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Question: Remove algebraic common factor from matrix

Question:Remove algebraic common factor from matrix

Ronan 1441 Maple 2026
factor
1 hour ago
0

This the part of a procedure I have for removing common factors from matrices and vectors used in a projective geometry setting.
How can a get it to work so it handles aglebraic denominators? Such as 

x__1*y__2 - x__1*y__3 - x__2*y__1 + x__2*y__3 + x__3*y__1 - x__3*y__2
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restart
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ReduceM:= proc(C::{Matrix, :-_ProjM3})
            local i, tgdc, dnm, V1, Ml, SV1, ns, M, r;
            option overload;
            if C::'Matrix' then
                M := LinearAlgebra:-Copy(C);
            elif C::{':-_ProjM3'} then
                M := LinearAlgebra:-Copy(C[1]);
            end if;
            V1 := convert(M, list);
            SV1 := convert(M, set) minus {0};
            ns := nops(SV1);
            dnm := frontend(lcm, [seq(denom(SV1[i]), i = 1 .. ns)]);
            Ml := simplify(dnm *~ M);
            V1 := convert(Ml, list);
            tgdc := SV1[1];
            if tgdc = 0 then
                tgdc := 1;
            end if;
            for i from 2 to ns do
                tgdc := frontend(gcd, [tgdc, SV1[i]]);
            end do;
            r := `if`(C::'Matrix', simplify(factor~(Ml /~ tgdc)),
                <[simplify(factor~(Ml /~ tgdc))]>);
            return r;
        end proc:
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A:=Matrix(3, 3, [[(x__2 - x__1)*(-y__2 + y__3) + (-y__1 + y__2)*(-x__3 + x__2), (x__2 - x__1)*(-y__1 + y__3) + (-y__1 + y__2)*(-x__3 + x__1), (x__2 - x__1)*(-y__1 + y__2) + (-y__1 + y__2)*(-x__2 + x__1)], [(x__3 - x__1)*(-y__2 + y__3) + (-y__1 + y__3)*(-x__3 + x__2), (x__3 - x__1)*(-y__1 + y__3) + (-y__1 + y__3)*(-x__3 + x__1), (x__3 - x__1)*(-y__1 + y__2) + (-y__1 + y__3)*(-x__2 + x__1)], [x__1*(-y__2 + y__3) + y__1*(-x__3 + x__2) - x__2*y__3 + x__3*y__2, (-y__1 + y__3)*x__1 + y__1*(-x__3 + x__1) - x__1*y__3 + x__3*y__1, (-y__1 + y__2)*x__1 + y__1*(-x__2 + x__1) - x__1*y__2 + x__2*y__1]])

Matrix(3, 3, {(1, 1) = (x__2-x__1)*(-y__2+y__3)+(-y__1+y__2)*(-x__3+x__2), (1, 2) = (x__2-x__1)*(-y__1+y__3)+(-y__1+y__2)*(-x__3+x__1), (1, 3) = (x__2-x__1)*(-y__1+y__2)+(-y__1+y__2)*(-x__2+x__1), (2, 1) = (x__3-x__1)*(-y__2+y__3)+(-y__1+y__3)*(-x__3+x__2), (2, 2) = (x__3-x__1)*(-y__1+y__3)+(-y__1+y__3)*(-x__3+x__1), (2, 3) = (x__3-x__1)*(-y__1+y__2)+(-y__1+y__3)*(-x__2+x__1), (3, 1) = x__1*(-y__2+y__3)+y__1*(-x__3+x__2)-x__2*y__3+x__3*y__2, (3, 2) = (-y__1+y__3)*x__1+y__1*(-x__3+x__1)-x__1*y__3+x__3*y__1, (3, 3) = (-y__1+y__2)*x__1+y__1*(-x__2+x__1)-x__1*y__2+x__2*y__1})

 (1)
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ReduceM(A)

Matrix(3, 3, {(1, 1) = 1, (1, 2) = 1, (1, 3) = 0, (2, 1) = 1, (2, 2) = 0, (2, 3) = -1, (3, 1) = -1, (3, 2) = 0, (3, 3) = 0})

 (2)
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B:=Matrix(3, 3, [[-(y__1 - y__3)/(x__1*y__2 - x__1*y__3 - x__2*y__1 + x__2*y__3 + x__3*y__1 - x__3*y__2) + (y__1 - y__2)/(x__1*y__2 - x__1*y__3 - x__2*y__1 + x__2*y__3 + x__3*y__1 - x__3*y__2), -(y__1 - y__3)/(x__1*y__2 - x__1*y__3 - x__2*y__1 + x__2*y__3 + x__3*y__1 - x__3*y__2), -(y__1 - y__2)/(x__1*y__2 - x__1*y__3 - x__2*y__1 + x__2*y__3 + x__3*y__1 - x__3*y__2)], [(-x__3 + x__1)/(x__1*y__2 - x__1*y__3 - x__2*y__1 + x__2*y__3 + x__3*y__1 - x__3*y__2) - (-x__2 + x__1)/(x__1*y__2 - x__1*y__3 - x__2*y__1 + x__2*y__3 + x__3*y__1 - x__3*y__2), (-x__3 + x__1)/(x__1*y__2 - x__1*y__3 - x__2*y__1 + x__2*y__3 + x__3*y__1 - x__3*y__2), (-x__2 + x__1)/(x__1*y__2 - x__1*y__3 - x__2*y__1 + x__2*y__3 + x__3*y__1 - x__3*y__2)], [-(x__1*y__3 - x__3*y__1)/(x__1*y__2 - x__1*y__3 - x__2*y__1 + x__2*y__3 + x__3*y__1 - x__3*y__2) + (x__1*y__2 - x__2*y__1)/(x__1*y__2 - x__1*y__3 - x__2*y__1 + x__2*y__3 + x__3*y__1 - x__3*y__2) - 1, -(x__1*y__3 - x__3*y__1)/(x__1*y__2 - x__1*y__3 - x__2*y__1 + x__2*y__3 + x__3*y__1 - x__3*y__2), -(x__1*y__2 - x__2*y__1)/(x__1*y__2 - x__1*y__3 - x__2*y__1 + x__2*y__3 + x__3*y__1 - x__3*y__2)]])

