Cartan matrix and Perron-Frobenius eigenvector

# define Cartan matrix - function of arbitrary number of vectors?

# find eigenvalues using SAGE routine

# find the Perron-Frobenius eigenvalue

def Cartan_matrix(Delta, symbolic=False):
    """
    Return the Cartan matrix for a simple system
    
    INPUT:
    
    - ``Delta`` -- list of vectors or any iterable that can be turned into a vector
    - 

    EXAMPLES::

        sage: Delta = ((1, 0, 0)), ((0, 1, 0)), ((0, 0, 1))
        sage: C = Cartan_matrix(Delta)
        sage: C
        [2 0 0]
        [0 2 0]
        [0 0 2]
    """
    Delta=[vector(v) for v in Delta]

    M = matrix([[2 / w.dot_product(w) * v.dot_product(w) for w in Delta] for v in Delta])
    
    if symbolic:
        M = M.apply_map(expand)
        
    return M

Delta = ((0, 1, 0)), ((-1/4*(1+sqrt(5)), -1/2, 1/4*(1-sqrt(5)))), ((1, 0, 0))

Delta

C = Cartan_matrix(Delta, symbolic=True)
C

E = C.eigenvectors_right()
E

  
A=QQbar(E[1][1][0][2])
A
E

# find PF eigenvector i.e. all coefficients of the same sign
for i in range(3):
    tmp = E[i][1][0]
    if all (x * tmp[0] > 0 for x in tmp):
        print(tmp)
 

# find duals to roots (weights), construct two vectors from two-colouring of the graph and the PF evec just found; orthonormalise and project root system into this plane