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Vector Spaces of Linearizations for Matrix Polynomials
TLDR
This work develops a systematic approach to generating large classes of linearizations for a matrix polynomial, and shows how to simply construct two vector spaces of pencils that generalize the companion forms of $P, and proves that almost all of these pencils are linearized for $P$.
Structured Polynomial Eigenvalue Problems: Good Vibrations from Good Linearizations
TLDR
This paper analyzes the existence and uniqueness of a special class of linearizations which reflect the structure of structured polynomials, and therefore preserve symmetries in their spectra, and shows how they may be systematically constructed.
Fiedler Companion Linearizations and the Recovery of Minimal Indices
TLDR
It is proved that these pencils are linearizations even when $P(\lambda)$ is a singular square matrix polynomial, and it is shown explicitly how to recover the left and right minimal indices and minimal bases of the polynomials from the minimum indices and bases of these linearizations.
The Conditioning of Linearizations of Matrix Polynomials
TLDR
This work investigates the conditioning of linearizations from a vector space $\mathbb{DL}(P)$ of pencils recently identified and studied and analyzes the eigenvalue conditioning of the widely used first and second companion linearizations.
Symmetric Linearizations for Matrix Polynomials
TLDR
This work provides a self-contained treatment of some of the key properties of $\mathbb{DL}(P)$ together with some new, more concise proofs.
LINEARIZATIONS OF SINGULAR MATRIX POLYNOMIALS AND THE RECOVERY OF MINIMAL INDICES
A standard way of dealing with a regular matrix polynomial P (λ) is to convert it into an equivalent matrix pencil - a process known as linearization. Two vector spaces of pencils L1(P ) and L2(P )
Structured Factorizations in Scalar Product Spaces
TLDR
A key feature of this analysis is the identification of two particular classes of scalar products, termed unitary and orthosymmetric, which serve to unify assumptions for the existence of structured factorizations.
Definite Matrix Polynomials and their Linearization by Definite Pencils
TLDR
For the important special case of quadratics, it is shown how a definite quadratic polynomial can be transformed into a definite linearization with a positive definite leading coefficient matrix—a form that is particularly attractive numerically.
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