m= Matrix([[1/3,1/2],[2/3,1/2]]) i= Matrix([[1,0],[0,1]]) #Comme l'espace propre est 1, je créé la matrice identité de diagonal 1. show("M =",m," , I =",i)
M = (31322121) , I = (1001)
show("valeurs propres de M =",m.eigenvalues()) #Valeur Propre show(m-i)
valeurs propres de M = [1, −61]
(−323221−21)
show("noyau = ",(m-i).right_kernel()) #Noyau
noyau = RowSpanQ(134)
var('x,y') show(1/3*x+1/2*y==x);show(2/3*x+1/2*y==y) #Système linéaire associé à m S=solve([1/3*x+1/2*y==x,2/3*x+1/2*y==y],[x,y]);show(S) #solution du système linéaire associé à m
(x, y)
31x+21y=x
32x+21y=y
[[x=43r2, y=r2]]
k=2 ; show("M^2 =",(m^k).n(digits=3)) k=16 ; show("M^16 =",(m^k).n(digits=3)) k=128 ; show("M^128 =",(m^k).n(digits=3)) #la matrice m
M^2 = (0.4440.5560.4170.583)
M^16 = (0.4290.5710.4290.571)
M^128 = (0.4290.5710.4290.571)
I? #un imaginaire
File: /ext/sage/sage-8.1/local/lib/python2.7/site-packages/sage/symbolic/expression.pyx Signature : I(SR, x=0) Docstring : Nearly all expressions are created by calling new_Expression_from_*, but we need to make sure this at least does not leave self._gobj uninitialized and segfault.
#Avec N : N=matrix([[1/2,0,0],[1/4,0,1],[1/4,1,0]]);show("N =",N)
N = 214141001010
#espace propre pour la valeur propre 1 #valeurs propres -> eigenvalue I=matrix.identity(3) show((N-I).right_kernel().basis())
[(0,1,1)]
var('x y z') c=['gray','darkkhaki','aquamarine'] show(sum(implicit_plot3d((N-I)[k]*vector([x,y,z])==0,(x,-1,1),(y,-1,1),(z,-1,1),color=c[k]) for k in range(3)))
(x, y, z)
3D rendering not yet implemented
show((N^16).n(digits=3)) show((N^17).n(digits=3))
0.00001530.5000.5000.0001.000.0000.0000.0001.00
7.63×10−60.5000.5000.0000.0001.000.0001.000.000
show("valeurs propres de N =",N.eigenvalues()) #Valeur Propre show((N-I).right_kernel())
valeurs propres de N = [1, 21, −1]
RowSpanQ(011)
var('x,y') show(1/2*x+0*y+0*z==x);show(1/4*x+0*y+1*z==y);show(1/4*x+1*y+0*z==x) #Système linéaire associé à N T=solve([1/2*x+0*y+0*z==x,1/4*x+0*y+1*z==y,1/4*x+1*y+0*z==z],[x,y,z]);show(T) #solution du système linéaire associé à N
(x, y)
21x=x
41x+z=y
41x+y=x
[[x=0, y=r5, z=r5]]
#r5 étant le paramètre r* qui est créé par Sage et change à chaque exécution #partie sur feuille : M= Matrix([[1/2,0,0],[-1/4,-1/2,-9/4],[1/6,2/3,2]]); show("M =",M) i=Matrix.identity(3) show("determinant M =",(M-1/2*i).determinant());show("Identitée =",i) show(M.rank())
M = 21−41610−21320−492
determinant M = 0
Identitée = 100010001
3
show("valeurs propres =",M.eigenvalues()) #multiplicité =2
valeurs propres = [1, 21, 21]
show((M-1/2*i).rank()) #d'après le thml des rangs c'est juste car 3-2 = 1
1
var('x y z') c=['gray','darkkhaki','aquamarine'] show(sum(implicit_plot3d((M-I)[k]*vector([x,y,z])==0,(x,-2,2),(y,-2,2),(z,-2,2),color=c[k]) for k in range(3)))
(x, y, z)
3D rendering not yet implemented
k=2 ; show("M^2 =",(M^k).n(digits=3)) k=16 ; show("M^16 =",(M^k).n(digits=3)) k=128 ; show("M^128 =",(M^k).n(digits=3)) #la matrice M tend vers M=([0,0,0],[-1/2,-2,-9/2],[1/6,2/3,3])
M^2 = 0.250−0.3750.2500.000−1.251.000.000−3.382.50
M^16 = 0.0000153−0.5000.3330.000−2.001.330.000−4.503.00
M^128 = 2.94×10−39−0.5000.3330.000−2.001.330.000−4.503.00
f=Matrix([[2,2,-1],[0,4,0],[-4,4,2]]);show("f =",f)
f = 20−4244−102
f.right_kernel()
Free module of degree 3 and rank 1 over Integer Ring
Echelon basis matrix:
[1 0 2]