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MathGroup Archive 2001

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Re: Solutions that are not solutions

  • To: mathgroup at smc.vnet.net
  • Subject: [mg32093] Re: [mg32049] Solutions that are not solutions
  • From: Andrzej Kozlowski <andrzej at tuins.ac.jp>
  • Date: Sat, 22 Dec 2001 04:23:25 -0500 (EST)
  • Sender: owner-wri-mathgroup at wolfram.com

You can see more clearly what is going on by using Reduce instead of 
Solve:

In[1]:=
X = {{0, y, z}, {y, x, t}, {u, v, w}};
A = {{1, 1, a}, {0, 1, 0}, {0, 0, 1}};
Reduce[{X . A == Transpose[A] . X, Det[X] == 1},
   {x, y, z, t, u, v, w}, VerifySolutions -> True]

Out[3]=
a == 0 && u == 0 && w == -(1/y^2) && z == 0 ||
   u == a*y && x == (-1 + a*t*y^2 + a*v*y^2 - w*y^2)/
      (a^2*y^2) && z == a*y && a != 0 && y != 0

You see that the first solution works only if a=0. So the "spurious" 
solution that you get actually is a solution that works only in the 
special case a==0. In general Solve tries to avoid such non-generic 
solutions. However, in some situations it is possible that certain 
variables get eliminated during the process of finding a solution and 
then one may obtain non-generic solutions with respect to these 
eliminated variables (parameters). Note that in your equation the number 
of solve variables is larger than the number independent variables. You 
can avoid getting this non-generic  solution  by reducing the number of 
solve variables in your equations to just three and treating all the 
others as parameters, e.g. :

In[35]:=
Solve[{X . A == Transpose[A] . X, Det[X] == 1}, {x, y, z},
   VerifySolutions -> True]

Out[35]=
{{x -> (-a^2 + a*t*u^2 + a*u^2*v - u^2*w)/(a^2*u^2), z -> u,
    y -> u/a}}

In[36]:=
Simplify[{X . A == Transpose[A] . X, Det[X] == 1} /. %]

Out[36]=
{{True, True}}

Andrzej Kozlowski
Toyama International University
JAPAN
http://platon.c.u-tokyo.ac.jp/andrzej/

On Friday, December 21, 2001, at 05:57  PM, PSi wrote:

> I'm trying to solve a system as follows:
> X={{0,y,z},{y,x,t},{u,v,w}}
> A={{1,1,a},{0,1,0},{0,0,1}}
> Solve[{X.A==Transpose[A].X, Det[X]==1},
> {x,y,z,t,u,v,w}]
>
> Mathematica 4.1 gives two solutions:
> X1={{0,y,0},{y,x,t},{0,v,-1/y^2}}
> and
> X2={{0,y,a*y},{y,(a*t*y^2-1+a*v*y^2-w*y^2)/(a*y)^2,t},
> {a*y,v,w}}
>
> However, X1 is not a solution:
> X1.A-Transpose[A].X1={{0,0,0},{0,0,a*y},{0,-a*y,0}}
>
> Could anybody explain this behaviour?
> Thanks
> PSi
>
>
>
>
>
>



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