Question

Show that if $\mathit{C}$is a matrix whose columns are the components $\left(x_1,y_1\right)$and $\left(x_2,y_2\right)$of two perpendicular vectors each of unit length, then $\mathit{C}$is an orthogonal matrix. Hint: Find${\mathit{C}}^{\mathbf{T}}\mathit{C}$

Short answer

$\mathit{C}$ is an orthogonal matrix.

Step-by-step solution

Given information

C is a matrix whose columns are the components $\left(x_1,y_1\right)$ and $\left(x_2,y_2\right)$ of two perpendicular vectors, each of unit length.

Orthogonal matrix

An orthogonal matrix, or orthonormal matrix, is a real square matrix whose columns and rows are orthonormal vectors.

Calculate the matrix  C

Write matrix $C$as $C=\left(r_1,r_2\right)$, where $r_1$and $r_2$ are orthogonal unit vectors, that is,

$$\begin{gathered} {r_1}^T{r}_2=0 \\ {r_1}^T{r}_1=1 \\ {r_2}^T{r}_2=1 \end{gathered}$$

Now, calculate $C^{T}C$.

$$\begin{gathered} C^{T}C=\left(\begin{array}{c}{r}_1^T\\ r_2^2\end{array}\right)\left(\begin{array}{cc}{r}_1& r_2\end{array}\right) \\ =\left(\begin{array}{c}{r}_1^T{r}_1\\ r_2^T{r}_1\end{array}\begin{array}{c}{r}_1^T{r}_2\\ r_1^T{r}_1\end{array}\right) \end{gathered}$$

Use the relations above for the orthonormality of $r_1$and $r_2$.

$C^{T}C=\left(\begin{array}{cc}1& 0\\ n& 1\end{array}\right)$

The above equation shows that $C$ is an orthogonal matrix.