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Many applications give rise to nonlinear eigenvalue problems with an underlying structured matrix polynomial. In this paper several useful classes of structured polynomial (e.g., palindromic, even, odd) are identified and the relationships between them explored. A special class of linearizations that reflect the structure of these polynomials, and therefore(More)
The classical approach to investigating polynomial eigenvalue problems is linearization, where the polynomial is converted into a larger matrix pencil with the same eigenvalues. For any polynomial there are infinitely many linearizations with widely varying properties, but in practice the companion forms are typically used. However, these companion forms(More)
We present structure-preserving numerical methods for the eigenvalue problem of complex palindromic pencils. Such problems arise in control theory, as well as from palindromic linearizations of higher degree palindromic matrix polynomials. A key ingredient of these methods is the development of an appropriate condensed form — the anti-triangular Schur form.(More)
In this paper we consider real or complex skew-Hamiltonian/Hamiltonian pencils λS − H, i.e., pencils where S is a skew-Hamiltonian and H is a Hamiltonian matrix. These pencils occur for example in the theory of continuous time, linear quadratic optimal control problems. We reduce these pencils to canonical and Schur-type forms under structure-preserving(More)
The asymptotic convergence behavior of cyclic versions of the nonsymmetric Jacobi algorithm for the computation of the Schur form of a general complex matrix is investigated. Similar to the symmetric case, the nonsymmetric Jacobi algorithm proceeds by applying a sequence of rotations that annihilate a pivot element in the strict lower triangular part of the(More)
The polar decomposition of a square matrix has been generalized by several authors to scalar products on R n or C n given by a bilinear or sesquilinear form. Previous work has focused mainly on the case of square matrices, sometimes with the assumption of a Hermitian scalar product. We introduce the canonical generalized polar decomposition A = W S, defined(More)
We derive a formula for the backward error of a complex number λ when considered as an approximate eigenvalue of a Hermitian matrix pencil or polynomial with respect to Hermit-ian perturbations. The same are also obtained for approximate eigenvalues of matrix pencils and polynomials with related structures like skew-Hermitian, *-even and *-odd. Numerical(More)
The classical singular value decomposition for a matrix A ∈ C m×n is a canonical form for A that also displays the eigenvalues of the Hermitian matrices AA * and A * A. In this paper, we develop a corresponding decomposition for A that provides the Jordan canonical forms for the complex symmetric matrices AA T and A T A. More generally, we consider the(More)
We derive formulas for the backward error of an approximate eigenvalue of a *-palindromic matrix polynomial with respect to *-palindromic perturbations. Such formulas are also obtained for complex T-palindromic pencils and quadratic polynomials. When the T-palindromic polynomial is real, then we derive the backward error of a real number considered as an(More)