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D.11.3.3 divideUnits

Procedure from library jacobson.lib (see jacobson_lib).

divideUnits(L); list L

matrix or list of matrices

L is an output of smith or a jacobson procedures, that is
either L contains one rectangular matrix with elements only on the main diagonal
or L consists of three matrices, where L[1] and L[3] are square invertible matrices
while L[2] is a rectangular matrix with elements only on the main diagonal

divide out units from the diagonal and reflect this in transformation matrices

LIB "jacobson.lib";
ring R=(0,m,M,L1,L2,m1,m2,g), D, lp; // two pendula example
matrix P[3][4]=m1*L1*D^2,m2*L2*D^2,(M+m1+m2)*D^2,-1,
list s=smith(P,1);  // returns a list with 3 entries
print(s[2]); // a diagonal form, close to the Smith form
==> (L1*L2*m2*g^2-L2^2*m2*g^2),0,   0,    0,
==> 0,                         (L2),0,    0,
==> 0,                         0,   (g^2),0 
print(s[1]); // U, left transformation matrix
==> 0,    (-L2*m2)/(L1*m1),    1,
==> (-L2),(M*L2+L2*m1)/(L1*m1),1,
==> 0,    1/(L1*m1),           0 
list t = divideUnits(s);
print(t[2]); // the Smith form of the matrix P
==> 1,0,0,0,
==> 0,1,0,0,
==> 0,0,1,0 
print(t[1]); // U', modified left transformation matrix
==> 0, -1/(L1^2*m1*g^2-L1*L2*m1*g^2),1/(L1*L2*m2*g^2-L2^2*m2*g^2),
==> -1,(M+m1)/(L1*m1),               1/(L2),                      
==> 0, 1/(L1*m1*g^2),                0