Description
Given a unimodular m \times \ n matrix over a polynomial ring with coefficients in
QQ,
ZZ, or
ZZ/p with
p a prime integer, this method returns the inverse of the matrix returned by
qsAlgorithm. The first m rows or columns (depending on whether m < n or m > n) of this matrix are equal to
U and the determinant of the matrix is a unit in the polynomial ring.
i1 : R = ZZ/101[x,y]
o1 = R
o1 : PolynomialRing
|
i2 : U = matrix{{x^2*y+1,x+y-2,2*x*y}}
o2 = | x2y+1 x+y-2 2xy |
1 3
o2 : Matrix R <-- R
|
i3 : isUnimodular U
o3 = true
|
i4 : M = completeMatrix U
o4 = {0} | x2y+1 x+y-2 2xy |
{0} | 50x 0 -1 |
{1} | 0 1 0 |
3 3
o4 : Matrix R <-- R
|
i5 : det M
o5 = 1
o5 : R
|
The method can also be used over a Laurent polynomial ring with coefficients in
QQ or
ZZ/p for
p a prime integer. The following example demonstrates how to construct a Laurent polynomial ring and also how to use the method on a unimodular matrix over the ring.
i6 : R = QQ[x,Inverses => true,MonomialOrder => Lex]
o6 = R
o6 : PolynomialRing
|
i7 : U = matrix{{3*x^-1-2-2*x+2*x^2, 3*x^-1-2*x,2*x},{6*x^-1+25-23*x-16*x^2+20*x^3, 6*x^-1+29-4*x-20*x^2,2+4*x+20*x^2}}
o7 = | 2x2-2x-2+3x-1 -2x+3x-1 2x |
| 20x3-16x2-23x+25+6x-1 -20x2-4x+29+6x-1 20x2+4x+2 |
2 3
o7 : Matrix R <-- R
|
i8 : M = completeMatrix U
o8 = | 2x2-2x-2+3x-1 -2x+3x-1 2x |
| 20x3-16x2-23x+25+6x-1 -20x2-4x+29+6x-1 20x2+4x+2 |
| -1/270x2+1/108x-1/540-1/180x-1 1/270x-1/180-1/180x-1 -1/270x+1/180 |
3 3
o8 : Matrix R <-- R
|
i9 : det M
1
o9 = ---
180
o9 : R
|