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Ch8_lib.py
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Sun, Apr 28, 20:35

Ch8_lib.py
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import sys, os
sys.path.append('../Librairie')
import AL_Fct as al
import numpy as np
import sympy as sp
from IPython.utils import io
from IPython.display import display, Latex, Markdown
import plotly
import plotly.graph_objects as go
def vector_plot_3D(v, b):
"""
Show 3D plot of a vector (v) and of b = A * v
@param v: numpy array of shape (3,)
@param b: numpy array of shape (3,)
@return:
"""
fig = go.Figure()
fig.add_trace(go.Scatter3d(x=[0, v[0]], y=[0, v[1]], z=[0, v[2]],
line=dict(color='red', width=4),
mode='lines+markers',
name='$v$'))
fig.add_trace(go.Scatter3d(x=[0, b[0]], y=[0, b[1]], z=[0, b[2]],
line=dict(color='royalblue', width=4, dash='dash'),
mode='lines+markers',
name='$A \ v$'))
fig.show()
def CheckEigenVector(A, v):
"""
Check if v is an eigenvector of A, display step by step solution
@param A: square sympy Matrix of shape (n,n)
@param v: 1D sympy Matrix of shape (n,1)
@return:
"""
# Check Dimensions
if A.shape[0] != A.shape[1] or v.shape[0] != A.shape[1]:
raise ValueError('Dimension problem, A should be square (n x n) and v (n x 1)')
if v == sp.zeros(v.shape[0], 1):
display(Latex("$v$ est le vecteur nul, il ne peut pas être un vecteur propre par définition."))
else:
# Matrix Multiplication
b = A * v
# Print some explanation about the method
display(Latex("On voit que $ b = A v = " + latexp(b) + "$"))
display(Latex("On cherche alors un nombre $\lambda \in \mathbb{R}$ tel que $b = \lambda v" \
+ "\Leftrightarrow" + latexp(b) + " = \lambda" + latexp(v) + '$'))
# Symbol for lambda
l = sp.symbols('\lambda', real=True)
# Check if there is a solution lambda of eq: A*v = lambda * v
eq = sp.Eq(b, l * v)
sol = sp.solve(eq, l)
# If there is l st b = l*v
if sol:
display(Latex("Il existe bien une solution pour $\lambda$. Le vecteur $v$ est donc un vecteur \
propre de la matrice $A$."))
display(Latex("La valeur propre associée est $\lambda = " + sp.latex(sol[l]) + "$."))
# Otherwise
else:
display(Latex("L'equation $b = \lambda v$ n'a pas de solution."))
display(Latex("Le vecteur $v$ n'est donc pas un vecteur propre de la matrice $A$."))
def ch8_1_exo_2(A, l, vp, v):
"""
Display step by step
@param A: Square sympy matrix
@param l: eigenvalue (float or int)
@param vp: Boolean, given answer to question is l an eigenvalue of A
@param v: proposed eigenvector
@return:
"""
# Check Dimensions
if A.shape[0] != A.shape[1] or v.shape[0] != A.shape[1]:
raise ValueError('Dimension problem, A should be square (n x n) and v (n x 1)')
n = A.shape[0]
eig = list(A.eigenvals().keys())
for i, w in enumerate(eig):
eig[i] = float(w)
eig = np.array(eig)
if np.any(abs(l-eig) < 10**-10):
if vp:
display(Latex("$\lambda = " + str(l) + "$ est bien une valeur propre de la matrice $A$."))
else:
display(Latex("Non, $\lambda = " + str(l) + "$ est bien une valeur propre de la matrice $A$."))
if v != sp.zeros(n, 1):
# Check the eigen vector v
z = sp.simplify(A * v - l * v)
if z == sp.zeros(n, 1):
display(Latex("$v$ est bien un vecteur propre de $A$ associé à $\lambda = " + str(l) + "$ car on a:"))
display(Latex("$$" + latexp(A) + latexp(v) + "= " + str(l) + "\cdot " + latexp(v) + "$$"))
else:
display(Latex("$v$ n'est pas un vecteur propre de $A$ associé à $\lambda = " + str(l) + "$ car on a:"))
display(Latex("$$" + latexp(A) + latexp(v) + "\\neq \lambda" + latexp(v) + "$$"))
else:
display(Latex("$v$ est le vecteur nul et ne peut pas être par définition un vecteur propre."))
