(*^
::[	Information =

	"This is a Mathematica Notebook file.  It contains ASCII text, and can be
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	everything from the line containing (*^ down to the line containing ^*)
	into a plain text file.  On some systems you may have to give the file a 
	name ending with ".ma" to allow Mathematica to recognize it as a Notebook.
	The line below identifies what version of Mathematica created this file,
	but it can be opened using any other version as well.";

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:[font = title; inactive; preserveAspect; startGroup]
Lorentz
:[font = subsection; inactive; preserveAspect; startGroup]
linéarisation
:[font = text; inactive; preserveAspect]
On se donne la matrice linéarisée
:[font = input; preserveAspect; startGroup]
L={{-s,s,0},{r,-1,0},{0,0,-b}}
:[font = output; output; inactive; preserveAspect; endGroup]
{{-s, s, 0}, {r, -1, 0}, {0, 0, -b}}
;[o]
{{-s, s, 0}, {r, -1, 0}, {0, 0, -b}}
:[font = input; preserveAspect; startGroup]
MatrixForm[L]
:[font = output; output; inactive; preserveAspect; endGroup]
MatrixForm[{{-s, s, 0}, {r, -1, 0}, {0, 0, -b}}]
;[o]
-s   s    0

r    -1   0

0    0    -b
:[font = text; inactive; preserveAspect]
ses vecteurs propres,
:[font = input; preserveAspect; startGroup]
ev=Eigenvectors[L]
:[font = output; output; inactive; preserveAspect; endGroup]
{{0, 0, 1}, {-(-1 + s + (1 - 2*s + 4*r*s + s^2)^(1/2))/
 
    (2*r), 1, 0}, {-(-1 + s - 
 
       (1 - 2*s + 4*r*s + s^2)^(1/2))/(2*r), 1, 0}}
;[o]
                                                2
             -(-1 + s + Sqrt[1 - 2 s + 4 r s + s ])
{{0, 0, 1}, {--------------------------------------, 1, 
                              2 r
 
                                           2
        -(-1 + s - Sqrt[1 - 2 s + 4 r s + s ])
   0}, {--------------------------------------, 1, 0}}
                         2 r
:[font = text; inactive; preserveAspect]
ses valeurs propres,
:[font = input; preserveAspect; startGroup]
el=Eigenvalues[L]
:[font = output; output; inactive; preserveAspect; endGroup]
{-b, (-1 - s - (-4*(1 - r)*s + (1 + s)^2)^(1/2))/2, 
 
  (-1 - s + (-4*(1 - r)*s + (1 + s)^2)^(1/2))/2}
;[o]
                                         2
     -1 - s - Sqrt[-4 (1 - r) s + (1 + s) ]
{-b, --------------------------------------, 
                       2
 
                                      2
  -1 - s + Sqrt[-4 (1 - r) s + (1 + s) ]
  --------------------------------------}
                    2
:[font = input; preserveAspect; startGroup]
Simplify[ev.(L.{x,y,0})-lbd2 Simplify[ev.{x,y,0}]]
:[font = output; output; inactive; preserveAspect; endGroup]
{0, ((r - s)*(2*r*x - y + s*y + 
 
       (1 - 2*s + 4*r*s + s^2)^(1/2)*y))/(2*r), 
 
  (r + (s*(-1 + s - (1 - 2*s + 4*r*s + s^2)^(1/2)))/
 
       (2*r))*x + ((-2*r + s - s^2 + 
 
        s*(1 - 2*s + 4*r*s + s^2)^(1/2))*y)/(2*r) + 
 
   ((1 + s + (1 - 2*s + 4*r*s + s^2)^(1/2))*
 
      (x - s*x + (1 - 2*s + 4*r*s + s^2)^(1/2)*x + 2*r*y))
 
     /(4*r)}
;[o]
    (r - s) (2 r x - y + s y + 
     
                                2
        Sqrt[1 - 2 s + 4 r s + s ] y)
{0, --------------------------------------------------------
                              2 r
 
                                              2
          s (-1 + s - Sqrt[1 - 2 s + 4 r s + s ])
   , (r + ---------------------------------------) x + 
                            2 r
 
                2                             2
   (-2 r + s - s  + s Sqrt[1 - 2 s + 4 r s + s ]) y
   ------------------------------------------------ + 
                         2 r
 
                                     2
   ((1 + s + Sqrt[1 - 2 s + 4 r s + s ]) 
 
                                         2
      (x - s x + Sqrt[1 - 2 s + 4 r s + s ] x + 2 r y)) / 
 
    (4 r)}
:[font = input; preserveAspect; startGroup]
 v3=ev[[3]] 
:[font = output; output; inactive; preserveAspect; fontLeading = 0; endGroup]
{-(-1 + s - (1 - 2*s + 4*r*s + s^2)^(1/2))/(2*r), 1, 0}
;[o]
                                    2
 -(-1 + s - Sqrt[1 - 2 s + 4 r s + s ])
{--------------------------------------, 1, 0}
                  2 r
:[font = input; preserveAspect; startGroup]
v2=ev[[2]];lbd2=el[[2]];
:[font = output; output; inactive; preserveAspect; endGroup]
(-1 - s - (-4*(1 - r)*s + (1 + s)^2)^(1/2))/2
;[o]
                                    2
-1 - s - Sqrt[-4 (1 - r) s + (1 + s) ]
--------------------------------------
                  2
:[font = input; preserveAspect; startGroup]
Simplify[L.v2/lbd2-v2]
:[font = output; output; inactive; preserveAspect; fontLeading = 0; endGroup]
{0, 0, 0}
;[o]
{0, 0, 0}
:[font = text; inactive; preserveAspect; fontLeading = 0]
List(-(-1 + s + Sqrt(1 - 2*s + 4*r*s + Power(s,2)))/(2*r),
 
  1,0)
;[o]
List(-(-1 + s + Sqrt(1 - 2*s + 4*r*s + Power(s,2)))/(2*r),
   1,0)
:[font = input; preserveAspect]
CForm[el[[3]]]
:[font = text; inactive; preserveAspect]
(-1 - s + Sqrt(-4*(1 - r)*s + Power(1 + s,2)))/2
;[o]
(-1 - s + Sqrt(-4*(1 - r)*s + Power(1 + s,2)))/2
:[font = text; inactive; preserveAspect; endGroup; endGroup]
(-1 - s - Sqrt(-4*(1 - r)*s + Power(1 + s,2)))/2
;[o]
(-1 - s - Sqrt(-4*(1 - r)*s + Power(1 + s,2)))/2
^*)
