Christian Mehl

Vector Spaces of Linearizations for Matrix Polynomials

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 are not always entirely satisfactory, and linearizations with special properties may sometimes be required. Given a matrix polynomial

Structured Polynomial Eigenvalue Problems: Good Vibrations from Good Linearizations

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Numerical methods for palindromic eigenvalue problems: Computing the anti‐triangular Schur form

, we develop a systematic approach to generating large classes of linearizations for

Smith forms of palindromic matrix polynomials

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Condensed Forms for Skew-Hamiltonian/Hamiltonian Pencils

. We show how to simply construct two vector spaces of pencils that generalize the companion forms of

On classification of normal matrices in indefinite inner product spaces.

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Schur-like forms for matrix Lie groups, Lie algebras and Jordan algebras

, and prove that almost all of these pencils are linearizations for

Canonical forms for doubly structured matrices and pencils.

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Symmetric matrices with respect to sesquilinear forms

. Eigenvectors of these pencils are shown to be closely related to those of

The Canonical Generalized Polar Decomposition

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