## Matrices that Generate the Same Krylov Residual Spaces (1994)

Venue: | in Recent Advances in Iterative Methods |

Citations: | 24 - 2 self |

### BibTeX

@INPROCEEDINGS{Greenbaum94matricesthat,

author = {Anne Greenbaum and Zdenek Strakos},

title = {Matrices that Generate the Same Krylov Residual Spaces},

booktitle = {in Recent Advances in Iterative Methods},

year = {1994},

pages = {95--118},

publisher = {Springer}

}

### Years of Citing Articles

### OpenURL

### Abstract

Given an n by n nonsingular matrix A and an n-vector v, we consider the spaces of the form AK k (A; v), k = 1; :::; n, where K k (A; v) is the k th Krylov space, equal to spanfv; Av; :::; A k\Gamma1 vg. We characterize the set of matrices B that, with the given vector v, generate the same spaces; i.e., those matrices B for which BK k (B; v) = AK k (A; v), for all k = 1; :::; n. It is shown that any such sequence of spaces can be generated by a unitary matrix. If zero is outside the field of values of A, then there is a Hermitian positive definite matrix that generates the same spaces, and, moreover, if A is close to Hermitian then there is a nearby Hermitian matrix that generates the same spaces. It is also shown that any such sequence of spaces can be generated by a matrix having any desired eigenvalues. Implications about the convergence rate of the GMRES method are discussed. A new proof is given that if zero is outside the field of values of A, then convergence of...

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