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Ronan

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14 years, 168 days
East Grinstead, United Kingdom
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What type of matrix normalisation is thi...

Ronan 1486 Maple 2026
matrix normalization
August 14 2026
2 11

Could anyone tell me what type of matrix normalisation this is and it it built into Maple? I found this in a paper and Google AI said this is the process of normailsation being used.

> 

restart
> 

with(LinearAlgebra):
> 

J:=Matrix([[ 1 , 0 , 0 ],
        [ 0 , 1 , 0 ],
        [ 0 , 0 , -1 ]]);

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

 (1)

Matrix N has already been scaled, so the rows are of equal magnitude  x^2 + y^2 - z^2= k

> 

N:=Matrix(3, 3, [[4, -3, 2], [-1/2*sqrt(70), -3/10*sqrt(70), 1/5*sqrt(70)], [1/11*sqrt(77), 6/11*sqrt(77), 2/11*sqrt(77)]])

Matrix(3, 3, {(1, 1) = 4, (1, 2) = -3, (1, 3) = 2, (2, 1) = -(1/2)*sqrt(70), (2, 2) = -(3/10)*sqrt(70), (2, 3) = (1/5)*sqrt(70), (3, 1) = (1/11)*sqrt(77), (3, 2) = (6/11)*sqrt(77), (3, 3) = (2/11)*sqrt(77)})

 (2)

 

> 

for i to 3 do
add(N[i,j]^2,j=1..2)-N[i,3]^2;
end do

21

  

21

  

21

 (3)

Normalisation process to produce C

> 

A:=N.J.N^%T

Matrix(3, 3, {(1, 1) = 21, (1, 2) = -(3/2)*sqrt(70), (1, 3) = -(18/11)*sqrt(77), (2, 1) = -(3/2)*sqrt(70), (2, 2) = 21, (2, 3) = -(27/110)*sqrt(70)*sqrt(77), (3, 1) = -(18/11)*sqrt(77), (3, 2) = -(27/110)*sqrt(70)*sqrt(77), (3, 3) = 21})

 (4)
> 

dA:=DiagonalMatrix(1/~(sqrt~(abs(Diagonal((A))))));

Matrix(3, 3, {(1, 1) = (1/21)*sqrt(21), (1, 2) = 0, (1, 3) = 0, (2, 1) = 0, (2, 2) = (1/21)*sqrt(21), (2, 3) = 0, (3, 1) = 0, (3, 2) = 0, (3, 3) = (1/21)*sqrt(21)})

 (5)
> 

C:=(dA.A.dA);

Matrix(3, 3, {(1, 1) = 1, (1, 2) = -(1/14)*sqrt(70), (1, 3) = -(6/77)*sqrt(77), (2, 1) = -(1/14)*sqrt(70), (2, 2) = 1, (2, 3) = -(9/770)*sqrt(70)*sqrt(77), (3, 1) = -(6/77)*sqrt(77), (3, 2) = -(9/770)*sqrt(70)*sqrt(77), (3, 3) = 1})

 (6)
> 

evalf(C)

Matrix(3, 3, {(1, 1) = 1., (1, 2) = -.5976143047, (1, 3) = -.6837634587, (2, 1) = -.5976143047, (2, 2) = 1., (2, 3) = -.8581163304, (3, 1) = -.6837634587, (3, 2) = -.8581163304, (3, 3) = 1.})

 (7)
 

 

Download 2026-08-14_Q_What_Type_of_Matrix_Normalisation.mw

Type checking for a kind of set...

Ronan 1486 Maple 2026
type set
August 06 2026
1 4

I am trying to add this to my type list in a package, but cannot get it to work.

TypeTools:-AddType(_L3DP, set(satisfies(s -> type(s, [algebraic $ 3])),'Vector[column](3, algebraic)'));

when I test with this I get an error.

 type({[7,8,9],<1,2,3>},:-_L3DP);
Error, testing against an invalid type

or this 

type({[7,8,9],<1,2,3>},set(satisfies(s -> type(s, [algebraic $ 3])),'Vector[column](3, algebraic)'));
Error, testing against an invalid type

Remove common factor from Matrix...

