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H. Lin proved that "almost commuting" hermitian matrices are "nearly commuting." To be more precise, Lin showed that given $\epsilon > 0$ there exists a $\delta > 0$ such that if $A, B \in M_N$ are self-adjoint, with $|| AB - BA || < \delta$, and $\|A\|, \|B\|\le 1$, then there exists $X, Y \in M_N$ with $XY = YX$ such that $ || A - X || + || B - Y || < \epsilon$. Here, $|| . ||$ is the usual operator norm and $\delta = \delta(\epsilon)$ does not depend on the dimension N.

Does anyone know of any applications of this theorem?

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It is a little surprising that this is a named theorem. Is the corresponding result without hermitian assumption false? –  Igor Rivin Mar 3 '12 at 14:25
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Yes, this is false even if one matrix is Hermitian and the other is a general matrix. A recent summary of this sort of result is "Local operator theory, random matrices and Banach spaces" by Davidson and Szarek in the "Handbook of the geometry of Banach spaces." <p> I will edit the original question in a minute, as it is missing the needed hypothesis that $A$ and $B$ are contractions. –  Terry Loring Mar 3 '12 at 17:02
    
Just as an aside to Igor Rivin's comment: see some of the remarks and references in "Almost commuting unitary matrices", R. Exel, T. Loring, Proc. Amer. Math. Soc. 106 (1989), no. 4, 913--915 –  Yemon Choi Mar 7 '12 at 2:51
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1 Answer

up vote 14 down vote accepted

Lin's theorem shows the existence of a localized basis for the low-energy space in models of non-interacting fermions on a finite lattice on a disk. This was observed by Matt Hastings.

See "Topology and phases in fermionic systems" in the Journal of Statistical Mechanics: Theory and Experiment, 2008, L01001, especially the next to last paragraph.

Other norms come up in applications. It can be hard to figure what norm is relevant in engineering and science papers, but it is often clearly not the operator norm. Generically speaking, approximating two commuting hermitian matrices by commuting hermitian matrices is equivalent to joint approximate diagonalization of those matrices.

A famous algorithm in blind source separation if by Cardoso and Souloumiac, "Jacobi angles for simultaneous diagonalization" in SIAM Journal on Matrix Analysis and Applications, vol 17, no. 1, 161--164, 1996.

The Cardoso and Souloumiac algorithm was used later in computational quantum chemistry, as in Francois Gygi, Jean-Luc Fattebert and Eric Schwegler, "Computation of Maximally Localized Wannier Functions using a simultaneous diagonalization algorithm," Computer Physics Communications, Volume 155, Issue 1, 1 September-15 September 2003, 1--6.

Jon von Neumann considered almost commuting operators. See "Proof of the ergodic theorem and the H-theorem in quantum mechanics," The European Physical Journal, 1--37, 2010 and the commentary that accompanied the translation, "Long-time behavior of macroscopic quantum systems, by Goldstein, S. and Lebowitz, J.L. and Tumulka, R. and Zanghi, N., The European Physical Journal H, 1--28, 2010.

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Your first sentence is quite intriguing :) –  Mariano Suárez-Alvarez Mar 3 '12 at 2:35
    
Thank you Prof. Loring for your response to my post. L. Glebsky proved a similar theorem with the operator norm replaced by the Frobenius norm. Does Glebsky's result have any applications that you know of? –  Mustafa Said Mar 7 '12 at 7:32
    
I added some more references to my answer. –  Terry Loring Mar 7 '12 at 16:12
    
Regarding the question with the normalized Frobenius norm, much more is true, and this is relevant in von Neumann algebra theory (and the result was known before Glebsky's work). See my question mathoverflow.net/questions/68367 and especially the answer by Andreas Thom. –  Mikael de la Salle Mar 7 '12 at 16:33
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