Complete Algorithm for Unitary Hessenberg Matrices

Separable Type Representations of Matrices and Fast Algorithms - Operator Theory: Advances and Applications ◽

10.1007/978-3-0348-0612-1_8 ◽

2013 ◽

pp. 117-131

Author(s):

Yuli Eidelman ◽

Israel Gohberg ◽

Iulian Haimovici

Keyword(s):

Hessenberg Matrices

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On the similarity to nonnegative and Metzler Hessenberg forms

Special Matrices ◽

10.1515/spma-2020-0140 ◽

2021 ◽

Vol 10 (1) ◽

pp. 1-8

Author(s):

Christian Grussler ◽

Anders Rantzer

Keyword(s):

Standard Form ◽

Complete Characterization ◽

Nonnegative Matrices ◽

Positive Systems ◽

Third Order ◽

Hessenberg Form ◽

Hessenberg Matrices ◽

Open Question ◽

Order Continuous

Abstract We address the issue of establishing standard forms for nonnegative and Metzler matrices by considering their similarity to nonnegative and Metzler Hessenberg matrices. It is shown that for dimensions n 3, there always exists a subset of nonnegative matrices that are not similar to a nonnegative Hessenberg form, which in case of n = 3 also provides a complete characterization of all such matrices. For Metzler matrices, we further establish that they are similar to Metzler Hessenberg matrices if n 4. In particular, this provides the first standard form for controllable third order continuous-time positive systems via a positive controller-Hessenberg form. Finally, we present an example which illustrates why this result is not easily transferred to discrete-time positive systems. While many of our supplementary results are proven in general, it remains an open question if Metzler matrices of dimensions n 5 remain similar to Metzler Hessenberg matrices.

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Representation of doubly infinite matrices as non-commutative Laurent series

Special Matrices ◽

10.1515/spma-2017-0018 ◽

2017 ◽

Vol 5 (1) ◽

pp. 250-257 ◽

Cited By ~ 2

Author(s):

María Ivonne Arenas-Herrera ◽

Luis Verde-Star

Keyword(s):

Laurent Series ◽

Infinite Matrices ◽

General Properties ◽

Hessenberg Matrices ◽

Diagonal Representation

Abstract We present a new way to deal with doubly infinite lower Hessenberg matrices based on the representation of the matrices as the sum of their diagonal submatrices. We show that such representation is a simple and useful tool for computation purposes and also to obtain general properties of the matrices related with inversion, similarity, commutativity, and Pincherle derivatives. The diagonal representation allows us to consider the ring of doubly infinite lower Hessenberg matrices over a ring R as a ring of Laurent series in one indeterminate, with coefficients in the ring of R-valued sequences that don’t commute with the indeterminate.

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