The numerical measure of a complex matrix
We introduce and carefully study a natural probability measure over the numerical range of a complex matrix . This numerical measure can be defined as the law of the random variable when the vector is uniformly distributed on the unit sphere. If the matrix is normal, we show that has a piecewise polynomial density , which can be identified with a multivariate -spline. In the general (nonnormal) case, we relate the Radon transform of 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 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 dimensions. Finally, we prove under general assumptions that the numerical measure of a matrix concentrates to a Dirac mass as the size goes to infinity.
If is a complex square matrix of size , the numerical range of is the compact subset of the complex plane defined by
where is the usual scalar product in and . It is quite obvious that , where (the spectrum of ) is the collection of all eigenvalues of , and that for any unitary matrix . Moreover, a celebrated result due to Toeplitz  and Hausdorff  asserts that is always a convex subset of the complex plane. In particular, contains the convex hull of , and it is easy to verify that if the matrix is normal, namely . The interested reader is referred to Chapter 1 of  for a detailed discussion of the various properties of the numerical range, including complete proofs.
Let be the unit sphere in , considered as a real manifold of dimension . By definition, the numerical range is the image of the numerical map defined by
The algebraic and geometric properties of the map have been extensively studied, see [19, 26, 4, 9, 16, 17]. In particular, the set of all critical values of , which we denote by , has received a lot of attention, because this is an interesting object which contains a lot of information on the matrix . For instance, it is known that and . In addition, there exists a real algebraic curve with the property that , where denotes the set of all line segments joining pairs of points of at which has the same tangent line . Under generic assumptions on , the bitangent set is empty, and the critical set is therefore the union of a finite number of closed curves, one of which is the boundary of the numerical range . This distinguished curve is smooth, and encloses all the other ones in its interior. We refer to Section 5 below for more details on the geometry of the singular set, and to Section 7 for a few concrete examples.
Our purpose in this paper is to introduce another mathematical quantity which is naturally related to the numerical map . Given , the numerical measure of is the probability measure on defined by the formula
for all continuous functions . Here denotes the Euclidian measure on the unit sphere , normalized as a probability measure. In words, the numerical measure is thus the image under the numerical map of the normalized Euclidean measure on the unit sphere. Equivalently, if is a random variable that is uniformly distributed on , the numerical measure is just the distribution of the random variable . This probabilistic intepretation will be useful later, especially in Section 8.
Our first goal is to establish a few general properties of the numerical measure . It is clear by construction that is invariant under unitary conjugations of , namely for all . This is precisely the reason why we used the Euclidean measure on in the definition (1). It is also easy to verify that the support of is exactly the numerical range , see Section 2 below. Less obvious, perhaps, is the fact that is absolutely continuous with respect to the Lebesgue measure on , so that we can define the numerical density as the Radon-Nikodym derivative of with respect to (in the particular situation where reduces to a line segment , we understand as the one-dimensional Lebesgue measure on , see Section 2.) We also prove that the numerical density is strictly positive in the interior of , a property that can be interpreted as a strong version of Hausdorff’s theorem . Finally, we shall see that the singular support of is contained in the critical set , which means that the numerical density is smooth outside . In fact, we conjecture that for all , but this has not been proved yet.
After these general properties have been established, our next goal is to give a more precise description of the numerical density . For this purpose, it is convenient to distinguish between various cases:
1. (The scalar case) If is reduced to a single point , then (where denotes the identity matrix) and . In this trivial situation, there is of course no need to introduce a numerical density.
2. (The Hermitian case) Assume that and that is a line segment. Then we can find , , and a Hermitian matrix such that . If are the eigenvalues of , we shall see in Section 3 that the numerical measure is absolutely continuous with respect to Lebesgue’s measure on , and that the corresponding density is exactly the normalized -spline of degree with knots . In particular, is polynomial of degree on each interval , vanishes identically outside , and is continuous at each point together with its derivatives up to order , where is the multiplicity of as an eigenvalue of (if , then is discontinuous at .) This gives an explicit representation of the numerical measure , and the measure is the image of under the affine isometry .