else:
if vp:
display(Latex("En effet, $\lambda$ n'est pas une valeur propre de $A$."))
else:
display(Latex("Non, $\lambda = " + str(l) + "$ n'est pas une valeur propre de $A$."))
def red_matrix(A, i, j):
""" Return reduced matrix (without row i and col j)"""
row = [0, 1, 2]
col = [0, 1, 2]
row.remove(i - 1)
col.remove(j - 1)
return A[row, col]
def pl_mi(i, j, first=False):
""" Return '+', '-' depending on row and col index"""
if (-1) ** (i + j) > 0:
if first:
return ""
else:
return "+"
else:
return "-"
def brackets(expr):
"""Takes a sympy expression, determine if it needs parenthesis and returns a string containing latex of expr
with or without the parenthesis."""
expr_latex = sp.latex(expr)
if '+' in expr_latex or '-' in expr_latex:
return "(" + expr_latex + ")"
else:
return expr_latex
def Determinant_3x3(A, step_by_step=True, row=True, n=1):
"""
Step by step computation of the determinant of a 3x3 sympy matrix strating with given row/col number
@param A: 3 by 3 sympy matrix
@param step_by_step: Boolean, True: print step by step derivation of det, False: print only determinant
@param row: True to compute determinant from row n, False to compute determinant from col n
@param n: row or col number to compute the determinant from (int between 1 and 3)
@return: display step by step solution for
"""
if A.shape != (3, 3):
raise ValueError('Dimension of matrix A should be 3x3. The input A must be a sp.Matrix of shape (3,3).')
if n < 1 or n > 3 or not isinstance(n, int):
raise ValueError('n should be an integer between 1 and 3.')
# Construc string for determinant of matrix A
detA_s = sp.latex(A).replace('[', '|').replace(']', '|')
# To print all the steps
if step_by_step:
# If we compute the determinant with row n
if row:
# Matrix with row i and col j removed (red_matrix(A, i, j))
A1 = red_matrix(A, n, 1)
A2 = red_matrix(A, n, 2)
A3 = red_matrix(A, n, 3)
detA1_s = sp.latex(A1).replace('[', '|').replace(']', '|')
detA2_s = sp.latex(A2).replace('[', '|').replace(']', '|')
detA3_s = sp.latex(A3).replace('[', '|').replace(']', '|')
line1 = "$" + detA_s + ' = ' + pl_mi(n, 1, True) + brackets(A[n - 1, 0]) + detA1_s + pl_mi(n, 2) + \
brackets(A[n - 1, 1]) + detA2_s + pl_mi(n, 3) + brackets(A[n - 1, 2]) + detA3_s + '$'
line2 = '$' + detA_s + ' = ' + pl_mi(n, 1, True) + brackets(A[n - 1, 0]) + "\cdot (" + sp.latex(sp.det(A1)) \
+ ")" + pl_mi(n, 2) + brackets(A[n - 1, 1]) + "\cdot (" + sp.latex(sp.det(A2)) + ")" + \
pl_mi(n, 3) + brackets(A[n - 1, 2]) + "\cdot (" + sp.latex(sp.det(A3)) + ')$'
line3 = '$' + detA_s + ' = ' + sp.latex(sp.simplify(sp.det(A))) + '$'
# If we compute the determinant with col n
else:
# Matrix with row i and col j removed (red_matrix(A, i, j))
A1 = red_matrix(A, 1, n)
A2 = red_matrix(A, 2, n)
A3 = red_matrix(A, 3, n)
detA1_s = sp.latex(A1).replace('[', '|').replace(']', '|')
detA2_s = sp.latex(A2).replace('[', '|').replace(']', '|')
detA3_s = sp.latex(A3).replace('[', '|').replace(']', '|')
line1 = "$" + detA_s + ' = ' + pl_mi(n, 1, True) + brackets(A[0, n - 1]) + detA1_s + pl_mi(n, 2) + \
brackets(A[1, n - 1]) + detA2_s + pl_mi(n, 3) + brackets(A[2, n - 1]) + detA3_s + '$'
line2 = '$' + detA_s + ' = ' + pl_mi(n, 1, True) + brackets(A[0, n - 1]) + "\cdot (" + sp.latex(sp.det(A1))\
+ ")" + pl_mi(n, 2) + brackets(A[1, n - 1]) + "\cdot (" + sp.latex(sp.det(A2)) + ")" + \
pl_mi(n, 3) + brackets(A[2, n - 1]) + "\cdot (" + sp.latex(sp.det(A3)) + ')$'
line3 = '$' + detA_s + ' = ' + sp.latex(sp.simplify(sp.det(A))) + '$'
# Display step by step computation of determinant
display(Latex(line1))
display(Latex(line2))
display(Latex(line3))
# Only print the determinant without any step
else:
display(Latex("$" + detA_s + "=" + sp.latex(sp.det(A)) + "$"))
def valeurs_propres(A):
if A.shape[0] != A.shape[1]:
raise ValueError("A should be a square matrix")
l = sp.symbols('\lambda')
n = A.shape[0]
poly = sp.det(A - l * sp.eye(n))
poly_exp = sp.expand(poly)
poly_factor = sp.factor(poly)
det_str = sp.latex(poly_exp) + "=" + sp.latex(poly_factor)
display(Latex("On cherche les valeurs propres de la matrice $ A=" + latexp(A) + "$."))
