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Solve the following problem using Python, implementing the functions described below, one line at a time
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Description:
 def fingerprint_relaxation(T, p0, obs, tau=1, k=None, ncv=None):
 r"""Dynamical fingerprint for relaxation experiment. The dynamical fingerprint is given by the implied time-scale spectrum together with the corresponding amplitudes. Parameters T : (M, M) ndarray or scipy.sparse matrix Transition matrix obs1 : (M,) ndarray Observable, represented as vector on state space obs2 : (M,) ndarray (optional) Second observable, for cross-correlations k : int (optional) Number of time-scales and amplitudes to compute tau : int (optional) Lag time of given transition matrix, for correct time-scales ncv : int (optional) The number of Lanczos vectors generated, `ncv` must be greater than k; it is recommended that ncv > 2*k Returns ------- timescales : (N,) ndarray Time-scales of the transition matrix amplitudes : (N,) ndarray Amplitudes for the relaxation experiment See also -------- relaxation, fingerprint_correlation References .. [1] Noe, F, S Doose, I Daidone, M Loellmann, M Sauer, J D Chodera and J Smith. 2010. Dynamical fingerprints for probing individual relaxation processes in biomolecular dynamics with simulations and kinetic experiments. PNAS 108 (12):
 4822-4827. Notes ----- Fingerprints are a combination of time-scale and amplitude spectrum for a equilibrium correlation or a non-equilibrium relaxation experiment. **Relaxation** A relaxation experiment looks at the time dependent expectation value of an observable for a system out of equilibrium .. math:: \mathbb{E}_{w_{0}}[a(x, t)]=\sum_x w_0(x) a(x, t)=\sum_x w_0(x) \sum_y p^t(x, y) a(y). The fingerprint amplitudes :math:`\gamma_i` are given by .. math:: \gamma_i=\langle w_0, r_i\rangle \langle l_i, a \rangle. And the fingerprint time scales :math:`t_i` are given by .. math:: t_i=-\frac{\tau}{\log \lvert \lambda_i \rvert}. Examples -------- array([ inf, 9.49122158, 0.43429448]) array([ 0.45454545, 0.5 , 0.04545455]) """

    # check if square matrix and remember size
    T = _types.ensure_ndarray_or_sparse(T, ndim=2, uniform=True, kind='numeric')
    n = T.shape[0]
    # will not do fingerprint analysis for nonreversible matrices
    if not is_reversible(T):
        raise ValueError('Fingerprint calculation is not supported for nonreversible transition matrices. ')
    p0 = _types.ensure_ndarray(p0, ndim=1, size=n, kind='numeric')
    obs = _types.ensure_ndarray(obs, ndim=1, size=n, kind='numeric')
    # go
    if _issparse(T):
        return sparse.fingerprints.fingerprint_relaxation(T, p0, obs, tau=tau, k=k, ncv=ncv)
    else:
        return dense.fingerprints.fingerprint_relaxation(T, p0, obs, tau=tau, k=k)