Show that $x$ and $Q x$ are equidistant

Question

$$Q= \begin{bmatrix} \cos x & -\sin x\\ \sin x & \cos x\end{bmatrix}$$

Given x belongs to $\mathbb{R^2}$, show $Qx$ and $x$ are equidistant.

I've tried dot producting $Qx$ and seeing whether they are equal. I just can't seem to get it.

Answer 1

Firstly, we can easily prove that $Q $ is an orthogonal matrix, i.e. $Q^{-1} = Q^T$. Now, for every orthogonal matrix we have that $\|Qx\| = \|x \|$.

Proof:

$$\|Qx\|^2 =(Qx)^T (Qx) = x^TQ^TQx = x^T(Q^TQ) x = x^TIx = x^Tx= \|x\|^2$$

Answer 2

Of course, @thanasissdr answer is the well educated one, but arguing more naïvely:

As written in my comment: $$d((a,b),(0,0)=\sqrt{(a-0)^2+(b-0)^2}=\sqrt{a^2+b^2}=\|(a,b)\|$$

Thus we compare the norms.

We have:

$$\|(a,b)\|=\sqrt{a^2+b^2}$$ and \begin{align*}\|Q(x,y)\|& =\|(\cos(x)a+\sin(x)b,-\sin(x)a+\cos(x)b)\|\\ & =\sqrt{(\cos(x)a+\sin(x)b)^2+(-\sin(x)a+\cos(x)b)^2}\\ & =\sqrt{\cos^2(x)a^2+\sin^2(x)b^2+2\sin(x)\cos(x)ab\cos^2(x)b^2+\sin^2(x)a^2\color{red}{-}2\sin(x)\cos(x)ab}\\ & =\sqrt{[\sin^2(x)+ \cos^2(x)]a^2+[\sin^2(x)b^2+\cos^2(x)]b^2}\\ & =\sqrt{a^2+b^2}\end{align*} as claimed, where we have used the classic identity $\sin^2(x)+\cos^2(x)=1$.

Note: Any matrix $Q$ which fulfills this property is called orthogonal matrix. This is equivalent to the above cited criterion $Q^\top Q=1$.

Answer 3

Kuldeep Singh's Linear Algebra: Step by Step (2013) p 327 details this more.

There's a typo. "Exercise 4" ought "Exercise 9".

Unlike other textbooks, Singh gainfully posted all solutions!