3. (The normal case) Suppose now that and that is a normal matrix whose spectrum is not contained in a line segment. Then is a convex polygon with nonempty interior, and it turns out that the numerical density is the bivariate B-spline of degree whose knots are the eigenvalues of . Here we refer to the work of W. Dahmen  for the definition and the main properties of multivariate -splines. In this particular case, the critical set is thus the collection of all line segments joining pairs of eigenvalues of , and the density is polynomial of degree in each connected component of . In the generic situation where no straight line contains more than two eigenvalues of , one can show that is continuous together with its derivatives up to order (and is discontinuous on if .)
4. (The nonnormal case) Finally, we consider the most interesting situation where the matrix is not normal. In that case, there is no explicit formula for the numerical density, but the problem can be reduced in some sense to the Hermitian case by the following simple observation. For any , let be the Hermitian matrix defined by
where and . Then for all . Now, if the random variable is uniformly distributed on , the distribution of is by definition the numerical measure , whereas the distribution of is easily identified as the two-dimensional Radon transform of the numerical measure , evaluated at . We thus have
where denotes the two-dimensional Radon transformation. Since the numerical density of is known to be the -spline based on the eigenvalues of , we can reconstruct the numerical measure by inverting the Radon transformation in (3), using the the well-known backprojection method which plays an important role in tomography . This provides a useful representation formula for the numerical density, as well as an efficient algorithm for numerical calculations, see Section 4 for more details.
It is worth mentioning here that the critical set can be conveniently characterized using the family of Hermitian matrices associated with . Indeed, if we define the eigenvalues in such a way that they depend analytically on , one can shown that the algebraic curve which generates is given by
As was already mentioned, the numerical density is smooth (in fact, real-analytic) on each connected component of . The regularity across is more difficult to study, but we shall show in Section 6.2 that is everywhere of class if and satisfies some generic hypotheses, which exclude in particular the case of normal matrices. In addition, for an arbitrary matrix of size , we shall prove that all derivatives of of order vanish identically in some distinguished regions, which have the following geometric characterization. For any , let be the number of straight lines containing which are tangent to the curve , see (39) below for a precise definition where possible multiplicities are taken into account. It is easy to verify that is constant in each connected component of , and that . The distinguished regions where is polynomial of degree (if ) or (if ) are exactly those connected components of on which takes its maximal value . This remarkable property of the numerical density, which is one of our main results, will be established in Section 6.1. The geometric condition is always satisfied in the complement of the numerical range, where vanishes identically, but for many matrices of size is it also met in some regions inside . For instance, in the three-dimensional example represented in Fig. 1, it is easy to verify that if is outside or inside the cuspidal triangle, and in the intermediate region where the numerical density is not constant.
At this point, it is necessary to make a connection with the theory of lacunas of symmetric hyperbolic systems of partial differential equations in variables. Given , we consider the following system of linear PDE’s in :
One can show that is homogeneous of degree in and , it is thus sufficient to consider the time-one trace , which is a distribution on . Due to the finite speed of propagation, it is well-known that is zero outside a compact set of , but it may also happen that vanishes identically in some regions inside the domain of influence of the origin. Such regions are called lacunas of the hyperbolic system (4).
The properties of the fundamental solution of symmetric hyperbolic systems have been studied by many authors, see e.g. [22, 26, 4, 2, 3]. In the particular case of system (4), J. Bazer and D. Yen have shown that, if one identifies with , the singular support of the distribution is contained in the critical set , and the (stable) lacunas of system (4) are exactly the regions described above where the numerical density is polynomial of degree . This remarkable coincidence is of course not fortuitous. In Section 6.3, we explain it by showing that the fundamental solution can be expressed as a linear combination of derivatives of order of a homogeneous extension of the numerical density . This connection allows us to recover some of the main results of , and therefore confirms that the numerical measure is a natural quantity attached to the matrix . One might even argue that contains more information than , since for instance while is in general strictly smaller and not necessarily convex, see Section 7. Similarly, we believe that always coincide with , while is usually smaller.
