Question

Let each of the following matrices represent an active transformation of vectors in (x,y)plane (axes fixed, vector rotated or reflected).As in Example 3, show that each matrix is orthogonal, find its determinant and find its rotation angle, or find the line of reflection.

$\frac{\mathbf{1}}{\sqrt{\mathbf{2}}}\left(\begin{array}{cc}-1& -1\\ 1& -1\end{array}\right)$

Short answer

The matrix $\frac{1}{\sqrt{2}}\left(\begin{array}{cc}-1& -1\\ 1& -1\end{array}\right)$ is orthogonal, and its determinant is 1 with its rotation angle $315°$.

Step-by-step solution

Step 1:Definition of an orthogonal matrix, reflection, and rotation.

A Square matrix is orthogonal, if the product of the square matrix (c)and the transpose of the matrix $\left(c^T\right)$, are equal to the identity matrix$\left(\mathrm{I}\right)$.

The transpose of the matrix role="math" localid="1658982923269" $\left({\mathrm{c}}^{\mathrm{T}}\right)$ is the inverse of the square matric $\left(c\right)$ .

If the determinant of the square matrix is equal to the1, then the matrix represents a rotation.

If the determinant of the square matrix is equal to the-1, then the matrix represents a reflection.

Given parameters

A matrix $\frac{1}{\sqrt{2}}\left(\begin{array}{cc}-1& -1\\ 1& -1\end{array}\right)$is given.

The determinant, rotation angle, or the line of reflection of the given matrix need to be determined and also prove that it is orthogonal.

Find whether the matrix is orthogonal or not.

The matrix A given by A=$\frac{1}{\sqrt{2}}\left(\begin{array}{cc}-1& -1\\ 1& -1\end{array}\right)$is orthogonal matrix when ${\mathrm{AA}}^{\mathrm{T}}$=I , where I represent the identity matrix.

$$\begin{gathered} A{A}^T=\frac{1}{\sqrt{2}}\left(\begin{array}{cc}-1& -1\\ 1& -1\end{array}\right)\times \frac{1}{\sqrt{2}}{\left(\begin{array}{cc}-1& -1\\ 1& -1\end{array}\right)}^T \\ =\left(\frac{1}{\sqrt{2}}\right)\left(\frac{1}{\sqrt{2}}\right){\left(\begin{array}{cc}-1& -1\\ 1& -1\end{array}\right)}_{2\times 2}{\left(\begin{array}{cc}-1& -1\\ 1& -1\end{array}\right)}_{2\times 2}^T \\ =\frac{1}{2}\left(\begin{array}{cc}-1& -1\\ 1& -1\end{array}\right){\left(\begin{array}{cc}-1& -1\\ 1& -1\end{array}\right)}^T \\ =\frac{1}{2}\left(\begin{array}{cc}-1& -1\\ 1& -1\end{array}\right)\left(\begin{array}{cc}-1& 1\\ -1& -1\end{array}\right) \end{gathered}$$

Further, solve.

$$\begin{gathered} A{A}^T=\frac{1}{2}\left(\begin{array}{cc}\left(-1\right)\left(-1\right)+\left(-1\right)\left(-1\right)& \left(-1\right)\left(1\right)+\left(-1\right)\left(-1\right)\\ \left(1\right)\left(-1\right)+\left(-1\right)\left(-1\right)& \left(1\right)\left(1\right)+\left(-1\right)\left(-1\right)\end{array}\right) \\ =\frac{1}{2}\left(\begin{array}{cc}1+1& 1-1\\ -1+1& 1+1\end{array}\right) \\ =\frac{1}{2}\left(\begin{array}{cc}2& 0\\ 0& 2\end{array}\right) \end{gathered}$$

Factor out 2 from the matrix.

$$\begin{gathered} A{A}^T=\left(\frac{1}{2}\right)\left(2\right)\left(\begin{array}{cc}1& 0\\ 0& 1\end{array}\right) \\ =\left(\begin{array}{cc}1& 0\\ 0& 1\end{array}\right) \\ =\mathrm{I} \end{gathered}$$

Thus, it has been proved that A is an orthogonal matrix as $A{A}^T=I$, where I represent the identity matrix.

Step 4: Find the determinant of the matrix

The determinant of order two is determined by using the formula

$\left|\begin{array}{cc}{a}_{11}& a_{12}\\ a_{21}& a_{22}\end{array}\right|=a_{11}{a}_{22}-a_{12}{a}_{21}.$

Find the determinant of A.

$$\begin{gathered} det\left(A\right)=\left|\begin{array}{cc}-\frac{1}{\sqrt{2}}& -\frac{1}{\sqrt{2}}\\ \frac{1}{\sqrt{2}}& -\frac{1}{\sqrt{2}}\end{array}\right| \\ =\left[\left(-\frac{1}{\sqrt{2}}\right)\left(-\frac{1}{\sqrt{2}}\right)-\left(-\frac{1}{\sqrt{2}}\right)\left(\frac{1}{\sqrt{2}}\right)\right] \\ =\left[\frac{1}{2}-\left(-\frac{1}{2}\right)\right] \\ =\frac{1}{2}+\frac{1}{2} \end{gathered}$$

Further, solve.

$det\left(A\right)=1$

Find the axis of rotation

Any $2\times 2$orthogonal matrix with a determinant equal to 1 corresponds to a rotation, thus, it has to be shown that the following orthogonal matrix A is a rotation matrix.

Since the matrix A represents an active transformation of vectors in the (x,y) plane (axes fixed, vectors rotated, that must use the following definition of the general form of the vector representation of an active rotation in two dimensions defined as

The vector $\overrightarrow{r}$from the origin to the point (x,y) has been rotated by an angle $\theta$to become the vector $\overrightarrow{R}$from the origin to the point (X,Y) written in the matrix form $\left(\begin{array}{c}X\\ Y\end{array}\right)=\left(\begin{array}{cc}\cos \theta & -\sin \theta \\ \sin \theta & \cos \theta \end{array}\right)\left(\begin{array}{c}x\\ y\end{array}\right)$, vector rotated $\%$called rotation equation which relates the components of $\overrightarrow{r}$and$\overrightarrow{R}$ .

Compare this rotation matrix A with the general form of the matrix representation (rotation matrix) of an active rotation in two dimensions $M=\left(\begin{array}{cc}\cos \theta & -\sin \theta \\ \sin \theta & \cos \theta \end{array}\right),$it Is deduced that$\left(\cos \theta \right)=-\frac{1}{\sqrt{2}}and\sin \left(\theta \right)=\frac{1}{\sqrt{2}}$.

Find the rotation angle.

$$\begin{gathered} \theta ={\tan }^{-1}\frac{1/\sqrt{2}}{-1/\sqrt{2}} \\ ={\tan }^{-1}\left(-1\right) \\ =2\pi -\frac{\pi }{4} \\ =\frac{7\pi }{4} \\ =315° \end{gathered}$$

Further, solve.

So, this is a rotation of$315°$.