Numerical Measure of a Complex Matrix

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.