Numerical Measure of a Complex Matrix

Abstract

AbstractWe introduce a natural probability measure over the numerical range of a complex matrix A ∈ Mn( \input amssym $\Bbb C$). This numerical measure μA 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 μA has a piecewise polynomial density fA, which can be identified with a multivariate B‐spline. In the general (nonnormal) case, we relate the Radon transform of μA 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 fA 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 ∈ Mn (\input amssym $\Bbb C$) concentrates to a Dirac mass as the size n goes to infinity. © 2011 Wiley Periodicals, Inc.