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There does not appear to be any documentation on how to use Method in Eigensystem, or at least not in Version 9.0.1 of Mathematica (unless I'm looking in the wrong place). The documentation page for Eigensystem just gives a single example of using an Arnoldi shift as

Method -> {"Arnoldi", Shift -> -4}

and the internal implementation page at tutorial/SomeNotesOnInternalImplementation#5107 just says that Eigensystem uses an ARPACK Arnoldi solver for iterative computing of the first $k$ eigenvectors.

The documentation page on Method likewise makes no mention of what options are available for use in eigendecomposition problems. Is there any documentation on what possible syntax options are available for Method in the context of Eigensystem?

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One of my big complaints about the Mathematica documentation is that the Method option of various functions is typically documented only in a very limited way. Enough information to make me interested, but not enough information to make it usable. However, I have good news for you in this case: the v10 documentation has added a nice description of the possible method options of Eigenvalues. You'll have to try if these work in v9. –  Szabolcs Feb 22 at 16:05
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The as of 2014 February work-in-progress v10 documentation has added detailed information on the possible Method options for Eigenvalues. See it here, under Details and Options, and also under Options.

I'm not sure how much of this applies to v9 as well because I haven't actually tried using all these options.

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A bit surprising that Wolfram Research has posted such intimate details of the as-yet unreleased Mathematica 10. I believe pre-release testers of Wolfram products have to sign confidentiality agreements, and if somebody violates it, don't they have to kill him? –  murray Feb 22 at 17:11
    
@murray if you query the version number on a raspberry it will tell you already v10. –  Stefan Feb 22 at 19:14
    
@murray The documentation is online and public because a v10 beta has already been released for the Raspberry Pi computer and it's freely available (!). I haven't posted any information that wasn't already public. –  Szabolcs Feb 22 at 23:02
    
@Szabolcs: Aha! I forgot about the Raspberry Pi beta. –  murray Feb 23 at 16:44
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