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Numerical Measure of a Complex Matrix
Author(s) -
Gallay Thierry,
Serre Denis
Publication year - 2012
Publication title -
communications on pure and applied mathematics
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 3.12
H-Index - 115
eISSN - 1097-0312
pISSN - 0010-3640
DOI - 10.1002/cpa.20374
Subject(s) - numerical range , measure (data warehouse) , mathematics , hermitian matrix , complex plane , piecewise , polynomial , matrix (chemical analysis) , unit circle , probability measure , random matrix , mathematical analysis , unit sphere , pure mathematics , combinatorics , eigenvalues and eigenvectors , physics , quantum mechanics , materials science , database , computer science , composite material
We introduce a natural probability measure over the numerical range of a complex matrix A ∈ M n ( \input amssym $\Bbb C$ ). This numerical measure μ A can be defined as the law of the random variable 〈 AX , X 〉 ∈ \input amssym $\Bbb C$ when the vector X ∈ \input amssym $\Bbb C$ n is uniformly distributed on the unit sphere. If the matrix A is normal, we show that μ A has a piecewise polynomial density f A , which can be identified with a multivariate B ‐spline. In the general (nonnormal) case, we relate the Radon transform of μ A to the spectrum of a family of Hermitian matrices, and we deduce an explicit representation formula for the numerical density that is appropriate for theoretical and computational purposes. As an application, we show that the density f A is polynomial in some regions of the complex plane that can be characterized geometrically, and we recover some known results about lacunae of symmetric hyperbolic systems in 2 + 1 dimensions. Finally, we prove under general assumptions that the numerical measure of a matrix A ∈ M n ( \input amssym $\Bbb C$ ) concentrates to a Dirac mass as the size n goes to infinity. © 2011 Wiley Periodicals, Inc.