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Fiedler Companion Linearizations and the Recovery of Minimal Indices
TLDR
A standard way of dealing with a matrix polynomial $P(\lambda)$ is to convert it into an equivalent matrix pencil—a process known as linearization. Expand
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Spectral equivalence of matrix polynomials and the index sum theorem
Abstract The concept of linearization is fundamental for theory, applications, and spectral computations related to matrix polynomials. However, recent research on several important classes ofExpand
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Block Kronecker linearizations of matrix polynomials and their backward errors
TLDR
We introduce a new family of strong linearizations of matrix polynomials—which we call “block Kronecker pencils”— and perform a backward stability analysis of complete polynomial eigenproblems. Expand
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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 )Expand
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Weyl-type relative perturbation bounds for eigensystems of Hermitian matrices
We present a Weyl-type relative bound for eigenvalues of Hermitian perturbations A+E of (not necessarily definite) Hermitian matrices A. This bound, given in function of the quantityExpand
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Implicit standard Jacobi gives high relative accuracy
TLDR
We prove that the Jacobi algorithm applied implicitly on a decomposition A = XDXT of the symmetric matrix A, where D is diagonal, and X is well conditioned, computes all eigenvalues of A to high relative accuracy. Expand
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Low Rank Perturbation of Jordan Structure
TLDR
We show that for most matrices B satisfying ${\rm rank}\,(B)\leq g$, the Jordan blocks of A+B with eigenvalue $\lambda_0$ are just the smallest Jordan blocks in the Jordan canonical form of A, while the rest remain unchanged. Expand
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Complementary bases in symplectic matrices and a proof that their determinant is one
New results on the patterns of linearly independent rows and columns among the blocks of a symplectic matrix are presented. These results are combined with the block structure of the inverse of aExpand
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Accurate solution of structured linear systems via rank-revealing decompositions
Linear systems of equations Ax = b, where the matrix A has some particular structure, arise frequently in applications. Very often structured matrices have huge condition numbers κ(A) = ‖A−1‖‖A‖ and,Expand
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Matrix Polynomials with Completely Prescribed Eigenstructure
TLDR
We present necessary and sufficient conditions for the existence of a matrix polynomial when its degree, its finite and infinite elementary divisors, and its left and right minimal indices are prescribed. Expand
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