Ronan 1486 Maple 2026
factor gcd
August 03 2026
2 5

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 polynomial 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
> 

restart
> 

 

> 

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:
> 

 

> 

 

> 

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)
> 

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)
> 

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)
> 

ReduceM(B)

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

  
> 

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)
> 

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)
 

 

Download 2026-08-03_Q_Remove_Common_Factors_from_Matrix.mw

Simplifications and substitutions...

Ronan 1486 Maple 2026
substitution
July 29 2026
1 1

I am trying to follow a series of substitutions used in a paper. I have given some examples of the substitutions. But I am having difficulty getting them to work. I know if all else fails I can manually recreate an expression with the substitution variables and then check backwards using eval(f,vals).
Would appreciate any insight on substitution techniques here.
 

> 

restart
> 

 

> 

sidrels := [a*c - b^2 = Delta, a + c - 2*b = d, a - b = c__bar, a - c = b__bar, c - b = a__bar];

[a*c-b^2 = Delta, a+c-2*b = d, a-b = c__bar, a-c = b__bar, c-b = a__bar]

 (1)
> 

vals:='rhs=lhs'~(sidrels)

[Delta = a*c-b^2, d = a+c-2*b, c__bar = a-b, b__bar = a-c, a__bar = c-b]

 (2)
> 

t1:=a + c - 2*b;
simplify(t1,sidrels); #this should be d

a+c-2*b

  

d

 (3)
> 

t2:=Vector[column](3, [-a + b, c - b, 0]);
simplify(t2,sidrels) ; # This should be <-c__bar,a__bar,0> or <c__bar,-a__bar,0,>r

Vector(3, {(1) = -a+b, (2) = c-b, (3) = 0})

  

Vector[column](%id = 36893490321805397396)

 (4)
> 

t3:=normal(1 - b^2/(a*c));

(a*c-b^2)/(a*c)

 (5)
> 

simplify((5),sidrels); # This should be Delta/(a*c)

Delta/(b^2+Delta)

 (6)
> 

simplify(numer((5)),sidrels,'mindeg')/(denom((5)) ); # This works
> 

 

Delta/(a*c)

 (7)
> 

t4:=[(-b^2 + b*c)/(a*c - b^2), (a*b - b^2)/(a*c - b^2)]

[(-b^2+b*c)/(a*c-b^2), (a*b-b^2)/(a*c-b^2)]

 (8)
> 

simplify(t4,sidrels,'mindeg');# this should be [(b*a__bar)/Delta, (b*b__bar)/Delta]

[-b*(-c+b)/Delta, b*(a-b)/Delta]

 (9)
> 

t5:=simplify(Vector[column](3, [3*b*c__bar - Delta, -3*a__bar*b + Delta, b*(-c__bar + a__bar)]))

Vector(3, {(1) = 3*b*c__bar-Delta, (2) = -3*a__bar*b+Delta, (3) = b*(-c__bar+a__bar)})

 (10)
> 

t5[3]:=simplify(t5[3],sidrels)  ;#This should be b*b__bar
> 

 

-a__bar^2+(2*b+d)*a__bar-Delta

 (11)
> 

t5

Vector(3, {(1) = 3*b*c__bar-Delta, (2) = -3*a__bar*b+Delta, (3) = -a__bar^2+(2*b+d)*a__bar-Delta})

 (12)
> 

subs(sidrels[-2],b*(-c__bar + a__bar))

b*(-c__bar+a__bar)

 (13)
> 

algsubs(sidrels[-2],b*(-c__bar + a__bar))

b*(-c__bar+a__bar)

 (14)
> 

 


 

Download 2026-07-29_Q_Simplifications_and_Substitutions.mw

eBook help video?...

Ronan 1486 Maple 2026
ebook incomplete-question
July 07 2026
0 0

Are there any demonstration help videos on creating an eBook? Currently I am struggling with the pages Having a laid out example would really help.  

I would like to do an eBook version of my help pages to send to some people. Hopefully I can use the current help worksheets. The are formatted based on the Maple help structure.

My currrent structure in the help section is:

Rational Trigonometry

      (about 40 help topics)

       RTProjective

       (about 20 help topics)

       UHG

       (about 20 help topics)

     Edit:- I have made a small step of progress using the " Assistant eBook template" but I am getting this error on build. I don't know how to find the cause of the error.

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