Get all the updates for this publication

Articles

How Many Eigenvalues of a Product of Truncated Orthogonal Matrices are Real?Open Access

Published in Taylor and Francis Inc.

2020

Volume: 29

Issue: 3

Pages: 276 - 290

A truncation of a Haar distributed orthogonal random matrix gives rise to a matrix whose eigenvalues are either real or complex conjugate pairs, and are supported within the closed unit disk. This is also true for a product Pm of m independent truncated orthogonal random matrices. One of most basic questions for such asymmetric matrices is to ask for the number of real eigenvalues. In this paper, we will exploit the fact that the eigenvalues of Pm form a Pfaffian point process to obtain an explicit determinant expression for the probability of finding any given number of real eigenvalues. We will see that if the truncation removes an even number of rows and columns from the original Haar distributed orthogonal matrix, then these probabilities will be rational numbers. Finally, based on exact finite formulas, we will provide conjectural expressions for the asymptotic form of the spectral density and the average number of real eigenvalues as the matrix dimension tends to infinity. © 2018 Taylor & Francis.

Topics: Orthogonal matrix (66)%66% related to the paper, Matrix (mathematics) (63)%63% related to the paper, Complex conjugate (58)%58% related to the paper, Random matrix (58)%58% related to the paper and Eigenvalues and eigenvectors (57)%57% related to the paper

View more info for "How Many Eigenvalues of a Product of Truncated Orthogonal Matrices are Real"

Preprint Version

Content may be subject to copyright.This is a green open access article.This is a green open access article.

About the journal

Published in Taylor and Francis Inc.

Open Access

yesImpact factor

N/A