Timeline for Approximate action of unitary matrix with permutation matrix
Current License: CC BY-SA 3.0
7 events
| when toggle format | what | by | license | comment | |
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| Nov 24, 2011 at 22:24 | comment | added | Suvrit | How large can $n$ be? it seems like some kind of graph-matching might resolve it, maybe not! | |
| Nov 24, 2011 at 1:28 | comment | added | Charles Bailey | I apologize for not being clear in my edit. P1 is a minimizer of $||Q - P||$ and P2 is a minimizer of $||QBQ^{T} - PBP^{T}||$. I say "a minimizer" because there are multiple minimizers, but choosing a different representative does not necessarily make P1 behave as desired. | |
| Nov 23, 2011 at 22:35 | comment | added | John Jiang | Your counterexample only shows that the map from $SU(n)$ to the conjugacy class doesn't preserve monotonicity of distance. It could still be true that the minimum in the conjugacy orbit is always attained by the minimizer in $SU(n)$, unless you can demonstrate that $P_2$ is the minimizer in your example. | |
| Nov 23, 2011 at 20:36 | comment | added | Charles Bailey | I have included a counterexample in the edit to my question. Thanks for your help. | |
| Nov 23, 2011 at 20:02 | history | edited | Charles Bailey | CC BY-SA 3.0 |
Added a counterexample, per responders request
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| Nov 23, 2011 at 19:20 | comment | added | John Jiang | Why doesn't the straightforward method work? Did you find a counterexample? If so could you show us? I think the minimum distance is bounded by a constant independent of the dimension of the matrix, but would depend on the spectral norm of $B$. | |
| Nov 23, 2011 at 19:13 | history | asked | Charles Bailey | CC BY-SA 3.0 |