Question

Find all orthogonal 2 × 2 matrices.

Short answer

Any$2\times 2$ orthogonal matrix has one of the following forms,

role="math" localid="1660124575708" $\left[\begin{array}{cc}a& b\\ -b& a\end{array}\right]v\left[\begin{array}{cc}a& b\\ b& -a\end{array}\right],a,b\in R,a^2+b^2=1$

Step-by-step solution

Form of the matrix A.

Let’s assume that${\stackrel{r}{v}}_1={\left(a,c\right)}^T,{\stackrel{r}{v}}_2={\left(b,d\right)}^T$ are the column vectors of an orthogonal matrix A.

Therefore, orthogonal conditions are$\left|\left|{\stackrel{\perp }{v}}_1\right|\right|=\left|\left|{\stackrel{\perp }{v}}_2\right|\right|=1$ and${\stackrel{\perp }{v}}_1.{\stackrel{\perp }{v}}_2=0$.

Let’s find the form of matrix.

$$\begin{gathered} \left|\left|{\stackrel{r}{v}}_1\right|\right|=1\Rightarrow \sqrt{a^2+c^2}=1 \\ \Rightarrow a^2+c^2=1 \\ \left|\left|{\stackrel{r}{v}}_2\right|\right|=1\Rightarrow \sqrt{b^2+d^2}=1 \\ \Rightarrow b^2+d^2=1 \\ {\stackrel{r}{v}}_1.{\stackrel{r}{v}}_2=0\Rightarrow \left[ac\right]\left[\begin{array}{c}b\\ d\end{array}\right] \\ \Rightarrow ab+cd=0 \end{gathered}$$

Now use these equations to make conclusion about the entries of A.

$\begin{array}{r}{a}^2+c^2=1\wedge b^2+d^2=1\wedge ab+cd=0\\ \Rightarrow a^2=1-c^2\wedge b^2=1-d^2\wedge ab=-cd\\ \Rightarrow a^2=1-c^2\wedge d^2=1-b^2\wedge (ab{)}^2=(-cd{)}^2\\ \Rightarrow a^2=1-c^2\wedge d^2=1-b^2\wedge a^2{b}^2=c^2{d}^2\\ \Rightarrow a^2=1-c^2\wedge d^2=1-b^2\wedge \left(1-c^2\right)b=c^2\left(1-b^2\right)\\ \Rightarrow a^2=1-c^2\wedge d^2=1-b^2\wedge b^2-c^2{b}^2=c^2-c^2{b}^2\\ \Rightarrow a^2=1-c^2\wedge d^2=1-b^2\wedge b^2=c^2\\ \Rightarrow a^2=1-b^2\wedge d^2=1-b^2\wedge b^2=c^2\\ \Rightarrow a^2=d^2\wedge b^2=c^2\wedge ab+cd=0\\ \Rightarrow d=\pm a\wedge b^2=c^2\wedge ab+cd=0\\ \Rightarrow d=\pm a\wedge b^2=c^2\wedge ab+c(\pm a)=0\\ \Rightarrow d=\pm a\wedge b^2=c^2\wedge a(b+\pm c)=0\\ \Rightarrow d=\pm a\wedge b^2=c^2\wedge (b+\pm c)=0,(a\ne 0)\\ \Rightarrow d=\pm a\wedge c=\mathrm{mb}\end{array}$

Hence, the solution is any $2\times 2$orthogonal matrix has one of the following forms is

$\left[\begin{array}{cc}a& b\\ -b& a\end{array}\right]v\left[\begin{array}{cc}a& b\\ b& -a\end{array}\right],a,b\in R,a^2+b^2=1$.