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.