How can I solve A*X + X*B = C equation for X? where A, B&C are 2*2 matrixes.

A = [1 1;2 3] B= [5 6;7 8] C = [11 12;13 14] Syms X; Eqn = A*X + X*B == C; solve(Eqn,X); % I'm facing difficulties to define matrix X as syms

Answer 1

[Steven Lord]

If you're using release R2014a or later, use the sylvester function in MATLAB.

If you're using an older release but have access to Control System Toolbox, use the lyap function.

[HEmant kuralkar]

Thank you very much Steven. It will be very helpful to me..

[Andrei Bobrov]

+1

Answer 2

[Andrei Bobrov]

>> A = [1 1;2 3];
  B= [5 6;7 8];
  C = [11 12;13 14];
  X = reshape(sym('x',[4,1],'real'),2,[]);
  v = A*X + X*B - C;
  v = v(:);
  f = matlabFunction(v,'Vars',{X});
  x = fsolve(f,[1;1;1;1])

Equation solved.

fsolve completed because the vector of function values is near zero
as measured by the default value of the function tolerance, and
the problem appears regular as measured by the gradient.

<stopping criteria details>

x =
    2.1809
   -0.1489
   -0.2766
    1.4043
>> X = reshape(x,2,[]);
>> v = A*X + X*B - C
v =
   1.0e-14 *
   -0.3553   -0.3553
   -0.1776   -0.5329
>>

[KSSV]

That means problem using solve eh?

[Andrei Bobrov]

No. Just other variant of solution with fsolve.

[KSSV]

But result is different..isn't it?

[Andrei Bobrov]

Same

>> A = [1 1;2 3] ;
B= [5 6;7 8] ;
C = [11 12;13 14] ;

syms x1 y1 x2 y2
X = [x1 y1 ; x2 y2] ;
eqn = A*X+X*B == C ;
D = solve(eqn);

X = reshape(sym('x',[4,1],'real'),2,[]);
v = A*X + X*B - C;
v = v(:);
f = matlabFunction(v,'Vars',{X});
x = fsolve(f,[1;1;1;1]);

Equation solved.

fsolve completed because the vector of function values is near zero
as measured by the default value of the function tolerance, and
the problem appears regular as measured by the gradient.

<stopping criteria details>

>> structfun(@double,D) - x
ans =
   1.0e-14 *
    0.6217
   -0.4469
   -0.4219
    0.3775
>>

[KSSV]

okay.....fine...:)

Answer 3

[Karan Gill]

You can solve this with the symbolic toolbox if you declare "X" as a matrix:

>> X = sym('x%d',[2 2])
X =
[ x11, x12]
[ x21, x22]

Then solve as usual:

>>  A = [1 1;2 3];
  B= [5 6;7 8];
  C = [11 12;13 14];

>> eqn = A*X + X*B == C
eqn =
[   6*x11 + 7*x12 + x21 == 11,    6*x11 + 9*x12 + x22 == 12]
[ 2*x11 + 8*x21 + 7*x22 == 13, 2*x12 + 6*x21 + 11*x22 == 14]

>> sol = solve(eqn,X)
sol = 
  struct with fields:

      x11: [1×1 sym]
      x21: [1×1 sym]
      x12: [1×1 sym]
      x22: [1×1 sym]

 >> solX = [sol.x11 sol.x12; sol.x21 sol.x22]
  solX =
  [ 205/94, -13/47]
  [  -7/47,  66/47]

>> vpa(solX)
ans =
[  2.1808510638297872340425531914894, -0.27659574468085106382978723404255]
[ -0.1489361702127659574468085106383,   1.4042553191489361702127659574468]