       Re: Solutions that are not solutions

• To: mathgroup at smc.vnet.net
• Subject: [mg32184] Re: [mg32049] Solutions that are not solutions
• From: Daniel Lichtblau <danl at wolfram.com>
• Date: Fri, 4 Jan 2002 05:04:01 -0500 (EST)
• References: <200112210857.DAA25144@smc.vnet.net>
• Sender: owner-wri-mathgroup at wolfram.com

```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

Simpler form:

polys = {a*y-z, u-a*y, a*u-a*z, -1+t*u*y-w*y^2-u*x*z+v*y*z};
vars = {x,y,z,t,u,v,w};
InputForm[sol1 = Solve[polys==0, vars, Sort->False]]

I use Sort->False so there can be no confusion over which variables
should be solved in terms of which.

In:= InputForm[sol1 = Solve[polys==0, vars, Sort->False]]

Solve::svars: Equations may not give solutions for all "solve"
variables.

Out//InputForm=
{{y -> (-I)/Sqrt[w], z -> 0, u -> 0}, {y -> I/Sqrt[w], z -> 0, u -> 0},
{x -> (-a^2 + a*t*u^2 + a*u^2*v - u^2*w)/(a^2*u^2), y -> u/a, z -> u}}

The first two solutions are not generically correct insofar as they
force a parameter equation a==0. The fact that these are returned at all
is a bug (alas) which will be fixed.

Daniel Lichtblau
Wolfram Research

```

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