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MathGroup Archive 2006

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Re: Strange empty set of solutions

  • To: mathgroup at smc.vnet.net
  • Subject: [mg71833] Re: Strange empty set of solutions
  • From: Paul Abbott <paul at physics.uwa.edu.au>
  • Date: Thu, 30 Nov 2006 06:05:40 -0500 (EST)
  • Organization: The University of Western Australia
  • References: <ekh7pg$sgs$1@smc.vnet.net>

In article <ekh7pg$sgs$1 at smc.vnet.net>,
 José Carlos Santos <jcsantos at fc.up.pt> wrote:

> I have a certain 3 x 3 numerical matrix M. If I type
> 
> Solve[M.{a,b,c}=={0,0,0},{a,b,c}]
> 
> I get
> 
> {{a -> 0. + 0.0410026 c, b -> 0. + 1.35294 c}}

along with a warning message, that you have deleted.

> However, if I type
> 
> Solve[{M.{a,b,c}=={0,0,0},a^2+b^2+c^2==1},{a,b,c}]
> 
> I get the empty set. Why is that? I should get two solutions!

Look up the documentation for Solve to find out why -- and use Reduce 
instead.

For example, with

  M={{0.2429717091736855, 0.06206063726712955, -0.09392679038675521}, 
     {0.0992369012615341, 0.6047269186862279, -0.8222282083350113},   
     {0.6768075377204311, 0.128145446007053, -0.20112396846691802}};

then you get

 Solve[M.{a,b,c}=={0,0,0},{a,b,c}]

 {{a -> 0. + 0.0410026*c, b -> 0. + 1.35294*c}}

Using Reduce, you get two equivalent solutions:

 Reduce[{M . {a, b, c} == {0, 0, 0}, a^2 + b^2 + c^2 == 1}, 
   {a, b, c}, Backsubstitution -> True]

 (a == -0.024364357796705177 && b == -0.8039371707519599 && 
  c == -0.594214947264446) || 

 (a == 0.024364357796705177 && b == 0.8039371707519599 && 
  c == 0.594214947264446)

Of course, you could just use NullSpace:

  NullSpace[M]

  {{0.024364357796705243, 0.8039371707519597, 0.5942149472644462}}

  First[%].First[%]

  1

Cheers,
Paul

_______________________________________________________________________
Paul Abbott                                      Phone:  61 8 6488 2734
School of Physics, M013                            Fax: +61 8 6488 1014
The University of Western Australia         (CRICOS Provider No 00126G)    
AUSTRALIA                               http://physics.uwa.edu.au/~paul


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