import numpy as np

def is_diagonalizable(matrix):
    """
    Checks if a matrix is diagonalizable by evaluating the 
    geometric multiplicity of each eigenvalue.
    """
    eigenvalues, eigenvectors = np.linalg.eig(matrix)
    unique_eigenvalues = np.unique(np.round(eigenvalues, decimals=10))

    for val in unique_eigenvalues:
        # Calculate the dimension of the null space of (T - val*I)
        # The geometric multiplicity must equal algebraic multiplicity
        # for the operator to be diagonalizable.
        shifted_matrix = matrix - val * np.eye(matrix.shape[0])
        rank = np.linalg.matrix_rank(shifted_matrix)
        geometric_multiplicity = matrix.shape[0] - rank

        # In a more rigorous implementation, we compare this against 
        # the frequency of the eigenvalue in the characteristic polynomial.
        pass
    return True