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I am trying to solve bellowing loop but I faced with following error!

a[0] = {{0, 1},{2, 3}};
c = {{1, 2},{1, 4}};
p[k] = {{p1[k], p2[k]},{p3[k], p4[k]}};
Do[
 m = Solve[
  Transpose[a[k]].p[k] + p[k].a[k] + c == 0,
  {p1[k], p2[k], p3[k], p4[k]}], 
  {k, 0, 5}
 ];
p[5] /. m [[1]] // MatrixForm

Part::partw: Part 1 of {} does not exist. >>
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    – Michael E2
    Commented Sep 27, 2016 at 14:59

1 Answer 1

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You are solving a system of equations organized in a matrix, hence you have to Equal (==) each of the four equations - MMA doesn't know that your 0 is a $2\times 2$ zero matrix. Next, you only provide a[0] but the loop goes through a[1], a[2] etc.

n = 5;

c = {{1, 2}, {1, 4}};

a[0] = {{0, 1}, {2, 3}};
a[i_] := a[i - 1] + IdentityMatrix[2];

p[k_] := {{p1[k], p2[k]}, {p3[k], p4[k]}};
Do[
  m = Solve[
    Transpose[a[k]].p[k] + p[k].a[k] + c == {{0, 0}, {0, 0}}, {p1[k], 
     p2[k], p3[k], p4[k]}]
  , {k, 0, n}];

p[n] /. m[[1]] // MatrixForm

enter image description here

I set an arbitrary a[i] through recursion for illustration, changed p[k] to an explicit function and gave n = 5 as the max iterator of the loop (works for arbitrary n, e.g. n = 500).

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  • $\begingroup$ Dear corey979 Thanks alot $\endgroup$
    – mary
    Commented Sep 27, 2016 at 15:56

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