Abstract
We study the average condition number for polynomial eigenvalues of collections of matrices drawn from some random matrix ensembles. In particular, we prove that polynomial eigenvalue problems defined by matrices with random Gaussian entries are very well conditioned on the average.
Highlights
Following the ideas in [3,7,19], we note that many different numerical problems can be described within the following simple general framework
Definition 1 (Condition number in the algebraic setting) Let I, O and V ⊆ I × O be real algebraic varieties such that the smooth loci of I, O are endowed with Finsler structures and let (i, o) ∈ V be a smooth point of V such that i ∈ I, o ∈ O are smooth points of I and O, respectively
In Corollary 3, we provide an analogous formula in the case when A0, . . . , Ad are independent Gaussian Orthogonal Ensemble (GOE)(n)-distributed matrices
Summary
Following the ideas in [3,7,19], we note that many different numerical problems can be described within the following simple general framework. Definition 1 (Condition number in the algebraic setting) Let I, O and V ⊆ I × O be real algebraic varieties such that the smooth loci of I, O are endowed with Finsler structures and let (i, o) ∈ V be a smooth point of V such that i ∈ I, o ∈ O are smooth points of I and O, respectively. The most important result in this paper is a very general theorem which is designed to provide exact formulas for the expected value of the condition number in the PEVP and other problems. Remark 2 Recently in [1] Armentano and the first author of the current article investigated the expectation of the squared condition number for polynomial eigenvalues of complex Gaussian matrices.
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callDisclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
