%%%% Sta of "RREF-Mod-p.Lisp.txt"

;; Uses a semicolon ";" to start a comment.
;; Use "::::" to indicate a new problem or section.
;; Use "%" for the Lisp prompt.


  :::: Reducing a Zed_{17} matrix ::::
% (set-ring-modulus 17)
% (setq *REWid* 4)  ;;Printing-width can be less

% (use-ring Mod-M-ring)

% (setq m17 (mat-make-random 4 6 100))
  [   6   4  14   7  16   9  ]
  [  16   8  12  10  13  11  ]
  [  10  14  12   2   6  16  ]
  [   6   0   2  12   3  15  ]



% (rref-mtab-beforecol m17 :stepwise t)

  Reducing column-0 produces

      c0  c1  c2  c3  c4  c5     Row operations
   |---------------------------|
r0 |   1  12   8   4  14  10   |   3   0   0   0
r1 |   0   3   3  14  10   4   |   3   1   0   0
r2 |   0  13   0  13   2   1   |   4   0   1   0
r3 |   0  13   5   5   4   6   |  16   0   0   1
   |---------------------------|


  Reducing column-1 produces

      c0  c1  c2  c3  c4  c5     Row operations
   |---------------------------|
r0 |   1   0  13  16   8  11   |   8  13   0   0
r1 |   0   1   1  16   9   7   |   1   6   0   0
r2 |   0   0   4   9   4  12   |   8   7   1   0
r3 |   0   0   9   1   6   0   |   3   7   0   1
   |---------------------------|


  Reducing column-2 produces

      c0  c1  c2  c3  c4  c5     Row operations
   |---------------------------|
r0 |   1   0   0   8  12   6   |  16   3   1   0
r1 |   0   1   0   1   8   4   |  16   0   4   0
r2 |   0   0   1  15   1   3   |   2   6  13   0
r3 |   0   0   0   2  14   7   |   2   4   2   1
   |---------------------------|


  Reducing column-3 produces

      c0  c1  c2  c3  c4  c5     Row operations
   |---------------------------|
r0 |   1   0   0   0   7  12   |   8   4  10  13
r1 |   0   1   0   0   1   9   |  15  15   3   8
r2 |   0   0   1   0  15  10   |   4  10  15   1
r3 |   0   0   0   1   7  12   |   1   2   1   9
   |---------------------------|

  JK: Found 4 pivots before the sixth column.



;; Reducing again, but using symmetric-residues.

% (use-ring Modsym-M-ring)
% (rref-mtab-beforecol m17 :stepwise t)


  Reducing column-0 produces

      c0  c1  c2  c3  c4  c5     Row operations
   |---------------------------|
r0 |   1  -5   8   4  -3  -7   |   3   0   0   0
r1 |   0   3   3  -3  -7   4   |   3   1   0   0
r2 |   0  -4   0  -4   2   1   |   4   0   1   0
r3 |   0  -4   5   5   4   6   |  -1   0   0   1
   |---------------------------|


  Reducing column-1 produces

      c0  c1  c2  c3  c4  c5     Row operations
   |---------------------------|
r0 |   1   0  -4  -1   8  -6   |   8  -4   0   0
r1 |   0   1   1  -1  -8   7   |   1   6   0   0
r2 |   0   0   4  -8   4  -5   |   8   7   1   0
r3 |   0   0  -8   1   6   0   |   3   7   0   1
   |---------------------------|


  Reducing column-2 produces

      c0  c1  c2  c3  c4  c5     Row operations
   |---------------------------|
r0 |   1   0   0   8  -5   6   |  -1   3   1   0
r1 |   0   1   0   1   8   4   |  -1   0   4   0
r2 |   0   0   1  -2   1   3   |   2   6  -4   0
r3 |   0   0   0   2  -3   7   |   2   4   2   1
   |---------------------------|


