[MUG] Rings ...

[MUG] Re: Rings ...
Author: Maple Group    Posted: 20/11/2000 16:24:00 GMT
>> From: Maple Group | >> From: L C G Rogers | | Is there any way in Maple to do symbolic manipulation | without the assumption that a*b=b*a? | -=*=-=*=-=*=-=*=-=*=-=*=-=*=-=*=-=*=-=*=-=*=-=*=-=*=-=*=-=*=-=*=-=*=-=*=-=*=- Date: Thu, 16 Nov 2000 11:05:51 +0100 From: Maerivoet Roland To: Subject: Rings ... You could use the matrixoperator &* Roland Maerivoet Applied Informatics Hogeschool Gent -=*=-=*=-=*=-=*=-=*=-=*=-=*=-=*=-=*=-=*=-=*=-=*=-=*=-=*=-=*=-=*=-=*=-=*=-=*=- Date: Thu, 16 Nov 2000 13:01:39 +0000 From: "Dr Francis J. Wright" To: Subject: Rings ... The standard non-commutative multiplication operator is &* (and . in maple6). Francis -- Dr Francis J. Wright | mailto: School of Mathematical Sciences | tel: (020) 7882 5453 (direct) Queen Mary, University of London | fax: (020) 8981 9587 (dept.) Mile End Road, London E1 4NS, UK | http://centaur.maths.qmw.ac.uk/ -=*=-=*=-=*=-=*=-=*=-=*=-=*=-=*=-=*=-=*=-=*=-=*=-=*=-=*=-=*=-=*=-=*=-=*=-=*=- From: Douglas Wilhelm Harder Subject: Rings ... To: Date: Thu, 16 Nov 2000 11:38:15 -0500 (EST) Hello, In Maple6, you can use modules to override operators. Here is an example of a ring of 2x2 matrices module 19 which overrides +, * and -, and it recognizes the special ring members 0 and 1. Once you have done 'with(Matrix22mod19);', the call: > A * B; is immediately converted to `*`( A, B ) and so there you may decide whether or not * should be commutative. > Matrix22mod19 := module() export `*`, `+`, `-`; `*` := proc( A::{Matrix(integer), 0, 1}, B::{Matrix(integer), 0, 1} ) local result; if (member( A, {0, 1} ) or LinearAlgebra[Dimensions]( A ) = (2, 2)) and (member( B, {0, 1} ) or LinearAlgebra[Dimensions]( B ) = (2, 2)) then if A = 0 then 0; elif B = 0 then 0; elif A = 1 and B = 1 then 1; elif A = 1 then LinearAlgebra[Map]( `modp`, B, 19 ); elif B = 1 then LinearAlgebra[Map]( `modp`, A, 19 ); else result := Matrix( [[(A[1,1] * B[1,1] + A[1,2] * B[2,1]) mod 19, (A[1,1] * B[1,2] + A[1,2] * B[2,2]) mod 19], [(A[1,1] * B[1,1] + A[1,2] * B[2,1]) mod 19, (A[1,1] * B[1,2] + A[1,2] * B[2,2]) mod 19]] ); if LinearAlgebra[Equal]( result, Matrix([[0,0],[0,0]]) ) then 0; elif LinearAlgebra[Equal]( result, Matrix([[1,0],[0,1]]) ) then 1; else result; end if; end if; else error "expecting 2 by 2 matrices"; end if; end proc; `-` := proc( A::{Matrix(integer), 0, 1} ) local result; if A = 0 then 0; elif A = 1 then Matrix( [[18, 0], [0, 18]] ); elif LinearAlgebra[Dimensions]( A ) = (2, 2) then result := map( `modp`, map( :-`-`, A ), 19 ); if LinearAlgebra[Equal]( result, Matrix([[0,0],[0,0]]) ) then 0; elif LinearAlgebra[Equal]( result, Matrix([[1,0],[0,1]]) ) then 1; else result; end if; else error "expecting a 2 by 2 matrix"; end if; end proc; `+` := proc( A::{Matrix(integer), 