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Jacobians and rank 1 perturbations relating to unitary Hessenberg matrices

Forrester, Peter J. and Rains, Eric M. (2006) Jacobians and rank 1 perturbations relating to unitary Hessenberg matrices. International Mathematics Research Notices, 2006 . Art. No. 48306. ISSN 1073-7928. doi:10.1155/IMRN/2006/48306.

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Killip and Nenciu gave random recurrences for the characteristic polynomials of certain unitary and real orthogonal upper Hessenberg matrices. The corresponding eigenvalue probability density functions (p.d.f's) are β-generalizations of the classical groups. Left open was the direct calculation of certain Jacobians. We provide the sought direct calculation. Furthermore, we show how a multiplicative rank 1 perturbation of the unitary Hessenberg matrices provides a joint eigenvalue p.d.f. generalizing the circular β-ensemble, and we show how this joint density is related to known interrelations between circular ensembles. Projecting the joint density onto the real line leads to the derivation of a random three-term recurrence for polynomials with zeros distributed according to the circular Jacobi β-ensemble.

Item Type:Article
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Additional Information:© 2006 Hindawi Publishing Corporation. Received: 27 May 2005; Revision Received: 06 January 2006; Accepted: 08 February 2006; Published: 01 January 2006. Supported by the Australian Research Council.
Funding AgencyGrant Number
Australian Research CouncilUNSPECIFIED
Record Number:CaltechAUTHORS:20171106-151903451
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Official Citation:Peter J. Forrester, Eric M. Rains; Jacobians and rank 1 perturbations relating to unitary Hessenberg matrices, International Mathematics Research Notices, Volume 2006, 1 January 2006, 48306,
Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:83004
Deposited By: Tony Diaz
Deposited On:06 Nov 2017 23:24
Last Modified:15 Nov 2021 19:54

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