  Reducing column-3 produces

      c0  c1  c2  c3  c4  c5     Row operations
   |---------------------------|
r0 |   1   0   0   0   7  -5   |   8   4  -7  -4
r1 |   0   1   0   0   1  -8   |  -2  -2   3   8
r2 |   0   0   1   0  -2  -7   |   4  -7  -2   1
r3 |   0   0   0   1   7  -5   |   1   2   1  -8
   |---------------------------|

  JK: Found 4 pivots before the sixth column.


      c0  c1  c2  c3  c4  c5     Row operations
   |---------------------------|
r0 |   1   0   0   0   7  -5   |   8   4  -7  -4
r1 |   0   1   0   0   1  -8   |  -2  -2   3   8
r2 |   0   0   1   0  -2  -7   |   4  -7  -2   1
r3 |   0   0   0   1   7  -5   |   1   2   1  -8
   |---------------------------|




  :::: Change-of-Basis ::::
;; Over Zed_{17},  we have this mystery matrix:

% (setq  Mys (mat-make-initseq 3 3 '( -3 7 6 2 4 -7 0 -1 -1 )))
    [  -3   7   6  ]
    [   2   4  -7  ]
    [   0  -1  -1  ]


;; By sheer luck, we discover that Mys cubed, is the Identity-matrix.

% (mat-mul Mys Mys Mys)
    [  1  0  0  ]
    [  0  1  0  ]
    [  0  0  1  ]

;; So Mys is, relative to the std. basis, the matrix-name of some
;; abstract trn Y (for "mYstery") which will likely have a simple
;; description relative to a better basis.  Let's rename our given
;; matrix as the standard-basis-name of Y:

;; Matrix of trn Y, going To basis E, From basis E.
% (setq  Y-To-E-Fr-E  Mys)

      [  -3   7   6  ]
      [   2   4  -7  ]
      [   0  -1  -1  ]



    ;; Some nice person tells us that ordered-basis (b0, b1, b2) is
    ;; better for Y, where

% b0 , b1 , b2

    [ -8 ]     [ -6 ]    [ -6 ]
    [  0 ] ,   [ -5 ] ,  [  2 ]
    [ -5 ]     [ -7 ]    [  5 ]




    ;; Hence the Change-of-Basis matrix we need is

;; Matrix of Identity-trn, going To basis E, From basis B ...
% (setq Id-To-E-Fr-B  (mat-Horiz-concat b0 b1 b2))

    [  -8  -6  -6  ]
    [   0  -5   2  ]
    [  -5  -7   5  ]


    ;; ... together with its inverse:

% (setq Id-To-B-Fr-E   (mat-inverse?  Id-To-E-Fr-B))

    [  -5   8   1  ]
    [  -3  -4  -2  ]
    [   1  -1  -5  ]


  ;; We now conjugate Mys to get trn Y w.r.t basis B.

% (setq Y-To-B-Fr-B (mat-mul  Id-To-B-Fr-E  Y-To-E-Fr-E  Id-To-E-Fr-B))

    [  0  1  0  ]
    [  0  0  1  ]
    [  1  0  0  ]

;;; Ah ha!  Our Y is simply cyclic permution of the three Basis-B vectors.
;;  Its lefthand-action sends
;;     b0 -> b2 -> b1 ->
;;  and back to b0.
;;     Matrix  Y-To-B-Fr-B  is an example
;;  of a PERMUTATION MATRIX.  In the ring of NxN matrices, there
;;  are  N! (N factorial) many permutation-matrices.



;;; The ring makes a difference:

% (with-ring Rational-ring (setq Wrong-ring (mat-mul  Id-To-B-Fr-E  Y-To-E-Fr-E  Id-To-E-Fr-B)))

    [  187   443  -629  ]
    [  -68    85   -16  ]
    [  -50  -136   136  ]


;;; Although, we CAN recover the correct result:

% (mat-mul Wrong-ring (mat-eye 3))

    [  0  1  0  ]
    [  0  0  1  ]
    [  1  0  0  ]



%%%% End of "RREF-Mod-p.Lisp.txt"