def tau_column(tau, k, j):
    """
    Determine the column index for the non-zero elements of the matrix for a
    particular row `k` and the value of `j` from the Dicke space.

    Parameters
    ----------
    tau: str
        The tau function to check for this `k` and `j`.

    k: int
        The row of the matrix M for which the non zero elements have
        to be calculated.

    j: float
        The value of `j` for this row.
    """
    k = k + 1
    mapping = {'tau3': k - (2 * j + 3), 'tau2': k - 1, 'tau4': k + (2 * j -
        1), 'tau5': k - (2 * j + 2), 'tau1': k, 'tau6': k + 2 * j, 'tau7': 
        k - (2 * j + 1), 'tau8': k + 1, 'tau9': k + (2 * j + 1)}
    return int(mapping[tau] - 1)
