Isometries with complex variables

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March 31st 2011, 07:04 PM
Pinkk

Isometries with complex variables

Write the following isometries in terms of a complex variable $z = x + iy$ .

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

2. $\rho_{\theta}(x) = \left[$\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. $r(x) = \left[$\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 $z$ instead of the $x$ ? 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.
April 2nd 2011, 12:00 AM
Opalg

Quote:

Originally Posted by Pinkk

Write the following isometries in terms of a complex variable $z = x + iy$ .

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

2. $\rho_{\theta}(x) = \left[$\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. $r(x) = \left[$\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 $z$ instead of the $x$ ? 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 " $z=x+iy$ " really means " $z=x_1+ix_2$ ". In other words, the three isometries are expressed in terms of the vector $x=(x_1,x_2)\in\mathbb{R}^2$ , and the question is asking to to re-word them in terms of $z = x_1+ix_2\in\mathbb{C}$ . So for example the answer to 1. would be $\tau_a(z) = z+a$ , where $a = a_1+ia_2$ .
April 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?
April 3rd 2011, 01:31 PM
Pinkk

It's multiplication by $e^{i\theta}$ 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.
April 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 $-\theta.$