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MathGroup Archive 2006

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Re: Re: Strange empty set of solutions

  • To: mathgroup at smc.vnet.net
  • Subject: [mg71871] Re: [mg71814] Re: Strange empty set of solutions
  • From: Andrzej Kozlowski <akoz at mimuw.edu.pl>
  • Date: Fri, 1 Dec 2006 06:22:29 -0500 (EST)
  • References: <ekh7pg$sgs$1@smc.vnet.net> <ekjfrf$d7h$1@smc.vnet.net> <200611301105.GAA08424@smc.vnet.net>

On 30 Nov 2006, at 20:05, José Carlos Santos wrote:

> On 29-11-2006 8:19, Jens-Peer Kuska wrote:
>
>> and we can't read you mind and the memory of your computer
>> so we must imagine a matrix M and write down
>
> Do you want an example? Here it is:
>
> M = {{4/5(c - 1), -2/Sqrt[5]s, 2/5(1 - c)},
>       {2/5(1 - c), -1 + s/Sqrt[5], 4/5 + c/5},
>       {-2s/Sqrt[5], -c, -1 + s/Sqrt[5]}} // N
>
> with c = Cos[Sqrt[5]] and s = Sin[Sqrt[5]].
>
> If I type
>
> Solve[{M.{x, y, z} == {0, 0, 0}}, {x, y, z}]
>
> I get
>
> {{x -> 0. - 0.0438861 z, y -> 0. + 1. z}}
>
> but if I type
>
> Solve[{M.{x, y, z} == {0, 0, 0}, x^2 + y^2 + z^2 == 1}, {x, y, z}]
>
> then I get the empty set. Why is that?
>
> Best regards,
>
> Jose Carlos Santos
>

First,  is a bad idea to mix arbitrary precision numbers with  
algebraic functions such as Solve. Use the definition:

M = {{4/5(c - 1), -2/Sqrt[5]s, 2/5(1 - c)},
       {2/5(1 - c), -1 + s/Sqrt[5], 4/5 + c/5},
       {-2s/Sqrt[5], -c, -1 + s/Sqrt[5]}} ;


Second, you have three equations and 5 "unknowns". which you can  
arbitrarily divide into "variables" (in your case x,y,z} and  
parameters (s,c}.  This means that you will only get solutions  
provided certain relations hold between the parameters. But Solve  
gives only "generic" solutions for the variables, where generic means  
solutions valid for any values of the parameters. But your equation  
obviously has no such solutions.

There are two things you can do. One is to use Reduce:

Reduce[{M.{x, y, z} == {0, 0, 0}, x^2 + y^2 + z^2 == 1}, {x, y, z}]

will show you also the relations that have to hold between the  
parameters. The other possibility is to treat one of the "parameters"  
as a variable:

Solve[{M.{x, y, z} == {0, 0, 0}, x^2 + y^2 + z^2 == 1}, {x, y, z, c}]

will give you a (pretty awful) list of solutions for the "variables"  
x, y, z and c in terms of the "parameter" s.

Andrzej Kozlowski
Tokyo, Japan


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