Compute the qr factorization of a matrix.
    Factor the matrix `a` as `qr`, where `q` is orthonormal
    (:math:`dot( q_{:,i}, q_{:,j}) = \delta_{ij}`, the Kronecker delta) and
    `r` is upper-triangular.

Arrggghhhh... Totally. Unreadable. Why not say the columns of q are orthonormal vectors and r is upper triangular? The math tutorial should go in the notes, if anywhere. Might mention that this is a 'thin' factorization (Golub). Let me propose a rule: no math markup in the summary, ever.

    a : array_like, shape (M, N)
        Matrix to be factored.
    mode : {'full', 'r', 'economic'}
        Specifies the information to be returned. 'full' is the default.
        mode='r' returns a "true" `r`, while 'economic' returns a "polluted"
        `r` (albeit slightly faster; see Returns below).

Oh, come now, "true", "polluted"? Sounds a bit political... Actually, 'economic' contains info on the Householder reflections. In any case, why mention it at all, just refer to the return documentation. And wouldn't values be a better word than information?
    * If mode = 'full':
        * q : ndarray of float or complex, shape (M, K)
        * r : ndarray of float or complex, shape (K, N)
      Size K = min(M, N)
    * If mode = 'r':
      * r : ndarray of float or complex, shape (K, N)
    * If mode = 'economic':
      * a2 : ndarray of float or complex, shape (M, N)
      The diagonal and the upper triangle of a2 contains r,
      while the rest of the matrix is undefined.

WTF? I'm seeing stars.

I may be old and crotchety, and I don't mean to be mean to the folks who have done the hard work to bring the docstring to its current state, but I think things have gotten out of hand.