STA: "CharPoly-vs-MinPoly.txt"   Wednesday, 08Nov2023:

  =========================================
  Matrices B1 B2 B3 with the same Char-Poly
         but different Min-Polys
  =========================================

(setq *POLY-var* "t")
(progn (set-ring-modulus 7) (use-ring Modsym-M-ring))

;;; Making some Jordan-blocks:
(defaliasqq mms mat-make-square)

(setq  J1 (mms -1 0 0   0 -1 0   0 0 -1)
       J2 (mms -1 1 0   0 -1 0   0 0 -1)
       J3 (mms -1 1 0   0 -1 1   0 0 -1) )


;;Let's look at the three matrices.
(mat-Horiz-concat  J1 J2 J3)
   [ -1   0   0     -1   1   0     -1   1   0 ]
   [  0  -1   0      0  -1   0      0  -1   1 ]
   [  0   0  -1      0   0  -1      0   0  -1 ]


;; Make a random invertible  matrix.
(setq U (mms  2  -3  -1
             -1  -3   0
              1   0  -1  ) )

(setq InvU (mat-inverse? U))
(defun Conj (matrix) (mat-mul InvU matrix U))  ;;Conjugation fnc...
(setq  B1 (conj J1)  B2 (conj J2)  B3 (conj J3)) ;; ...applied.






;; Our new matrices, similar to J1 J2 J3, respectively.
(mat-Horiz-concat  B1 B2 B3)
  [ -1   0   0       2   2   0      -2   2  -3 ]
  [  0  -1   0      -1   3   0       0   3  -1 ]
  [  0   0  -1       3   2  -1      -1   2   3 ]


;;For convenience, multiply CharPolys by -1.
(defun NegCP (matrix) (poly-scale-by (mat-CharPoly matrix) -1))

;;Compute CharPolys (essentially).
(setq p1 (NegCP B1)  p2 (NegCP B2)  p3 (NegCP B3))

;;Compute MinPolys.
(setq m1 (mat-MinPoly B1)  m2 (mat-MinPoly B2)  m3 (mat-MinPoly B3))


;; Compare CharPolys to MinPolys:
(list p1 p2 p3) => t^3 + 3t^2 + 3t + 1,  t^3 + 3t^2 + 3t + 1,  t^3 + 3t^2 + 3t + 1
(list m1 m2 m3) =>               t + 1,         t^2 + 2t + 1,  t^3 + 3t^2 + 3t + 1

;; Written differently:
(list p1 p2 p3)   =>   [t + 1]^3 ,  [t + 1]^3 ,  [t + 1]^3
(list m1 m2 m3)   =>    t + 1    ,  [t + 1]^2 ,  [t + 1]^3

END: "CharPoly-vs-MinPoly.txt"













;; Compare CharPolys to MinPolys:
(list p1 p2 p3) => t³+3t²+3t+1 ,   t³+3t²+3t+1 ,    t³+3t²+3t+1
(list m1 m2 m3) =>         t+1 ,       t²+2t+1 ,    t³+3t²+3t+1

;; Written differently:
(list p1 p2 p3)   =>   [t+1]³ ,     [t+1]³ ,      [t+1]³
(list m1 m2 m3)   =>    t+1   ,     [t+1]² ,      [t+1]³