A = {
   {2, 1, 1, 3 , 2},
   {1, 2 , 2 , 1 , 1},
   {1, 2 , 9, 1, 5},
   {3, 1, 1, 7, 1},
   {2, 1, 5, 1, 8}
   };
"A="  MatrixForm[A]

L = Table[0, {i, Dimensions[A][[1]]}, {j, Dimensions[A][[1]]}];
n = Dimensions[A][[1]];
For[ k = 1, k <= n, k++,
 L[[k, k]] = Sqrt[A[[k, k]] - Sum[L[[k, m]]^2, {m, k - 1}]];
 For[ i = k + 1, i <= n, i++,
  L[[i, k]] = (A[[i, k]]  - Sum[L[[i, m]]*L[[k, m]], {m, k - 1}])/
    L[[k, k]]
  ]
 ]
"L="  MatrixForm[L]
"L transposed ="  MatrixForm[Transpose[L]]
"L . \!\(\*SuperscriptBox[\(L\), \(T\)]\) ="  MatrixForm[
  L.Transpose[L]]
Print[" A == L.\!\(\*SuperscriptBox[\(L\), \(T\)]\) "]
 A == L.Transpose[L]