A Fast QR Algorithm for Companion Matrices

Operator Theory: Advances and Applications - Recent Advances in Matrix and Operator Theory ◽

10.1007/978-3-7643-8539-2_7 ◽

2007 ◽

pp. 111-143 ◽

Cited By ~ 23

Author(s):

Shiv Chandrasekaran ◽

Ming Gu ◽

Jianlin Xia ◽

Jiang Zhu

Keyword(s):

Qr Algorithm ◽

Companion Matrices

Download Full-text

Implicit double shift QR-algorithm for companion matrices

Numerische Mathematik ◽

10.1007/s00211-010-0302-y ◽

2010 ◽

Vol 116 (2) ◽

pp. 177-212 ◽

Cited By ~ 25

Author(s):

Marc Van Barel ◽

Raf Vandebril ◽

Paul Van Dooren ◽

Katrijn Frederix

Keyword(s):

Qr Algorithm ◽

Companion Matrices

Download Full-text

Toeplitz nonnegative realization of spectra via companion matrices

Special Matrices ◽

10.1515/spma-2019-0017 ◽

2019 ◽

Vol 7 (1) ◽

pp. 230-245

Author(s):

Macarena Collao ◽

Mario Salas ◽

Ricardo L. Soto

Keyword(s):

Eigenvalue Problem ◽

Half Plane ◽

Toeplitz Matrix ◽

Toeplitz Matrices ◽

Nonnegative Matrix ◽

Inverse Eigenvalue Problem ◽

Complex Numbers ◽

Inverse Eigenvalue ◽

Companion Matrices ◽

Solution Matrix

Abstract The nonnegative inverse eigenvalue problem (NIEP) is the problem of finding conditions for the existence of an n × n entrywise nonnegative matrix A with prescribed spectrum Λ = {λ1, . . ., λn}. If the problem has a solution, we say that Λ is realizable and that A is a realizing matrix. In this paper we consider the NIEP for a Toeplitz realizing matrix A, and as far as we know, this is the first work which addresses the Toeplitz nonnegative realization of spectra. We show that nonnegative companion matrices are similar to nonnegative Toeplitz ones. We note that, as a consequence, a realizable list Λ= {λ1, . . ., λn} of complex numbers in the left-half plane, that is, with Re λi≤ 0, i = 2, . . ., n, is in particular realizable by a Toeplitz matrix. Moreover, we show how to construct symmetric nonnegative block Toeplitz matrices with prescribed spectrum and we explore the universal realizability of lists, which are realizable by this kind of matrices. We also propose a Matlab Toeplitz routine to compute a Toeplitz solution matrix.

Download Full-text