def get_normalised_generator(frequencies, rate_matrix=None):
    """
    Calculates the normalised generator from the rate matrix and character state frequencies.

    :param frequencies: character state frequencies.
    :type frequencies: numpy.array
    :param rate_matrix: (optional) rate matrix (by default an all-equal-rate matrix is used)
    :type rate_matrix: numpy.ndarray
    :return: normalised generator 1/mu Q
    :rtype: numpy.ndarray
    """
    if rate_matrix is None:
        n = len(frequencies)
        rate_matrix = np.ones(shape=(n, n), dtype=np.float64) - np.eye(n)
    generator = rate_matrix * frequencies
    generator -= np.diag(generator.sum(axis=1))
    mu = -generator.diagonal().dot(frequencies)
    generator /= mu
    return generator