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There exist many iterative algorithms for computing eigenvalues of large sparse matrices, e.g. Power, Lanczos, and Arnoldi iteration. Take the Lanczos algorithm for instance. It assumes that the matrix is Hermitian, in the standard inner product space. Since it only requires matrix-vector multiplies and inner products as basic operations, is it possible to apply the algorithm to a non-Hermitian matrix which is self-adjoint under a non-standard inner product and still expect the algorithm to "work" in some sense? I'm imagining this in the case of discretizing self-adjoint operators with respect to a non-standard bilinear form (not necessarily positive definite).

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    $\begingroup$ I suspect that performing Lanczos with a different inner product would be equivalent (in some way) to preconditioned Lanczos. $\endgroup$
    – Paul Tupper
    Commented Jul 29, 2011 at 17:41

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Using inner product $(x,My)$ in Lanczos is equivalent to using $M^{-1}$ preconditioner in its preconditioned version. As far as I know, preconditioned version of Lanczos is used for generalized eigen-problems: $A x = \lambda M x$

Preconditioned Lanczos operates with two black-boxes: $Ax$ multiplier and $M^{-1}x$ multiplier.

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