Revision history [back]
 | 1 | initial version |
One equality for polynomials denotes three equations for coefficients
R.<x1,x2> = PolynomialRing(SR)
Vx=matrix([x1,x2])
p=2*(x1^2) - 4*x1*x2 + 5*(x2^2)
var('a00,a01,a11')
A=matrix([ [a00,a01],[a01,a11] ]) # symmetry!
VtAV=Vx*A*(Vx.transpose())
p2=(VtAV-p)[0][0] # scalar polynomial lhs-rhs
eq=[u==0 for u in p2.coefficients()] # equations
solve(eq,[a00,a01,a11]) # solving
[[a00 == 2, a01 == -2, a11 == 5]]
| | 2 | No.2 Revision |
One equality for polynomials denotes three equations for coefficients
R.<x1,x2> = PolynomialRing(SR)
Vx=matrix([x1,x2])
p=2*(x1^2) - 4*x1*x2 + 5*(x2^2)
var('a00,a01,a11')
A=matrix([ [a00,a01],[a01,a11] ]) # symmetry!
VtAV=Vx*A*(Vx.transpose())
p2=(VtAV-p)[0][0] # scalar polynomial lhs-rhs
eq=[u==0 for u in p2.coefficients()] # equations
solve(eq,[a00,a01,a11]) # solving
[[a00 == 2, a01 == -2, a11 == 5]]
Shortcut:
R.<x1,x2> = PolynomialRing(SR)
p=2*(x1^2) - 4*x1*x2 + 5*(x2^2)
q=Q.from_polynomial(p);
q.Gram_matrix()
[ 2 -2]
[-2 5]
| | 3 | No.3 Revision |
One equality for polynomials denotes three equations for coefficients
R.<x1,x2> = PolynomialRing(SR)
Vx=matrix([x1,x2])
p=2*(x1^2) - 4*x1*x2 + 5*(x2^2)
var('a00,a01,a11')
A=matrix([ [a00,a01],[a01,a11] ]) # symmetry!
VtAV=Vx*A*(Vx.transpose())
p2=(VtAV-p)[0][0] # scalar polynomial lhs-rhs
eq=[u==0 for u in p2.coefficients()] # equations
solve(eq,[a00,a01,a11]) # solving
[[a00 == 2, a01 == -2, a11 == 5]]
Shortcut:
Q=QuadraticForm(QQ,2)
R.<x1,x2> = PolynomialRing(SR)
p=2*(x1^2) - 4*x1*x2 + 5*(x2^2)
q=Q.from_polynomial(p);
q.Gram_matrix()
[ 2 -2]
[-2 5]
| | 4 | No.4 Revision |
One equality for polynomials denotes three equations for coefficients
R.<x1,x2> = PolynomialRing(SR)
Vx=matrix([x1,x2])
p=2*(x1^2) - 4*x1*x2 + 5*(x2^2)
var('a00,a01,a11')
A=matrix([ [a00,a01],[a01,a11] ]) # symmetry!
VtAV=Vx*A*(Vx.transpose())
p2=(VtAV-p)[0][0] # scalar polynomial lhs-rhs
eq=[u==0 for u in p2.coefficients()] # equations
solve(eq,[a00,a01,a11]) # solving
[[a00 == 2, a01 == -2, a11 == 5]]
Shortcut:
Q=QuadraticForm(QQ,2)
R.<x1,x2> = PolynomialRing(SR)
PolynomialRing(QQ)
p=2*(x1^2) - 4*x1*x2 + 5*(x2^2)
q=Q.from_polynomial(p);
q.Gram_matrix()
[ 2 -2]
[-2 5]
| | 5 | No.5 Revision |
One equality for polynomials denotes three equations for coefficients
R.<x1,x2> = PolynomialRing(SR)
Vx=matrix([x1,x2])
p=2*(x1^2) - 4*x1*x2 + 5*(x2^2)
var('a00,a01,a11')
A=matrix([ [a00,a01],[a01,a11] ]) # symmetry!
VtAV=Vx*A*(Vx.transpose())
p2=(VtAV-p)[0][0] # scalar polynomial lhs-rhs
eq=[u==0 for u in p2.coefficients()] # equations
solve(eq,[a00,a01,a11]) # solving
[[a00 == 2, a01 == -2, a11 == 5]]
Shortcut:
Q=QuadraticForm(QQ,2)
R.<x1,x2> = PolynomialRing(QQ)
p=2*(x1^2) - 4*x1*x2 + 5*(x2^2)
q=Q.from_polynomial(p);
q=QuadraticForm.from_polynomial(p);
q.Gram_matrix()
[ 2 -2]
[-2 5]
