We compare three methods for refining estimates of invariant subspaces, due to \textit{F. Chatelin} [Comput. Suppl. 5, 67-74 (1984; Zbl 0555.65023)], \textit{J. Dongarra}, \textit{C. Moler} and \textit{J. Wilkinsons} [SIAM J. Numer. Anal. 20, 23-45 (1983; Zbl 0523.65021)] and \textit{G. Stewart} [SIAM Rev. 15, 727-764 (1973; Zbl 0297.65030)]. Even though these methods all apparently solve different equations, we show by changing variables that they all solve the same equation, the Riccati equation. The benefit of this point of view is threefold. First, the same convergence theory applies to all three methods, yielding a single criterion under which the last two methods converge linearly, and a slightly stronger criterion under which the first algorithm converges quadratically. Second, it suggests a hybrid algorithm combining advantages of all three. Third, it leads to algorithms (and convergence criteria) for the generalized eigenvalue problem. These techniques are compared to techniques used in the control systems community.