For a given nonderogatory matrix A, formulas are given for functions of A in terms of Krylov matrices of A. Relations between the coefficients of a polynomial of A and the generating vector of a Krylov matrix of A are provided. With the formulas, linear transformations between Krylov matrices and functions of A are introduced, and associated algebraic properties are derived. Hessenberg reductio...

The rational Krylov sequence (RKS) method can be seen as a generalisation of Arnoldi’s method. It projects a matrix pencil onto a smaller subspace; this projection results in a small upper Hessenberg pencil. As for the Arnoldi method, RKS can be restarted implicitly, using the QR decomposition of a Hessenberg matrix. This restart comes with a projection of the subspace using a rational function...

We study a family of subvarieties of the flag variety defined by certain linear conditions, called Hessenberg varieties. We compare them to Schubert varieties. We prove that some Schubert varieties can be realized as Hessenberg varieties and vice versa. Our proof explicitly identifies these Schubert varieties by their permutation and computes their dimension. We use this to answer an open quest...

The Rational Krylov algorithm computes eigenvalues and eigenvectors of a regular not necessarily symmetric matrix pencil. It is a generalization of the shifted and inverted Arnoldi algorithm, where several factorizations with di erent shifts are used in one run. It computes an orthogonal basis and a small Hessenberg pencil. The eigensolution of the Hessenberg pencil approximates the solution of...

A class ξ of algebras of symmetric n × n matrices, related to Toeplitz-plus-Hankel structures and including the well-known algebra H diagonalized by the Hartley transform, is investigated. The algebras of ξ are then exploited in a general displacement decomposition of an arbitrary n× n matrix A. Any algebra of ξ is a 1-space, i.e., it is spanned by n matrices having as first rows the vectors of...

Some variants of the (block) Gauss–Seidel iteration for solution linear systems with M-matrices in Hessenberg form are discussed. Comparison results asymptotic convergence rate some regular splittings derived: particular, we prove that a lower-Hessenberg M-matrix $$\rho (P_{GS})\ge \rho (P_S)\ge (P_{AGS})$$ , where $$P_{GS}, P_S, P_{AGS}$$ matrices Gauss–Seidel, staircase, and anti-Gauss–Seidel...

We modify the customary approach to solving the algebraic eigenproblem. Instead of applying the QR algorithm to a Hessenberg matrix, we begin with the recent unitary similarity transform into a triangular plus rank-one matrix. Our novelty is nonunitary transforms of this matrix into similar arrow-head matrices, which we perform at a low arithmetic cost. The resulting eigenproblem can be effecti...

In this paper an implicit (double) shifted QR-method for computing the eigenvalues of companion and fellow matrices will be presented. Companion and fellow matrices are Hessenberg matrices, that can be decomposed into the sum of a unitary and a rank 1 matrix. The Hessenberg, the unitary as well as the rank 1 structures are preserved under a step of the QR-method. This makes these matrices suita...

Let H ∈ Cn×n be an n × n unitary upper Hessenberg matrix whose subdiagonal elements are all positive. Partition H as H = H11 H12 H21 H22 , (0.1) where H11 is its k×k leading principal submatrix; H22 is the complementary matrix of H11. In this paper, H is constructed uniquely when its eigenvalues and the eigenvalues of b H11 and b H22 are known. Here b H11 and b H22 are rank-one modifications of...

The objective of this paper is to extend and redesign the block matrix reduction applied for the family of two-sided factorizations, introduced by Dongarra et al. [9], to the context of multicore architectures using algorithms-by-tiles. In particular, the Block Hessenberg Reduction is very often used as a pre-processing step in solving dense linear algebra problems, such as the standard eigenva...