Matrix(3, 3, {(1, 1) = -(y__1-y__3)/(x__1*y__2-x__1*y__3-x__2*y__1+x__2*y__3+x__3*y__1-x__3*y__2)+(y__1-y__2)/(x__1*y__2-x__1*y__3-x__2*y__1+x__2*y__3+x__3*y__1-x__3*y__2), (1, 2) = -(y__1-y__3)/(x__1*y__2-x__1*y__3-x__2*y__1+x__2*y__3+x__3*y__1-x__3*y__2), (1, 3) = -(y__1-y__2)/(x__1*y__2-x__1*y__3-x__2*y__1+x__2*y__3+x__3*y__1-x__3*y__2), (2, 1) = (-x__3+x__1)/(x__1*y__2-x__1*y__3-x__2*y__1+x__2*y__3+x__3*y__1-x__3*y__2)-(-x__2+x__1)/(x__1*y__2-x__1*y__3-x__2*y__1+x__2*y__3+x__3*y__1-x__3*y__2), (2, 2) = (-x__3+x__1)/(x__1*y__2-x__1*y__3-x__2*y__1+x__2*y__3+x__3*y__1-x__3*y__2), (2, 3) = (-x__2+x__1)/(x__1*y__2-x__1*y__3-x__2*y__1+x__2*y__3+x__3*y__1-x__3*y__2), (3, 1) = -(x__1*y__3-x__3*y__1)/(x__1*y__2-x__1*y__3-x__2*y__1+x__2*y__3+x__3*y__1-x__3*y__2)+(x__1*y__2-x__2*y__1)/(x__1*y__2-x__1*y__3-x__2*y__1+x__2*y__3+x__3*y__1-x__3*y__2)-1, (3, 2) = -(x__1*y__3-x__3*y__1)/(x__1*y__2-x__1*y__3-x__2*y__1+x__2*y__3+x__3*y__1-x__3*y__2), (3, 3) = -(x__1*y__2-x__2*y__1)/(x__1*y__2-x__1*y__3-x__2*y__1+x__2*y__3+x__3*y__1-x__3*y__2)})

 (3)
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ReduceM(B)

Error, (in gcd/Freeze) arguments should be polynomials

  
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B*denom(B[1,1])

Matrix(3, 3, {(1, 1) = (x__1*y__2-x__1*y__3-x__2*y__1+x__2*y__3+x__3*y__1-x__3*y__2)*(-(y__1-y__3)/(x__1*y__2-x__1*y__3-x__2*y__1+x__2*y__3+x__3*y__1-x__3*y__2)+(y__1-y__2)/(x__1*y__2-x__1*y__3-x__2*y__1+x__2*y__3+x__3*y__1-x__3*y__2)), (1, 2) = -y__1+y__3, (1, 3) = -y__1+y__2, (2, 1) = (x__1*y__2-x__1*y__3-x__2*y__1+x__2*y__3+x__3*y__1-x__3*y__2)*((-x__3+x__1)/(x__1*y__2-x__1*y__3-x__2*y__1+x__2*y__3+x__3*y__1-x__3*y__2)-(-x__2+x__1)/(x__1*y__2-x__1*y__3-x__2*y__1+x__2*y__3+x__3*y__1-x__3*y__2)), (2, 2) = -x__3+x__1, (2, 3) = -x__2+x__1, (3, 1) = (x__1*y__2-x__1*y__3-x__2*y__1+x__2*y__3+x__3*y__1-x__3*y__2)*(-(x__1*y__3-x__3*y__1)/(x__1*y__2-x__1*y__3-x__2*y__1+x__2*y__3+x__3*y__1-x__3*y__2)+(x__1*y__2-x__2*y__1)/(x__1*y__2-x__1*y__3-x__2*y__1+x__2*y__3+x__3*y__1-x__3*y__2)-1), (3, 2) = -x__1*y__3+x__3*y__1, (3, 3) = -x__1*y__2+x__2*y__1})

 (4)
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simplify( (4) );

Matrix(3, 3, {(1, 1) = -y__2+y__3, (1, 2) = -y__1+y__3, (1, 3) = -y__1+y__2, (2, 1) = -x__3+x__2, (2, 2) = -x__3+x__1, (2, 3) = -x__2+x__1, (3, 1) = -x__2*y__3+x__3*y__2, (3, 2) = -x__1*y__3+x__3*y__1, (3, 3) = -x__1*y__2+x__2*y__1})

 (5)
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