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If given the following affine maps, how do I go about proving the existence of a unique fixed point, if and only if $(\lambda + \mu) \cos( \theta) \ne 1+ \lambda \mu$?

$$\begin{bmatrix} x_1 \\ x_2 \\ \end{bmatrix} = \begin{bmatrix} \lambda \cos \theta & -\lambda \sin \theta \\ \mu \sin \theta & \mu \cos \theta \\ \end{bmatrix} \begin{bmatrix} x_1 \\ x_2 \\ \end{bmatrix} + \begin{bmatrix} b_1 \\ b_2 \\ \end{bmatrix}$$

I recall a fixed point being present if there exists an $a_0$ such that $f(a_0) = a_0$, and that $b=0$ only at the fixed point. I am having issues tying this knowledge together, and I have not yet touched on the concept of unique fixed points in my computer graphics course.

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    I'm not sure what you mane by $b=0$ above. – 2017-02-05
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    The translation component,$b$ is only allowed to by equal to 0 at the fixed point if I am recalling correctly from linear algebra. It means the $b_1$ and $b_2$ can be eliminated from the equations, I think. – 2017-02-05
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    Well, under the above conditions you will have $x=0$ **iff** $b = 0$. – 2017-02-05

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Look at it as a system $x = Ax +b$, or $(A-I)x = -b$. Then this system has a unique solution iff $\det (A-I) \neq 0$. If you compute $\det (A-I)$ you will get the equation above.

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    Ah, so it's just simple algebra then? Thank you! – 2017-02-05