I am working on a problem that asks me to show a $2 \times 2$ orthogonal matrix is reflection matrix or the rotation matrix and I do not know the definitions of the rotation matrix and the reflection matrix, the unitary matrix and some good examples of them and could anyone tell me something about them?
[Math] Rotation Matrix and Reflection Matrix, Unitary Matrix
linear algebramatrices
Best Answer
Unitary matrices matrices $A$ are invertible matrices that have property that $A^T = A^{-1}$. This leads to a very neat geometric characterisation: a matrix $A$ is unitary if and only if the corresponding linear map $v \mapsto Av$ is an isometry, meaning that it preserves lengths and distances. In other words, $\|Av\| = \|v\|$ for all $v$.
To give you an idea of why this is true, consider the identity $\|w\|^2 = w^T w$, where $w$ is an $n \times 1$ column vector. Then, if $A$ is unitary, $$\|Aw\|^2 = (Aw)^TAw = w^T A^T A w = w^T I w = w^T w = \|w\|^2.$$ Proving the other direction is a little more involved, so I won't get into it, but this should give you an idea as to how these concepts are connected.
If you think about rotations and reflection transformations, they also preserve lengths and distances, so their matrices should indeed be unitary.
You can look up formulas for rotation and reflection matrices, but it's also possible to derive them. Let's begin with a rotation matrix. We want, for any angle $\theta$, to find the matrix corresponding to the (linear) rotation map that rotates the space anti-clockwise (as is usually the convention).
The way I usually like to do this is with complex numbers. If we take an arbitrary vector $(x, y) \in \mathbb{R}^2$, we can identify it with a complex number $x + iy \in \mathbb{C}$. In $\mathbb{C}$, rotation anti-clockwise by $\theta$ can be done by multiplying $x + iy$ by the complex number $e^{i\theta} = \cos(\theta) + i \sin(\theta)$, which has modulus $1$ and argument $\theta$. The resulting complex number is: $$e^{i\theta}(x + iy) = x\cos(\theta) - y \sin(\theta) + i(x \sin(\theta) + y \cos(\theta)).$$ If we bring this back to $\mathbb{R}^2$, then the transformation, in total, does the following: $$(x, y) \mapsto (x\cos(\theta) - y \sin(\theta), x \sin(\theta) + y \cos(\theta)).$$ Putting this as a matrix gives us $$\begin{pmatrix} \cos(\theta) & -\sin(\theta) \\ \sin(\theta) & \cos(\theta) \end{pmatrix}.$$ I'll leave it to you to verify that $A^T A = A A^T = I$, and hence $A^{-1} = A^T$.
As for reflections, again, I prefer to do it with complex numbers. Recall that the complex conjugate reflects about the $x$-axis. Let's say that we want to reflect about a line that makes an angle $\theta$ with the positive $x$-axis (again, anti-clockwise). Our plan is to rotate the space so that this line becomes the $x$-axis, reflect in the $x$-axis using the conjugate, and rotate the space back.
Take an arbitrary $z = x + iy \in \mathbb{C}$. First, to rotate the space, we rotate everything clockwise (not anti-clockwise) by $\theta$. Our point is now at $e^{-i\theta}z$. We now take the complex conjugate, giving us, $$\overline{e^{-i\theta}z} = \overline{e^{-i\theta}}\overline{z} = e^{i\theta}\overline{z}.$$ Finally, we rotate back, that is anti-clockwise by $\theta$. This gives us our final total of $$e^{i\theta}e^{i\theta}\overline{z} = (e^{i\theta})^2\overline{z} = e^{i2\theta}\overline{z}.$$ Putting $z = x + iy$, we get, $$e^{i2\theta}\overline{z} = (\cos(2\theta) + i \sin(2\theta))(x - iy) = (x\cos(2\theta)+ y\sin(2\theta)) + i(x \sin(2\theta) - y \cos(2\theta)).$$ As a transformation on $\mathbb{R}^2$, $$(x, y) \mapsto (x\cos(2\theta)+ y\sin(2\theta), x \sin(2\theta) - y \cos(2\theta)).$$ This gives us a matrix $$\begin{pmatrix} \cos(2\theta) & \sin(2\theta) \\ \sin(2\theta) & -\cos(2\theta) \end{pmatrix}.$$ Again, show that $A^T A = A A^T = I$, and you're done.