display(Latex("Le polynome caractéristique de $A$ est: $$\det(A- \lambda I)= " + det_str + "$$"))
eq = sp.Eq(poly, 0)
sol = sp.solve(eq, l)
if len(sol) > 1:
display(Latex("Les racines du polynôme caractéristique sont $" + sp.latex(sol) + "$."))
display(Latex("Ces racines sont les valeurs propres de la matrice $A$."))
else:
display(Latex("L'unique racine du polynôme caractéristique est" + str(sol[0])))
def texVector(v):
"""
Return latex string for vertical vector
Input: v, 1D np.array()
"""
n = v.shape[0]
return al.texMatrix(v.reshape(n, 1))
def check_basis(sol, prop):
"""
Checks if prop basis is equivalent to sol basis
@param sol: verified basis, 2D numpy array, first dim: vector indexes, second dim: idx of element in a basis vect
@param prop: proposed basis
@return: boolean
"""
prop = np.array(prop, dtype=np.float64)
# number of vector in basis
n = len(sol)
# Check dimension of proposed eigenspace
if n != len(prop):
display(Latex("Le nomber de vecteur(s) propre(s) donné(s) est incorrecte. " +
"La dimension de l'espace propre est égale au nombre de variable(s) libre(s)."))
return False
else:
# Check if the sol vector can be written as linear combination of prop vector
# Do least squares to solve overdetermined system and check if sol is exact
A = np.transpose(prop)
lin_comb_ok = np.zeros(n, dtype=bool)
for i in range(n):
x, _, _, _ = np.linalg.lstsq(A, sol[i], rcond=None)
res = np.sum((A @ x - sol[i]) ** 2)
lin_comb_ok[i] = res < 10 ** -13
return np.all(lin_comb_ok)
def eigen_basis(A, l, prop_basis=None, disp=True, return_=False, dispA=True):
"""
Display step by step method for finding a basis of the eigenspace of A associated to eigenvalue l
Eventually check if the proposed basis is correct. Display or not
@param A: Square sympy Matrix with real coefficients
@param l: real eigen value of A (float or int)
@param prop_basis: Proposed basis: list of base vector (type list of list of floats)
@param disp: boolean if display the solution. If false it displays nothing
@param return_: boolean if return something or nothing
@return: basis: a correct basis for the eigen space (2D numpy array)
basic_idx: list with indices of basic variables of A - l*I
free_idx: list with indices of free variables of A - l*I
"""
if not A.is_Matrix:
raise ValueError("A should be a sympy Matrix.")
# Check if A is square
n = A.shape[0]
if n != A.shape[1]:
raise ValueError('A should be a square matrix.')
