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Eigensystem[] for higher dimensions?

Can the matrix m in Eigensystem[m] be greater than two dimensional?

And the eigenvectors v correspondingly greater than one dimensional?

The problem of interest is finding the eigenvalues lambda and 
eigenarrays A of the equation

   M A = lambda A

with A being a (physically nonseparable) two dimensional array and M a 
specified four dimensional array.  I can of course use appropriate 
indexing to convert these to one and two dimensional quantities, but the 
index transformations can be a pain; the reverse index transformations 
even more ugly; the Mathematica Book doesn't seem to explicitly answer 
the question -- and it's easier to ask it here, and maybe also get some 
useful pointers, than futz around with experiments.

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