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It is frequently important to determine the coherence of a matrix by finding the maximum pairwise correlation between all its column vectors. Similarly, when working with a union of subspaces of a Hilbert space it would be convenient to have a measure of similarity between two different subspaces. Does such a measure exist? And how is it computed? Assume I have an orthonormal basis for each subspace, but the subspaces are not necessarily the same dimension.

Importantly, I do not want this to be a supremum measure; i.e., if two subspaces share a single basis vector, they should not necessarily have a correlation of 1.

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    $\begingroup$ How about $\| P_V - P_W \|$ where $P_V$ is the projection onto $V$? This is more of a distance and less of a similarity, but at least it's a reasonably canonical choice. $\endgroup$
    – Qiaochu Yuan
    Commented Nov 18, 2013 at 5:08
  • $\begingroup$ I'm a little confused about that statement. Is $P_V$ a projection matrix? Is it an operator? How do you compute $P_V - P_W$? $\endgroup$
    – rlbond
    Commented Nov 18, 2013 at 14:50
  • $\begingroup$ Also, what norm are you using? $\endgroup$
    – rlbond
    Commented Nov 18, 2013 at 15:19
  • $\begingroup$ $P_V$ is a projection operator (you're working in a Hilbert space, right?) and the norm is the operator norm. $\endgroup$
    – Qiaochu Yuan
    Commented Nov 18, 2013 at 18:10

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