# Compute eigenvals in symbolic
eig = A.eigenvals()
eig = list(eig.keys())
# Deal with complex number (removal)
complex_to_rm = []
for idx, el in enumerate(eig):
if not el.is_real:
complex_to_rm.append(idx)
for index in sorted(complex_to_rm, reverse=True):
del eig[index]
eig = np.array(eig)
# evaluate symbolic expression
eig_eval = np.array([float(el) for el in eig])
# Check that entered eigenvalue is indeed an eig of A
if np.all(abs(l - eig_eval) > 1e-10) and len(eig) > 0:
display(Latex("$\lambda$ n'est pas une valeur propre de $A$."))
return None, None, None
# Change value of entered eig to symbolic expression (for nice print)
l = eig[np.argmin(np.abs(l - eig))]
I = sp.eye(n)
Mat = A - l * I
b = np.zeros(n)
if disp:
if dispA:
display(Latex("On a $ A = " + latexp(A) + "$."))
display(Latex("On cherche une base de l'espace propre associé à $\lambda = " + str(l) + "$."))
# ER matrix
e_Mat, basic_idx = Mat.rref()
# Idx of basic and free varialbe
basic_idx = list(basic_idx)
basic_idx.sort()
free_idx = [idx for idx in range(n) if idx not in basic_idx]
free_idx.sort()
n_free = len(free_idx)
# String to print free vars
free_str = ""
for i in range(n):
if i in free_idx:
free_str += "x_" + str(i + 1) + " \ "
# Display echelon matrix
if disp:
display(Latex("On échelonne la matrice du système $A -\lambda I = 0 \Rightarrow "
+ al.texMatrix(np.array(Mat), np.reshape(b, (n, 1))) + "$"))
display(Latex("On obtient: $" + al.texMatrix(np.array(e_Mat[:, :n]), np.reshape(b, (n, 1))) + "$"))
display(Latex("Variable(s) libre(s): $" + free_str + "$"))
# Build a list of n_free basis vector:
# first dim: which eigenvector (size of n_free)
# second dim: which element of the eigenvector (size of n)
basis = np.zeros((n_free, n))
for i in range(n_free):
basis[i, free_idx[i]] = 1.0
for idx, j in enumerate(free_idx):
for i in basic_idx:
basis[idx, i] = - float(e_Mat[i, j])
# Show calculated basis
basis_str = ""
for idx, i in enumerate(free_idx):
basis_str += "x_" + str(i + 1) + " \cdot" + texVector(basis[idx])
if idx < n_free - 1:
basis_str += " + "
if disp:
display(Latex("On peut donc exprimer la base de l'espace propre comme: $" + basis_str + "$"))
if prop_basis is not None and disp:
correct_answer = check_basis(basis, prop_basis)
if correct_answer:
display(Latex("La base donnée est correcte car on peut retrouver la base calculée ci-dessus" \
" avec une combinaison linéaire de la base donnée. "
"Aussi les deux bases ont bien le même nombre de vecteurs."))
else:
display(Latex("La base donnée est incorrecte."))
if return_:
return basis, basic_idx, free_idx
def generate_eigen_vector(basis, l, limit):
"""
Function to generate a random eigenvector associated to a eigenvalue given a basis of the eigenspace
The returned eigenvector is such that itself and its multiplication with the matrix will stay in range of limit
in order to have a nice plot
@param basis: basis of eigenspace associated to eigenvalue lambda
@param l: eigenvalue
@param limit: limit of the plot: norm that the engenvector or its multiplication with the matrix will not exceed
@return: eigen vector (numpy array)
"""
n = len(basis)
basis_mat = np.array(basis).T
basis_mat = basis_mat.astype(np.float64)
coeff = 2 * np.random.rand(n) - 1
vect = basis_mat @ coeff
if abs(l) <= 1:
vect = vect / np.linalg.norm(vect) * (limit - 1)
else:
vect = vect / np.linalg.norm(vect) * (limit - 1) / l
return vect
def plot3x3_eigspace(A, xL=-10, xR=10, p=None, plot_vector=False):
# To have integer numbers
if p is None:
p = xR - xL + 1
n = A.shape[0]
# Check 3 by 3
if n != 3 or n != A.shape[1]:
raise ValueError("A should be 3 by 3")
w = A.eigenvals()
w = list(w.keys())
# Deal with complex number (removal)
complex_to_rm = []
for idx, el in enumerate(w):
if not el.is_real:
complex_to_rm.append(idx)
for index in sorted(complex_to_rm, reverse=True):
del w[index]
display("Des valeurs propres sont complexes, on les ignore.")
if len(w)==0:
display("Toute les valeurs propres sont complexes.")
