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Re: Strange empty set of solutions
*To*: mathgroup at smc.vnet.net
*Subject*: [mg71791] Re: Strange empty set of solutions
*From*: Jean-Marc Gulliet <jeanmarc.gulliet at gmail.com>
*Date*: Wed, 29 Nov 2006 02:56:33 -0500 (EST)
*Organization*: The Open University, Milton Keynes, UK
*References*: <ekh7pg$sgs$1@smc.vnet.net>
José Carlos Santos wrote:
> Hi all,
>
> 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}}
>
> 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?
According to the online help, "Solve gives {} if there are no possible
solutions to the equations. "
> I should get two solutions!
Not knowing your matrix M, I cannot tell whether your expectation are
legitimate; but here is a simple system that returns several solutions:
In[1]:=
m = {{1, 2, 3}, {4, 5, 6}, {7, 8, 9}};
In[2]:=
sol = Solve[m . {a, b, c} == {0, 0, 0}, {a, b, c}]
From In[2]:=
Solve::svars: Equations may not give solutions for all "solve" variables.
Out[2]=
{{a -> c, b -> -2*c}}
In[3]:=
a^2 + b^2 + c^2 == 1 /. sol
Out[3]=
{6*c^2 == 1}
In[4]:=
sol = Solve[m . {a, b, c} == {0, 0, 0} &&
a^2 + b^2 + c^2 == 1, {a, b, c}]
Out[4]=
{{a -> -(1/Sqrt[6]), b -> Sqrt[2/3],
c -> -(1/Sqrt[6])}, {a -> 1/Sqrt[6],
b -> -Sqrt[2/3], c -> 1/Sqrt[6]}}
Regards,
Jean-Marc
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