0, 1}, B::{Matrix(integer), 0, 1} ) local result; if (member( A, {0, 1} ) or LinearAlgebra[Dimensions]( A ) = (2, 2)) and (member( B, {0, 1} ) or LinearAlgebra[Dimensions]( B ) = (2, 2)) then if A = 0 and B = 0 then 0; elif A = 1 then procname( Matrix([[1,0],[0,1]]), B ); elif B = 1 then procname( A, Matrix([[1,0],[0,1]]) ); elif A = 0 then LinearAlgebra[Map]( `modp`, B, 19 ); elif B = 0 then LinearAlgebra[Map]( `modp`, A, 19 ); else result := Matrix( [[(A[1,1] + B[1,1]) mod 19, (A[1,2] + B[1,2]) mod 19], [(A[2,1] + B[2,1]) mod 19, (A[2,2] + B[2,2]) mod 19]] ); if LinearAlgebra[Equal]( result, Matrix([[0,0],[0,0]]) ) then 0; elif LinearAlgebra[Equal]( result, Matrix([[1,0],[0,1]]) ) then 1; else result; end if; end if; else error "expecting 2 by 2 matrices"; end if; end proc; end module: So now you can do: > with(Matrix22mod19); [*, +, -] > <<1,5>|<7,3>> + 1; # New short form for entering the new Matrices. [2 7] [ ] [5 4] > <<1,5>|<7,3>> * 1; [1 7] [ ] [5 3] > 1 + 1; [2 0] [ ] [0 2] > <<1,5>|<7,3>> * <<7,2>|<5,2>>; [2 0] [ ] [2 0] > <<7,2>|<5,2>> * <<1,5>|<7,3>>; [13 7] [ ] [13 7] > <<1,5>|<7,3>> * <<7,2>|<5,2>> + <<18, 17>|<0, 1>>; 1 > <<7,2>|<5,2>> * <<1,5>|<7,3>> - <<12, 13>|<7, 6>>; 1 > -<<1,5>|<7,3>>; [18 12] [ ] [14 16] > -1; [18 0] [ ] [ 0 18] Note that since you have overridden +, -, and *, you can no longer do very much else: > -3; Error, - expects its 1st argument, A, to be of type {0, 1, Matrix(integer)}, but received 3 > 3 + 4; Error, + expects its 1st argument, A, to be of type {0, 1, Matrix(integer)}, but received 3 so you might want to not just return an error if you get something other than 0, 1, or a Matrix of type integer. Hopefully this is somewhat useful. Cheers, Douglas -- Douglas Wilhelm Harder ||\_ _\ /_/\_ _/\_\ /_ _/|| MM BSc MCpl (Retd) University of Waterloo ||\ _\ /_/\_\ /_ _/ Waterloo Fractal Compression Project _\ / _/|| _/\_\||\_ _/ /_/\_ http://links.uwaterloo.ca/~douglas _/||\_ _/ /_/\ -=*=-=*=-=*=-=*=-=*=-=*=-=*=-=*=-=*=-=*=-=*=-=*=-=*=-=*=-=*=-=*=-=*=-=*=-=*=- From: "Willard, Daniel Dr DUSA-OR" To: Subject: Rings ... Date: Thu, 16 Nov 2000 09:30:23 -0500 In the help section "evalm" (evaluate matrix expressions) you will find the paragraph: To indicate non-commutative matrix multiplication, use the operator &*. The matrix product ABC may be entered as A &* B &* C or as &*(A,B,C), the latter being more efficient. Automatic simplifications such as collecting constants and powers will be applied. Do NOT use the * to indicate purely matrix multiplication, as this will result in an error. The operands of &* must be matrices (or names) with the exception of 0. Unevaluated matrix products are considered to be matrices. The operator &* has the same precedence as the * operator.

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