restart; with(LinearAlgebra):
S:=Matrix(5,5,[[a(1),b(1),0,0,0],[c(1),a(2),b(2),0,0],
               [0,c(2),a(3),b(3),0],[0,0,c(3),a(4),b(4)],[0,0,0,c(4),a(5)]]);
D1:=DiagonalMatrix([1,b(1),b(1)*b(2),b(1)*b(2)*b(3),b(1)*b(2)*b(3)*b(4)]);
B1:=D1.S.MatrixInverse(D1);
L:=Matrix(5,5,[[1,0,0,0,0],[e(1),1,0,0,0],[0,e(2),1,0,0],
               [0,0,e(3),1,0],[0,0,0,e(4),1]]);
U:=Matrix(5,5,[[q(1),1,0,0,0],[0,q(2),1,0,0],[0,0,q(3),1,0],
               [0,0,0,q(4),1],[0,0,0,0,q(5)]]);
T:=L.U;
F:=DiagonalMatrix([-1,q(1),-q(1)*q(2),q(1)*q(2)*q(3),-q(1)*q(2)*q(3)*q(4)]);
A:=MatrixInverse(F).T.F;
X:=Matrix(5,5,[[q(1),0,0,0,0],[-e(1),q(2),0,0,0],
  [0,-e(2),q(3),0,0],[0,0,-e(3),q(4),0],[0,0,0,-e(4),q(5)]]);
Y:=Matrix(5,5,[[1,-1,0,0,0],[0,1,-1,0,0],
               [0,0,1,-1,0],[0,0,0,1,-1],[0,0,0,0,1]]);
X.Y;   #  = A
B:=Y.X;
D2:=DiagonalMatrix([1,sqrt(e(1)*q(1)),sqrt(e(1)*q(1)*e(2)*q(2)),
  sqrt(e(1)*q(1)*e(2)*q(2)*e(3)*q(3)),
  sqrt(e(1)*q(1)*e(2)*q(2)*e(3)*q(3)*e(4)*q(4))]);
H:=MatrixInverse(D2).T.D2: H:= simplify(H,symbolic);
R:=Matrix(5,5,[[sqrt(q(1)),sqrt(e(1)),0,0,0],[0,sqrt(q(2)),sqrt(e(2)),0,0],
               [0,0,sqrt(q(3)),sqrt(e(3)),0],[0,0,0,sqrt(q(4)),sqrt(e(4))],
               [0,0,0,0,sqrt(q(5))]]);
Transpose(R).R;