return
gr = 'rgb(102,255,102)'
org = 'rgb(255,117,26)'
# red = 'rgb(255,0,0)'
blue = 'rgb(51, 214, 255)'
colors = [blue, gr, org]
s = np.linspace(xL, xR, p)
t = np.linspace(xL, xR, p)
tGrid, sGrid = np.meshgrid(s, t)
data = []
A_np = np.array(A).astype(np.float64)
for i, l in enumerate(w):
l_eval = float(l)
basis, basic_idx, free_idx = eigen_basis(A, l_eval, disp=False, return_=True)
n_free = len(basis)
if n_free != len(free_idx):
raise ValueError("len(basis) and len(free_idx) should be equal.")
gr = 'rgb(102,255,102)'
colorscale = [[0.0, colors[i]],
[0.1, colors[i]],
[0.2, colors[i]],
[0.3, colors[i]],
[0.4, colors[i]],
[0.5, colors[i]],
[0.6, colors[i]],
[0.7, colors[i]],
[0.8, colors[i]],
[0.9, colors[i]],
[1.0, colors[i]]]
X = [None] * 3
if n_free == 2:
X[free_idx[0]] = tGrid
X[free_idx[1]] = sGrid
X[basic_idx[0]] = tGrid * basis[0][basic_idx[0]] + sGrid * basis[1][basic_idx[0]]
plot_obj = go.Surface(x=X[0], y=X[1], z=X[2],
showscale=False, showlegend=True, colorscale=colorscale, opacity=1,
name="$ \lambda= " + sp.latex(l) + "$")
elif n_free == 1:
plot_obj = go.Scatter3d(x=t * basis[0][0], y=t * basis[0][1], z=t * basis[0][2],
line=dict(colorscale=colorscale, width=4),
mode='lines',
name="$\lambda = " + sp.latex(l) + "$")
elif n_free == 3:
display(Latex("La dimension de l'espace propre de l'unique valeur propre est 3: tous les vecteurs" \
"$v \in \mathbb{R}^3 $ appartiennent à l'espace propre de la matrice $A$." \
"On ne peut donc pas reprensenter sous la forme d'un plan ou d'une droite."))
return
else:
print("error")
return
data.append(plot_obj)
if (plot_vector):
v1 = generate_eigen_vector(basis, l_eval, xR)
v2 = A_np @ v1
data.append(go.Scatter3d(x=[0, v1[0]], y=[0, v1[1]], z=[0, v1[2]],
line=dict(width=6),
marker=dict(size=4),
mode='lines+markers',
name='$v_{' + sp.latex(l) + '}$'))
data.append(go.Scatter3d(x=[0, v2[0]], y=[0, v2[1]], z=[0, v2[2]],
line=dict(width=6, dash='dash'),
marker=dict(size=4),
mode='lines+markers',
name="$A \ v_{" + sp.latex(l) + "}$"))
layout = go.Layout(
showlegend=True, # not there WHY???? --> LEGEND NOT YET IMPLEMENTED FOR SURFACE OBJECTS!!
legend=dict(orientation="h"),
autosize=True,
width=800,
height=800,
scene=go.layout.Scene(
xaxis=dict(
gridcolor='rgb(255, 255, 255)',
zerolinecolor='rgb(255, 255, 255)',
showbackground=True,
backgroundcolor='rgb(230, 230,230)',
range=[xL, xR]
),
yaxis=dict(
gridcolor='rgb(255, 255, 255)',
zerolinecolor='rgb(255, 255, 255)',
showbackground=True,
backgroundcolor='rgb(230, 230,230)',
range=[xL, xR]
),
zaxis=dict(
gridcolor='rgb(255, 255, 255)',
zerolinecolor='rgb(255, 255, 255)',
showbackground=True,
backgroundcolor='rgb(230, 230,230)',
range=[xL, xR]
),
aspectmode="cube",
)
)
fig = go.Figure(data=data, layout=layout)
plotly.offline.iplot(fig)
return
def plot2x2_eigspace(A, xL = -10, xR = 10, p=None):
if p is None:
p = xR - xL + 1
w = A.eigenvals()
w = list(w.keys())
# Deal with complex number (removal)
complex_to_rm = []
for idx, el in enumerate(w):
if not el.is_real:
complex_to_rm.append(idx)
for index in sorted(complex_to_rm, reverse=True):
del w[index]
display("Une valeur propre est complexe, on l'ignore.")
