Consider the matrix $$B_{\theta}= \begin{bmatrix} \cos(\theta) & \sin(\theta) \\ \sin(\theta) & -\cos{\theta} \end{bmatrix}$$
What are its eigen vectors?
Ofcourse, since this is a rotation matrix, i know that there are only eigen-vectors given that $\theta = n\pi$, whose direction points along the x and y axis.
However how would I actually calculate this?
The characteristic equation is:
$$\lambda^2-1 = 0$$ $$\lambda_{1} = 1 , \lambda_{2} = -1$$
Solving $$[B_{\theta} - 1I]\vec{V} = 0$$
$$\begin{bmatrix} \cos(\theta)-1 & \sin(\theta) \\ \sin(\theta) & -\cos{\theta}-1 \end{bmatrix} \begin{bmatrix} x \\ y \end{bmatrix} = \begin{bmatrix} 0 \\ 0 \end{bmatrix} $$
Writing the extended matrix to use gaussian elimination
$$\left( \begin{array}{cc|c} \cos(\theta)-1 & \sin(\theta) & 0 \\ \sin(\theta) & -\cos{\theta}-1 & 0 \\ \end{array} \right)$$
Row operation:
$$R_{1} \rightarrow R_{1} - \frac{\cos(\theta)-1}{\sin(\theta)} R_{2}$$
In order to get this in gaussian form, i need to divide by $sin(\theta)$ which is zero when theta is $n\pi$, which is the values of which i suspect that there should be solutions.
How do I go about solving for the eigen values? Without using gaussian elimination i cannot find the solutions to the equations. can anyone help?