D. Steven Mackey

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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Symmetric Linearizations for Matrix Polynomials

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

The Conditioning of Linearizations of Matrix Polynomials

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Fiedler Companion Linearizations and the Recovery of Minimal Indices

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

Linearizations of singular matrix polynomials and the recovery of minimal indices

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STRUCTURED TOOLS FOR STRUCTURED MATRICES

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

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

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Smith forms of palindromic matrix polynomials

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

Definite Matrix Polynomials and their Linearization by Definite Pencils

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