Most of the standard eigenvalue algorithms exploit projection processes in order to extract approximate eigenvectors from a given subspace. The basic idea of a projection method is to extract an approximate eigenvector from a specified low-dimensional subspace. Certain conditions are required in order to be able to extract these approximations. Once these conditions are expressed, a small matrix eigenvalue problem is obtained.
A somewhat trivial but interesting illustration of the use of projection is when the dominant eigenvalue of a matrix is computed by the power method as follows:
for until convergence do:
In the simplest case when the dominant eigenvalue, i.e., the
eigenvalue with largest modulus, is real and simple,
converges to the eigenvector and associated (largest in modulus) eigenvalue
under mild assumptions on the initial vector.
If this eigenvalue
is complex and real arithmetic is used, then the algorithm will fail to
converge. It is possible to work in complex arithmetic by introducing
a complex initial vector, and in this way convergence will be recovered. However,
it is also possible to extract these eigenvectors by working in real
arithmetic. This is based on the observation that two successive
iterates and will tend to belong to the
subspace spanned by the two complex conjugate eigenvectors associated
with the desired eigenvalue. Let and be
denoted by and , respectively. To extract the approximate
eigenvectors, we write them as linear combinations of and
The (complex) approximate pair of conjugate eigenvalues obtained from this problem will now converge under the same conditions as those stated for the real case. The iteration uses real vectors, yet the process is able to extract complex eigenvalues. Clearly, the approximate eigenvectors are complex but they need not be computed in complex arithmetic. This is because the (real) basis is also a suitable basis for the complex plane spanned by and .
The above idea can now be generalized from two dimensions to dimensions. Thus, a projection method consists of approximating the exact eigenvector , by a vector belonging to some subspace referred to as the subspace of approximants or the right subspace. If the subspace has dimension , this gives degrees of freedom and the next step is to impose constraints to be able to extract a unique solution. This is done by imposing the so-called Petrov-Galerkin condition that the residual vector of be orthogonal to some subspace , referred to as the left subspace. Though this may seem artificial, we can distinguish between two broad classes of projection methods. In an orthogonal projection technique the subspace is the same as . In an oblique projection method is different from and can be completely unrelated to it.
The construction of can be done in different ways. The above-suggested
generalization of the power method leads, if one works with all generated
vectors, to the Krylov subspace methods, that is, methods that seek the
solution in the subspace