A convenient way to provide such a spectral transformation is to note that

ThusA moments reflection will reveal the advantage of such a spectral transformation. Eigenvalues that are near will be transformed to eigenvalues that are at the extremes and typically well separated from the rest of the transformed spectrum. The corresponding eigenvectors remain unchanged. Perhaps more important is the fact that eigenvalues far from the shift are mapped into a tight cluster in the interior of the transformed spectrum. We illustrate this by showing the transformed spectrum of the matrix from Figure 4.8 with a shift (here ). Again, we show the total filter polynomial that was constructed during an IRA iteration on the transformed matrix . Here we compute the six eigenvalues of largest magnitude. These will transform back to eigenvalues of nearest to through the formula . The surface shown in Figure 4.9 is again but plotted over a region containing the spectrum of . Here, is the product of all of the filter polynomials constructed during the course of the iteration. Since the extrem eigenvalues are well separated the iteration converges much faster and degree of is only 45 in this case. Here, the ``+" signs are the eigenvalues of in the complex plane and the contours are the level curves of . The circled plus signs are the converged eigenvalues. The figure illustrates how much easier it is to isolate desired eigenvalues after a spectral transformation.

If is symmetric then one can maintain symmetry in the Arnoldi/Lanczos process by taking the inner product to be

It is easy to verify that the operator is symmetric with respect to this inner product if is symmetric. In the Arnoldi/Lanczos process the matrix-vector product is replaced by and the step is replaced by .If is symmetric then the matrix is symmetric and tridiagonal. Moreover, this process is well defined even when is singular and this can have important consequences even if is non-symmetric. We shall refer to this process as the -Arnoldi process.
If is singular
then the operator has a non-trivial null space and the bilinear
function is a semi-inner product and
is a semi-norm. Since is assumed to be
nonsingular,
Vectors in are generalized eigenvectors corresponding to *infinite* eigenvalues.
Typically, one is only interested in the finite eigenvalues
of () and these will correspond to the non-zero eigenvalues of *S*.
The invariant subspace corresponding to these non-zero eigenvalues
is easily corrupted by components of vectors from during the Arnoldi process. However, using the M-Arnoldi process
with some refinements can provide a solution.

In order to better understand the situation, it is convenient to note that since is positive semi-definite, there is an orthogonal matrix such that

where is a positive definite diagonal matrix of order