if len(w) == 0:
display("Toute les valeurs propres sont complexes.")
return
data = []
for i, l in enumerate(w):
l_eval = float(l)
basis, basic_idx, free_idx = eigen_basis(A, l_eval, disp=False, return_=True)
n_free = len(basis)
if n_free != len(free_idx):
raise ValueError("len(basis) and len(free_idx) should be equal.")
if n_free == 2:
display(Latex("Tous les vecteurs du plan appartiennent à l'espace propre de A associé à $\lambda = " \
+ sp.latex(l) + "$. On ne peut donc pas le représenter."))
return
else:
t = np.linspace(xL, xR, p)
trace = go.Scatter(x=t*basis[0][0], y=t*basis[0][1], marker=dict(size=6),
mode='lines+markers', name="$\lambda = " + sp.latex(l) + "$")
data.append(trace)
layout = go.Layout(showlegend=True, autosize=True)
fig = go.Figure(data=data, layout=layout)
plotly.offline.iplot(fig)
return
def plot_eigspace(A, xL=-10, xR=10, p=None):
"""
Plot the eigenspaces associated to all eigenvalues of A
@param A: Sympy matrix of shape (2,2) or (3,3)
@param xL: Left limit of plot
@param xR: Right limit of plot
@param p: Number of points to use
"""
n = A.shape[0]
# Check 3 by 3 or 2 by 2
if (n != 2 and n!=3) or n != A.shape[1]:
raise ValueError("A should be 2 by 2 or 3 by 3.")
if not A.is_Matrix:
raise ValueError("A should be a sympy Matrix.")
if n==2:
plot2x2_eigspace(A, xL, xR, p)
else:
plot3x3_eigspace(A, xL, xR, p)
def latexp(A):
"""
Function to output latex expression of a sympy matrix but with round parenthesis
@param A: sympy matrix
@return: latex string
"""
return sp.latex(A, mat_delim='(', mat_str='matrix')
def ch8_8_ex_1(A, prop_answer):
"""
Check if a matrix is diagonalisable.
@param A: sympy square matrix
@param prop_answer: boolean, answer given by the student
@return:
"""
if not A.is_Matrix:
raise ValueError("A should be a sympy Matrix.")
n = A.shape[0]
if n != A.shape[1]:
raise ValueError('A should be a square matrix.')
eig = A.eigenvects()
dim_geom = 0
for x in eig:
dim_geom += len(x[2])
answer = dim_geom == n
if answer:
display(Latex("Oui la matrice $A = " + latexp(A) + "$ est diagonalisable."))
else:
display(Latex("Non la matrice $A = " + latexp(A) + "$ n'est pas diagonalisable."))
if answer == prop_answer:
display(Latex("Votre réponse est correcte !"))
else:
display(Latex("Votre réponse est incorrecte."))
def isDiagonalizable(A):
"""
Step by step method to determine if a given matrix is diagonalizable. This methods uses always (I think)
the easiest way to determine it (as seen in the MOOC)
@param A: sympy matrix
@return: nothing
"""
if not A.is_Matrix:
raise ValueError("A should be a sympy Matrix.")
n = A.shape[0]
if n != A.shape[1]:
raise ValueError('A should be a square matrix.')
display(Latex("On cherche à déterminer si la matrice $A=" + latexp(A) + "$ de taille $n \\times n$ avec $n = " +
str(n) + "$ est diagonalisable."))
if A.is_lower or A.is_upper:
display(Latex("Les valeurs propres sont simple à trouver, ce sont les éléments diagonaux."))
else:
valeurs_propres(A)
# Check if eigenvalue are all distincts
eig = A.eigenvects()
if len(eig) == n:
display(Latex("On a $n$ valeurs propres distinctes. La matrice est donc diagonalisable."))
return
else:
display(Latex("Les valeurs propres ne sont pas toutes distinctes. On va donc vérifier la multiplicité " +
"géométrique des valeurs propres ayant une multiplicité algébrique supérieur à 1."))
