Potter, Wielandt, and Drazin on the Matrix Equation <i>AB</i> = <i>ωBA</i>: New Answers to Old Questions

Holtz, Olga; Mehrmann, Volker; Schneider, Hans

doi:10.1080/00029890.2004.11920127

Cited by 10 publications

(16 citation statements)

References 16 publications

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“…If we take a one dimensional subspace V 1 = CU for any unitary U ∈ C n×n , then the problems are equivalent. 16 However, if V 1 is a subspace of C n×n over C of dimension two at least, 15 Called MINRES [25]. 16 The sparse approximate inverse minimization problem (1.6) belongs to this category with U = I .…”

Section: Two Approximation Problems and Real Factoringmentioning

confidence: 99%

“…16 However, if V 1 is a subspace of C n×n over C of dimension two at least, 15 Called MINRES [25]. 16 The sparse approximate inverse minimization problem (1.6) belongs to this category with U = I . then the equalities cannot be expected to hold since there are always arbitrarily poorly conditioned (as well as singular) elements in V 1 [19].…”

Section: Two Approximation Problems and Real Factoringmentioning

confidence: 99%

“…for ω ∈ C [16]. With A (or, analogously, B) invertible, this is equivalent to looking at the eigenproblem for the linear operator X → AX A −1 which is readily analyzed with the help of the Kronecker product A ⊗ A −T .…”

Section: Two Approximation Problems and Real Factoringmentioning

confidence: 99%

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Approximate factoring of the inverse

Byckling¹,

Huhtanen²

2010

Numer. Math.

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Computation of approximate factors for the inverse constitutes an algebraic approach to preconditioning large and sparse linear systems. In this paper, the aim is to combine standard preconditioning ideas with sparse approximate inverse approximation, to have dense approximate inverse approximations (implicitly). For optimality, the approximate factoring problem is associated with a minimization problem involving two matrix subspaces. This task can be converted into an eigenvalue problem for a Hermitian positive semidefinite operator whose smallest eigenpairs are of interest. Because of storage and complexity constraints, the power method appears to be the only admissible algorithm for devising sparse-sparse iterations. The subtle issue of choosing the matrix subspaces is addressed. Numerical experiments are presented.

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Section: Two Approximation Problems and Real Factoringmentioning

confidence: 99%

Section: Two Approximation Problems and Real Factoringmentioning

confidence: 99%

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Approximate factoring of the inverse

Byckling¹,

Huhtanen²

2010

Numer. Math.

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“…To this end we are concerned with the products J i.e., J j c and J l d are e −2πi jl/n -commutative; see [8] for this concept with canonical forms and interesting historical remarks. This kind of relaxed commutativity is rare.…”

Section: A Group Of Unitary Unipotentsmentioning

confidence: 99%

Averaging operators for exponential splittings

Huhtanen¹

2007

Numer. Math.

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Averaging operations are considered in connection with exponential splitting methods. Toeplitz plus Hankel related matrices are resplit by applying appropriate averaging operators leading to a hierarchy of structured matrices. With the resulting parts, the option of using exponential splitting methods becomes available. A related, seemingly important group of unitary unipotents is looked at. Based on a formula due to Lenard, a very fast iterative method to find the nearest Toeplitz plus Hankel matrix in the Frobenius norm is devised.

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“…In [5], for a field F , ω ∈ F , and n ∈ N, a pair of matrices A and B in M n (F ) satisfying AB = ωBA are called ω-commutative. By this definition we define a directed graph as follows.…”

Section: Introductionmentioning

confidence: 99%

On ω‐commuting graphs and their diameters

Raja

Vaezpour

2011

Mathematische Nachrichten

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Let F be a field, let ω ∈ F, and let n ⩾ 2 be a natural number. In this paper we define the ω‐commuting graph of Mn(F), denoted by Γω(Mn(F)) which is a directed graph. We prove some theorems about the strong connectivity of this graph. Also we show that the induced directed subgraphs on the set of all reducible matrices and the set of all triangularizable matrices are strongly connected graphs. Among other results we determine the diameters of the induced directed subgraphs on the sets of all non‐invertible and nilpotent matrices in Mn(F), exactly. Finally, we find good upper bounds for the diameters of some induced directed subgraphs of Γω(Mn(F)), such as the induced directed subgraphs on the set of all idempotent and diagonazable matrices in Mn(F). © 2011 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim

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Potter, Wielandt, and Drazin on the Matrix Equation AB = ωBA: New Answers to Old Questions

Cited by 10 publications

References 16 publications

Approximate factoring of the inverse

Approximate factoring of the inverse

Averaging operators for exponential splittings

On ω‐commuting graphs and their diameters

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