Krylov methods are typically used for transforming large scale prolems and matrices to
matrices of a manageable size. Krylov inspired algorithms are based on the principle: project -
solve - lift. In this talk we will focus on the projection step, the orthogonal bases involved, and the structured of the resulting projection.
Two bases are needed for the projection: a basis for a search Krylov subspace, say V, and
another basis for the constraint Krylov subspace, name it W. For a matrix A, the projected
small to medium sized matrix will be W^*AV. There is now a wide variety of possibilities
for constructing the matrices W and V. We could go from classical Krylov to extended and
rational Krylov subspaces. But we could also attempt to construct subspaces with variants
of the matrix A, such as for instance its inverse, its conjugate transpose, or the inverse
of its conjugate transpose.
In this lecture we will examine some of these possibilities. A general framework is
proposed in which we will touch upon the following aspects: biorthogonal (rational)
Krylov, CMV factorizations, non-unitary CMV factorizations, quasiseparable Hessenberg
matrices, Hankel and Toeplitz Gramians.