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Re: Complex Analysis Question
Posted:
Feb 8, 2006 3:51 PM
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Notice that if z_1 and z_2 are both solutions of z' * x = 1, then z_1 -
z_2 is a solution to the equation z' * x = 0. What can you say about the
set of solutions z to the equation z' * x = 0?
________________________________
Eric J. Wingler (wingler@math.ysu.edu)
Dept. of Mathematics and Statistics
Youngstown State University
One University Plaza
Youngstown, OH 44555-0001
330-941-1817
"junoexpress" <mathimagical@netscape.net> wrote in message
news:1139416665.163490.311850@o13g2000cwo.googlegroups.com...
> Hi,
>
> I am curious if there is a general way to understand the solution to
> the following (simple) complex analysis problem.
>
> Suppose we have an n-dimensional vector space, and a fixed (i.e. known)
> vector x in C^n.
>
> The question is whether there is a way to describe the set of all
> vectors z such that:
> (i) z' * x = 1
> (where ' denotes conjugate transpose).
>
> If you sketch out this problem, it is not difficult to see that the
> conditions:
> (ii.a) Re(z' * x) = 1
> and
> (ii.b) Im(z' * x) = 0
> give you two linear equations in 2n unknowns, which you could then
> solve (in a least squares sense).
>
> This method of analysis, however, does little to describe what the
> solution set is like. I am curious if anyone else has another way of
> thinking about this problem. More general pictures for how to visualize
> this condition (like as a projection for example) do not seem that easy
> to conjur up.
>
> Thank you for any help you can provide,
>
> Juno
>
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