To find the fixed points of the bilinear map, we need to solve for the points where $w = z$. The given bilinear map is:

$w = \frac{2i z - 6}{z - 3}$

A fixed point occurs when $w = z$. So we set $w = z$ in the given bilinear map:

$z = \frac{2i z - 6}{z - 3}$

Now, we solve this equation for $z$:

1. Multiply both sides by $z - 3$ to clear the denominator:

$z(z - 3) = 2i z - 6$

2. Expand and rearrange the equation:

$z^2 - 3z = 2i z - 6$

$z^2 - 3z - 2i z + 6 = 0$

3. Combine like terms:

$z^2 - (3 + 2i)z + 6 = 0$

This is a quadratic equation in the form $az^2 + bz + c = 0$, where $a = 1$, $b = -(3 + 2i)$, and $c = 6$.

4. Solve for $z$ using the quadratic formula:

$z = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$

Substitute $a = 1$, $b = -(3 + 2i)$, and $c = 6$ into the formula:

$z = \frac{-(-3 - 2i) \pm \sqrt{(-3 - 2i)^2 - 4 \cdot 1 \cdot 6}}{2 \cdot 1}$

$z = \frac{3 + 2i \pm \sqrt{(3 + 2i)^2 - 24}}{2}$

5. Calculate $(3 + 2i)^2$:

$(3 + 2i)^2 = 9 + 12i + 4i^2 = 9 + 12i - 4 = 5 + 12i$

6. Now substitute back:

$z = \frac{3 + 2i \pm \sqrt{(5 + 12i) - 24}}{2}$

$z = \frac{3 + 2i \pm \sqrt{-19 + 12i}}{2}$

7. Finding the square root of $-19 + 12i$ can be done using the polar form. Let $r = \sqrt{(-19)^2 + 12^2}$ and $\theta = \tan^{-1}\left(\frac{12}{-19}\right)$. After calculating the square root, use:

$z_1 = \frac{3 + 2i + \sqrt{-19 + 12i}}{2}$

$z_2 = \frac{3 + 2i - \sqrt{-19 + 12i}}{2}$

These $z_1$ and $z_2$ values are the fixed points of the bilinear map.

Summary

The fixed points of the bilinear map $w = \frac{2iz - 6}{z - 3}$ are the solutions to the quadratic equation $z^2 - (3 + 2i)z + 6 = 0$. Solving this using the quadratic formula gives the fixed points.

Do you want details or have any questions?

Here are some related questions to deepen your understanding:

1. What is the geometric interpretation of fixed points in a bilinear map?
2. How do fixed points of a map relate to its critical points?
3. What is the significance of the discriminant in finding fixed points?
4. How do we find fixed points for a nonlinear transformation?
5. What are the applications of fixed points in complex dynamics?

Tip: When solving quadratic equations with complex coefficients, keep track of real and imaginary parts separately for accurate calculations.