# Isometries with complex variables

• Mar 31st 2011, 07:04 PM
Pinkk
Isometries with complex variables
Write the following isometries in terms of a complex variable $\displaystyle z = x + iy$.

1. $\displaystyle \tau_{a}(x) = \left[$\begin{array}{c}
x_{1}\\
x_{2}\end{array}$]\right + \left[$\begin{array}{c}
a_{1}\\
a_{2}\end{array}$]\right$

2. $\displaystyle \rho_{\theta}(x) = \left[$\begin{array}{cc}
\cos ( \theta) & -\sin (\theta) \\
\sin (\theta) & \cos (\theta) \\ \end{array}$]\right \left[$\begin{array}{c}
x_{1}\\
x_{2}\end{array}$]\right$

3. $\displaystyle r(x) = \left[$\begin{array}{cc}
1 & 0 \\
0 & -1 \\ \end{array}$]\right \left[$\begin{array}{c}
x_{1}\\
x_{2}\end{array}$]\right$

I'm not sure what the problem is asking for. Does it mean to plug in a complex variable $\displaystyle z$ instead of the $\displaystyle x$? Does it mean to change the matrices themselves or what? Wouldn't these matrices look exactly the same in the complex plane? Any help would be appreciated.
• Apr 2nd 2011, 12:00 AM
Opalg
Quote:

Originally Posted by Pinkk
Write the following isometries in terms of a complex variable $\displaystyle z = x + iy$.

1. $\displaystyle \tau_{a}(x) = \left[$\begin{array}{c}
x_{1}\\
x_{2}\end{array}$]\right + \left[$\begin{array}{c}
a_{1}\\
a_{2}\end{array}$]\right$

2. $\displaystyle \rho_{\theta}(x) = \left[$\begin{array}{cc}
\cos ( \theta) & -\sin (\theta) \\
\sin (\theta) & \cos (\theta) \\ \end{array}$]\right \left[$\begin{array}{c}
x_{1}\\
x_{2}\end{array}$]\right$

3. $\displaystyle r(x) = \left[$\begin{array}{cc}
1 & 0 \\
0 & -1 \\ \end{array}$]\right \left[$\begin{array}{c}
x_{1}\\
x_{2}\end{array}$]\right$

I'm not sure what the problem is asking for. Does it mean to plug in a complex variable $\displaystyle z$ instead of the $\displaystyle x$? Does it mean to change the matrices themselves or what? Wouldn't these matrices look exactly the same in the complex plane? Any help would be appreciated.

I think that the problem is incorrectly worded, and that "$\displaystyle z=x+iy$" really means "$\displaystyle z=x_1+ix_2$". In other words, the three isometries are expressed in terms of the vector $\displaystyle x=(x_1,x_2)\in\mathbb{R}^2$, and the question is asking to to re-word them in terms of $\displaystyle z = x_1+ix_2\in\mathbb{C}$. So for example the answer to 1. would be $\displaystyle \tau_a(z) = z+a$, where $\displaystyle a = a_1+ia_2$.
• Apr 2nd 2011, 02:26 AM
Ackbeet
Hint on the second one: that looks like rotation through some angle (what angle?). How do you do rotations in the complex plane?
• Apr 3rd 2011, 01:31 PM
Pinkk
It's multiplication by $\displaystyle e^{i\theta}$ and the reflection formula is conjugation of the complex variable. Thanks, just needed to clarify what the problem was asking for and figured it out once that was resolved.
• Apr 4th 2011, 02:04 AM
Ackbeet
I think if you compare your matrix with general rotation matrices, you'll find that it's actually rotation through the negative angle $\displaystyle -\theta.$