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Re: Solving an equation

  • To: mathgroup at smc.vnet.net
  • Subject: [mg34527] Re: Solving an equation
  • From: wempenj at asme.org (JDW)
  • Date: Mon, 27 May 2002 01:15:56 -0400 (EDT)
  • References: <aci5vg$3c7$1@smc.vnet.net>
  • Sender: owner-wri-mathgroup at wolfram.com

"PSi" <psino at tee.gr> wrote in message news:<aci5vg$3c7$1 at smc.vnet.net>...
> I want to solve the following equation with Mathematica 4.1:
> a*x+b*y=c
> where x, y are the unknown scalars,
> a={{1,0},{0,1}},
> b={{b1,b2},{b3,b4}},
> c={{c1,c2},{c3,c4}},
> the matrices b, c commute, and the matrix b is not a scalar multiple of the unit
> matrix a.
> Could anybody help?

The problem you are trying to solve is overdetermined you have 4
equations and only 2 varibles. To solve the problme you must determine
some constraint equations.

a = {{1, 0}, {0, 1}};
b = {{b1, b2}, {b3, b4}};
c = {{c1, c2}, {c3, c4}};

eqn = x*a + y*b == c
   {{x + b1 y, b2 y}, {b3 y, x + b4 y}} == {{c1, c2}, {c3, c4}}

sol1 = Solve[eqn]
   {{c1 -> x + b1 y, c2 -> b2 y, c3 -> b3 y, c4 -> x + b4 y}}

eq1 = c1 == Evaluate[c1 /. sol1[[1, 1]]]
eq2 = c2 == Evaluate[c2 /. sol1[[1, 2]]]
eq3 = c3 == Evaluate[c3 /. sol1[[1, 3]]]
eq4 = c4 == Evaluate[c4 /. sol1[[1, 4]]]
   c1 == x + b1 y
   c2 == b2 y
   c3 == b3 y
   c4 == x + b4 y

(*eq1 and eq4 can be sovled for x and y*)

Solve[{eq1, eq4}, {x, y}]
    {{x -> (-(b4*c1) + b1*c4)/(b1 - b4),
      y -> (c1 - c4)/(b1 - b4)}}
(* eq2 and eq3 can be solved for y)
Solve[{eq2}, y]
   (y -> c2/b2)

Solve[{eq3}, y]
   (y -> c3/b3)

(* a set of constrain equations can be developed*)

y=c3/b3; y=c2/b2; y= (c1-c4)/(b1-b4)
therfore

c3=c2=(c1-c4)
b3=b2=(b1-b4)

and the follwoing equations can be applied to the inital problem 

eq1 = b3 == b1 - b4;
eq2 = c3 == c1 - c4;
eq3 = b2 == b1 - b4;
eq4 = c2 == c1 - c4;

Solve[{eqn, eq1, eq2, eq3, eq4}, {x, y}]
   {{x->(-(b4*c1) + b1*c4)/(b1 - b4),y->(c1 - c4)/(b1 - b4)}}

Hope this helps
JDW


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