# Some list to have info about eigenvalues with algebraic mult > 1
idx = []
eigenvalues = []
mult_al = []
mult_geo = []
for i in range(len(eig)):
if eig[i][1] > 1:
idx.append(i)
eigenvalues.append(eig[i][0])
mult_al.append(eig[i][1])
mult_geo.append(len(eig[i][2]))
display(Latex("L'ensemble des valeurs propres ayant une multiplicité algébrique supérieur à 1 est " + str(
eigenvalues) + "."))
for i, l in enumerate(eigenvalues):
display(Markdown("**On calcule la multiplicité géométrique pour $\lambda= " + sp.latex(l) +
"$ ayant une multiplicité algébrique de " + str(mult_al[i]) + ".**"))
basis, basic, free = eigen_basis(A, l, prop_basis=None, disp=True, return_=True)
display(Markdown("**La multiplicité géométrique pour $\lambda= " + sp.latex(l) + "$ est de " +
str(len(free)) + ".**"))
if (len(free) < mult_al[i]):
display(Markdown("**La multiplicité géométrique est strictement inférieur à la multiplicité"
+ "algébrique pour cette valeur propre. La matrice n'est donc pas diagonalisable.**"))
return
else:
display(Latex("On a bien multiplicité algébrique = multiplicité géométrique pour cette valeur propre."))
display(Markdown("**Toutes les valeurs propres ont une multiplicité algébrique et géométrique égales." +
" La matrice $A$ est donc bien diagonalisable !**"))
def find_P_D(A, P_user, D_user, step_by_step=True):
"""
:param A: sympy square matrix
:param P_user: sympy square matrix
:param D_user: sympa sqaure matrix
:param step_by_step: Print step by step solution
:return:
"""
if not A.is_Matrix or not P_user.is_Matrix or not D_user.is_Matrix:
raise ValueError("A, P and D should be a sympy Matrix.")
n = A.shape[0]
if n != A.shape[1] or P_user.shape[0] != n or P_user.shape[1] != n or D_user.shape[0] != n or D_user.shape[1] != n:
raise ValueError('A, P and D should be a square matrix of the same size.')
if not D_user.is_diagonal():
raise ValueError("D should be a diagonal matrix.")
if not A.is_diagonalizable():
raise ValueError("A is not diagonalizable.")
if step_by_step:
display(Latex("On cherche à déterminer les matrices $P$ et $D$ telles que $A=" + latexp(A) + "= P D P^{-1}$."))
if A.is_lower or A.is_upper:
display(Latex("Les valeurs propres sont simple à trouver, ce sont les éléments diagonaux."))
else:
valeurs_propres(A)
display(
Latex("Pour chaque valeur propre $\lambda_i$, on cherche $n_i$ vecteurs propres linéairement indépendants"
+ " (avec $n_i = \dim E_{\lambda_i}$). On trouve ces vecteurs on calculant une base pour " +
"chaque espace propre. On peut ensuite utiliser les vecteurs de base comme colonnes pour la "
+ "matrice $P$."))
eig = A.eigenvects()
# Some list to have info about eigenvalues with algebraic mult > 1
idx = []
eigenvalues = []
mult_al = []
mult_geo = []
D = sp.zeros(n)
for i in range(len(eig)):
idx.append(i)
eigenvalues.append(eig[i][0])
mult_al.append(eig[i][1])
mult_geo.append(len(eig[i][2]))
P = []
k = 0
for i, l in enumerate(eigenvalues):
basis, _, _ = eigen_basis(A, l, return_=True, disp=step_by_step, dispA=False)
for j in range(len(basis)):
D[k, k] = l
k += 1
P.append(basis[j])
P = np.transpose(np.array(P))
if np.all(np.mod(P, 1) == 0):
P = P.astype(int)
P = sp.Matrix(P)
display(Latex("En utilisant les vecteurs de base trouvés ci dessus pour les colonnes de la matrice $P$ et en " +
"plaçant les valeurs propres de $A$ correspondantes sur la diagonal de $D$, on obtient " +
"les matrices $P$ et $D$."))
display(Latex("$P = " + latexp(P) + "$, " + "$D = " + latexp(D) + "$"))
P_1_user = P_user ** -1
if ((A - P_user * D_user * P_1_user).norm() < 1e-10):
display(Latex("Votre réponse est correcte, on a bien $A = PDP^{-1}$"))
else:
display(Latex("Votre réponse est incorrecte, $A \\neq PDP^{-1}$"))

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