_1,x_2) \ $ and $ \ y=(y_1,y_2)$. Then,

$\begin{bmatrix} y_1 \\ y_2 \end{bmatrix}=\begin{bmatrix}c & s \\ -s & c \end{bmatrix} \begin{bmatrix} x_1 \\ x_2 \end{bmatrix}$.

This gives us,

$y_1=cx_1+sx_2, \\ y_2=-sx_1+cx_2.$

Let $c=\cos \theta, \ s=\sin \theta$, then $ \ \sqrt{c^2+d^2}=1$.

But how to find the angles $\theta$ ?

Help me

Answer 1

HINT

show that $<x,y>=c$ then $cos \theta=c=<x,y>$