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Re: How to find more solutions for an periodical equation with infinity solutions

  • To: mathgroup at smc.vnet.net
  • Subject: [mg122928] Re: How to find more solutions for an periodical equation with infinity solutions
  • From: "Dr. Wolfgang Hintze" <weh at snafu.de>
  • Date: Wed, 16 Nov 2011 04:47:09 -0500 (EST)
  • Delivered-to: l-mathgroup@mail-archive0.wolfram.com
  • References: <j9tg76$n8d$1@smc.vnet.net>

Using Reduce will tell you much more about possible solutions (probably 
everything):

Reduce[{x, y, z} == Matrix[{0, 0, 1}, \[Theta]] . {x, y, z}, \[Theta]]

(C[1] \[Element] Integers && (\[Theta] == 2*Pi*C[1] || (x == 0 && y == 
0 && \[Theta] == Pi + 2*Pi*C[1]))) || ((-Pi + \[Theta])/(2*Pi) 
\[NotElement] Integers && x == 0 && y == 0)
Your equation to be solved for \[Theta] obviously is looking for 
Eigenvalues == 1 of the Matrix with Alpha = 0 and Beta = 0.

This can be accomplished more directly in the follwing way

ev = Eigenvalues[Matrix[{0, 0, 1}, \[Theta]]]
    {1, Cos[\[Theta]] - I*Sin[\[Theta]], Cos[\[Theta]] + 
I*Sin[\[Theta]]}

Reduce[ev[[1]] == 1, \[Theta]]
    True

Reduce[ev[[2]] == 1, \[Theta]]
    C[1] \[Element] Integers && \[Theta] == 2*Pi*C[1]

Reduce[ev[[3]] == 1, \[Theta]]
    C[1] \[Element] Integers && \[Theta] == 2*Pi*C[1]

PS: I don't see the 5 or 6 solutions you mentioned, but only 3.

Regards,
Wolfgang


"Gy Peng" <hitphyopt at gmail.com> schrieb im Newsbeitrag 
news:j9tg76$n8d$1 at smc.vnet.net...
> Dear all,
>
> I have a matrix defined as:
> Matrix[{\[Alpha]_, \[Beta]_, \[Gamma]_}, \[Theta]_] := {{\[Alpha]^2 \
> (1 - Cos[\[Theta]]) +
>    Cos[\[Theta]], \[Alpha] \[Beta] (1 -
>       Cos[\[Theta]]) - \[Gamma] Sin[\[Theta]], \[A lpha] \[Gamma] 
> (1 -
>        Cos[\[Theta]]) + \[Beta] Sin[\[Theta]]}, {\[Alpha] \[Beta] (1 
> \
> - Cos[\[Theta]]) + \[Gamma] Sin[\[Theta]], \[Beta]^2 (1 -
>       Cos[\[Theta]]) +
>    Cos[\[Theta]], \[Beta] \[Gamma] (1 -
>       Cos[\[Theta]]) - \[Alpha] Sin[\[Theta]]}, {\[Alpha] \[Gamma] \
> (1 - Cos[\[Theta]]) - \[Beta] Sin[\[Theta]], \[Beta] \[Gamma] (1 -
>       Cos[\[Theta]]) + \[Alpha] Sin[\[Theta]], \[Gamma]^2 (1 -
>       Cos[\[Theta]]) + Cos[\[Theta]]}}
>
> Solve[{x, y, z} == Matrix[{0, 0, 1}, \[Theta]].{x, y, z}, \[Theta]]
>
> I know this equation is periodical and has infinity solutions. So, 
> Mathematica only gave me only one solution: {{\[Theta] -> 0}} and 
> show me the message:
>
> Solve::ifun: Inverse functions are being used by Solve, so some 
> solutions
> may not be found; use Reduce for complete solution information. >>
>
> My question is, how could I see for example first 6 solution, because 
> I
> know the first 5 or 6 solutions should be different and then
> repeat themselves periodically. What should I do to find the first 5 
> or 6
> solutions?
>
> Thank you very much! 




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