This is an announcement for the paper "The numerical measure of a complex matrix" by Thierry Gallay and Denis Serre.
Abstract: We introduce and carefully study a natural probability measure over the numerical range of a complex matrix $A \in M_n(\C)$. This numerical measure $\mu_A$ can be defined as the law of the random variable $<AX,X> \in \C$ when the vector $X \in \C^n$ is uniformly distributed on the unit sphere. If the matrix $A$ is normal, we show that $\mu_A$ 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 $\mu_A$ to the spectrum of a family of Hermitian matrices, and we deduce an explicit representation formula for the numerical density which 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 which can be characterized geometrically, and we recover some known results about lacunas of symmetric hyperbolic systems in $2+1$ dimensions. Finally, we prove under general assumptions that the numerical measure of a matrix $A \in M_n(\C)$ concentrates to a Dirac mass as the size $n$ goes to infinity.
Archive classification: math.FA math.PR math.SP
Mathematics Subject Classification: 47A12, 28A33, 44A12, 65D07, 35L40, 60F05
Remarks: 41 pages, 5 figures
Submitted from: thierry.gallay@ujf-grenoble.fr
The paper may be downloaded from the archive by web browser from URL
http://front.math.ucdavis.edu/1009.1522
or