$2\times 2$ লম্বিক ম্যাট্রিক্সসমূহের শ্রেণিবিভাগ

An orthogonal matrix is a square matrix that has rows and columns equal to orthogonal unit vectors.

The classification of these matrices is more complicated in higher dimensions, but for $2\times 2$ matrices, there are only two types of orthogonal matrices, as we will now show.

Suppose that we have a $2\times 2$ orthogonal matrix $A$: $$A=\left[ \begin{array}{ccc} a & b \\ c & d \end{array} \right]$$ Then the following conditions hold: $$a^2+b^2=1$$ $$c^2+d^2=1$$ $$ac+bd=0$$ Without loss of generality, let $a=\cos\theta$ and $b=\sin\theta$. Then we have the following: $$c^2+d^2=1$$ $$c\cos\theta+d\sin\theta=0$$ Solving for $c$, we find that $c=-d\tan\theta$. Plugging into the first equation, $$d^2+(-d\tan\theta)^2=1$$ $$d^2\sec^2\theta=1$$ $$d=\cos\theta, -\cos\theta$$ $$c=-\sin\theta, \sin\theta$$ Thus, any $2\times 2$ orthogonal matrix has one of the following two forms: $$A=\left[ \begin{array}{ccc} \cos\theta & \sin\theta \\ -\sin\theta & \cos\theta \end{array} \right]$$ $$A=\left[ \begin{array}{ccc} \cos\theta & \sin\theta \\ \sin\theta & -\cos\theta \end{array} \right]$$ These are more than just simple expressions for orthogonal matrices; these forms both have distinct meanings in the language of linear transformations.

The first form is a rotation. This matrix will rotate every vector by an angle $-\theta$ (Whoops!). The second form is a reflection. This matrix will reflect every vector across the line defined by polar angle $\frac{\theta}{2}$ (Again, watch out!).