Let A be a $n\times n$ matrix

1. characteristic polynomial: calculate $det(A-\lambda I)=0$
2. eigenvalues: roots of characteristic polynomial
   • algebraische Multiplizität (AM): Vielfachheit von $\lambda$ als Nulstelle
3. eigenvectors: solve $A-\lambda I$ for $\lambda =$eigenvalue with e.g. gauss-elimination
   • geometrische Multipliztät (GM): Nummer von freien variablen in Eigenvektor
4. diagonalization:
   • $A=A^{\mathrm{\top }}⟹$ (Spektralsatz) es gibt ONB aus eigenvectoren $⟹$
     diagonalisierbar
   • h

from sympy import symbols, Matrix

# define the matrix
matrix = Matrix([
[3,-1],[0,2]
])

# Define the characteristic polynomial: det(A - x*I)
x = symbols('x')
char_poly = (matrix - x * Matrix.eye(2)).det()
# calculate eigenvals
eigenvalues = matrix.eigenvals()
# calculate eigenvalues, eigenvectors
vects = matrix.eigenvects()

print("Characteristic Polynomial:")
print(char_poly)
print("Eigenvalues")
print(eigenvalues)
print("Eigenvectors")
for i in vects:
        print(f"- {i[0]} ({i[1]})")
        print(*i[2])