Results 1 to 5 of 5

Math Help - Isometries with complex variables

  1. #1
    Senior Member Pinkk's Avatar
    Joined
    Mar 2009
    From
    Uptown Manhattan, NY, USA
    Posts
    419

    Isometries with complex variables

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

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

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

    3. r(x) = \left[$\begin{array}{cc}<br />
1 & 0 \\ <br />
0 & -1 \\ \end{array}$]\right \left[$\begin{array}{c}<br />
x_{1}\\ <br />
x_{2}\end{array}$]\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.
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor
    Opalg's Avatar
    Joined
    Aug 2007
    From
    Leeds, UK
    Posts
    4,041
    Thanks
    7
    Quote Originally Posted by Pinkk View Post
    Write the following isometries in terms of a complex variable z = x + iy.

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

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

    3. r(x) = \left[$\begin{array}{cc}<br />
1 & 0 \\ <br />
0 & -1 \\ \end{array}$]\right \left[$\begin{array}{c}<br />
x_{1}\\ <br />
x_{2}\end{array}$]\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.
    Follow Math Help Forum on Facebook and Google+

  3. #3
    A Plied Mathematician
    Joined
    Jun 2010
    From
    CT, USA
    Posts
    6,318
    Thanks
    4
    Awards
    2
    Hint on the second one: that looks like rotation through some angle (what angle?). How do you do rotations in the complex plane?
    Follow Math Help Forum on Facebook and Google+

  4. #4
    Senior Member Pinkk's Avatar
    Joined
    Mar 2009
    From
    Uptown Manhattan, NY, USA
    Posts
    419
    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.
    Follow Math Help Forum on Facebook and Google+

  5. #5
    A Plied Mathematician
    Joined
    Jun 2010
    From
    CT, USA
    Posts
    6,318
    Thanks
    4
    Awards
    2
    I think if you compare your matrix with general rotation matrices, you'll find that it's actually rotation through the negative angle -\theta.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Complex Variables
    Posted in the Advanced Math Topics Forum
    Replies: 2
    Last Post: February 5th 2010, 04:12 AM
  2. complex variables
    Posted in the Calculus Forum
    Replies: 4
    Last Post: November 12th 2009, 11:56 AM
  3. Isometries of the Complex Plane: Fixed Lines
    Posted in the Differential Geometry Forum
    Replies: 4
    Last Post: August 23rd 2009, 03:14 AM
  4. Complex Variables
    Posted in the Calculus Forum
    Replies: 1
    Last Post: March 12th 2009, 07:35 PM
  5. Complex Variables
    Posted in the Calculus Forum
    Replies: 1
    Last Post: January 23rd 2009, 07:17 PM

Search Tags


/mathhelpforum @mathhelpforum