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2020 Corank-1 projections and the randomised Horn problem
Peter Forrester, Jiyuan Zhang
Tunisian J. Math. 3(1): 55-73 (2020). DOI: 10.2140/tunis.2021.3.55

Abstract

Let x̂ be a normalised standard complex Gaussian vector, and project an Hermitian matrix A onto the hyperplane orthogonal to x̂. In a recent paper Faraut (Tunisian J. Math. 1 (2019), 585–606) has observed that the corresponding eigenvalue PDF has an almost identical structure to the eigenvalue PDF for the rank-1 perturbation A+bx̂x̂, and asks for an explanation. We provide one by way of a common derivation involving the secular equations and associated Jacobians. This applies also in a related setting, for example when x̂ is a real Gaussian and A Hermitian, and also in a multiplicative setting AUBU where A,B are fixed unitary matrices with B a multiplicative rank-1 deviation from unity, and U is a Haar distributed unitary matrix. Specifically, in each case there is a dual eigenvalue problem giving rise to a PDF of almost identical structure.

Citation

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Peter Forrester. Jiyuan Zhang. "Corank-1 projections and the randomised Horn problem." Tunisian J. Math. 3 (1) 55 - 73, 2020. https://doi.org/10.2140/tunis.2021.3.55

Information

Received: 14 June 2019; Revised: 10 July 2019; Accepted: 1 August 2019; Published: 2020
First available in Project Euclid: 25 June 2021

Digital Object Identifier: 10.2140/tunis.2021.3.55

Subjects:
Primary: 15A18‎, 15B52

Rights: Copyright © 2021 Mathematical Sciences Publishers

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