I am working on a project where I need to compute the eigenvalues of a symmetric real matrix **A** in Python. I encountered an issue where the eigenvalues returned by **eigvalsh**, **eigvals**, and **eigh** are different.

My code:

    import numpy as np

    def convert_Matrix(Var, dim):
        var_i = np.zeros((dim, dim))
        indices = np.tril_indices(dim)
        var_i[indices] = Var
        var = np.dot(var_i, var_i.T)
    return var

    Var = np.array([4.04640469e+05, 9.98970587e+07, 1.75522655e+02, 1.85808145e+08,
                2.88277248e+08, 7.50545040e+05])
    dim = 3

    A= convert_Matrix(Var, dim)

    det_A = np.linalg.det(A)
    eigvalsh_values = np.linalg.eigvalsh(A)
    eigvals_values = np.linalg.eigvals(A)
    eigh_values = np.linalg.eigh(A)[0]  # eigh returns (eigenvalues, eigenvectors), so we take the first element

    print("A:n", A)
    print("Eigenvalues from eigvalsh:", eigvalsh_values)
    print("Eigenvalues from eigvals:", eigvals_values)
    print("Eigenvalues from eigh:", eigh_values)
    print('det:', det_A)`

The output:

A:[[1.63733909e+11 4.04223927e+13 7.51854949e+13]
   [4.04223927e+13 9.97942234e+15 1.85617378e+16]
   [7.51854949e+13 1.85617378e+16 1.17629002e+17]]

Eigenvalues from eigvalsh:[2.43186146e-04 6.86886614e+15 1.20739722e+17]
Eigenvalues from eigvals: [1.20739722e+17 -5.60191504e-01  6.86886614e+15]
Eigenvalues from eigh:    [4.41546949e-04 6.86886614e+15 1.20739722e+17]

det: -1.4510608257526929e+28

The output eigenvalues are not consistent across these methods, and the determinant does not match the product of the eigenvalues. I would like to understand why these discrepancies occur and how I can ensure consistent results when working with symmetric real matrices.