A final question that is worth investigating is the behavior of the numerical measure when the size of the matrix goes to infnity. Here of course, specific assumptions have to be made in order to obtain convergence results. Suppose for instance that is a sequence of complex matrices with , , and for all . If, for each , is a random variable that is uniformly distributed on , we show in Section 8 that the complex variable converges almost surely to zero as . This is reminiscent of the strong law of large numbers in probability theory. Under slightly stronger assumptions, we also establish the analog of the central limit theorem in this context. Our convergence results mean that is very close to when is large, where is the barycenter of . This explains why plotting for randomly chosen points is a very unefficient algorithm for determining the numerical range if is a large matrix!
The rest of the paper is organized as follows. In Section 2, we establish some general properties of the numerical measure. Section 3 is devoted to the particular situations where the matrix is Hermitian or normal. The nonnormal case is treated in Sections 4–6, which constitute the core of the paper. In Section 4, we derive a representation formula for the numerical density using the inversion of the Radon transformation. Section 5 collects a few results on the geometry of the critical set , which are mainly borrowed from [19, 26, 17]. These informations are used in Section 6 to derive an explicit formula for the derivatives of order of the numerical density, which allows us to obtain generic regularity results and to express the fundamental solution of the hyperbolic system (4) in terms of derivatives of the numerical density. To illustrate our results, a few explicit examples are treated in Section 7. Finally, we investigate in Section 8 the concentration properties of the numerical density for large matrices, and we discuss in Section 9 a possible extension of our results to hyperbolic polynomials with an arbitrary number of variables.
Acknowledgements. This work has benefited of stimulating discussions with several of our colleagues, including Y. Colin de Verdière, F. Faure, and A. Joye.
2 General properties of the numerical measure
In this section, we establish a few general properties of the numerical measure of a complex matrix. In particular, we show that is absolutely continuous with respect to the Lebesgue measure on , and we prove a direct sum formula which will be useful later.
2.1 Support and regularity properties
We first show that the support of the numerical measure always coincides with the numerical range of the matrix.
For any , one has .
Proof. If , then , hence . This shows that . Conversely, if is any open set such that , then is a nonempty open subset of , hence . Thus .
Our next goal is to locate the singular support of . We recall that is a regular point of if the differential map is onto. Otherwise, we say that is a critical point. The following characterization will be useful:
In other words, the range of the differential has (real) dimension if and only if is an eigenvector of for a unique , and is reduced to if and only if is an eigenvector of for all . The proof is neither new nor difficult, but we shall repeat it here in order to introduce some notation that will be needed later on.
Proof. Since for all , we can consider the numerical map as acting on the quotient space . Thus, to detect the critical points of , we study the reduced map defined by
where and are the Hermitian matrices introduced in (2).
If , the tangent space to at is just the -dimensional affine subspace . Thus, using the definition above of , it is straightforward to verify that, for all with , one has
where denotes the real scalar product in , and
Of course, replacing , with , has no effect in (6) since , but after this substitution we can let run over the whole of without increasing the range. So our task is reduced to computing the rank of the -linear map , which is just the rank of the matrix
By the Cauchy-Schwarz inequality, the positive matrix is singular if and only if there exists such that , which exactly means that is an eigenvector of . Moreover, if and only if , which is equivalent to saying that is an eigenvector of both and , hence of both and .
Let denote the set of all critical values of , namely where is the set of all critical points of . Our next result is:
If , then .
Proof. If the numerical range is reduced to a line segment or to a single point, then , hence by Lemma 2.1. Thus, we assume from now on that has nonempty interior. By Sard’s lemma, the critical set is then a compact subset of with zero Lebesgue measure. We have to show that there exists a smooth density function such that on . Clearly, we must have on .
If , then is a compact submanifold of of codimension , which depends smoothly on . Using classical arguments, involving a partition of unity and the Implicit Function Theorem, it is not difficult to verify that, for any continuous function with , one has
where is the total measure of , is the -dimensional Euclidean measure on the submanifold , and where is the matrix defined in (7). Remark that , where are the singular values of the differential map . In view of (1), we conclude that on , where
It is easily verified that the density is smooth and strictly positive on .
The results obtained so far are summarized in the following proposition, which also asserts that the numerical measure is absolutely continuous with respect to Lebesgue’s measure on .
1) If the numerical range has nonempty interior, the numerical measure is absolutely continuous with respect to the (two-dimensional) Lebesgue measure on . The numerical density is smooth outside the critical set .
2) If is a nonscalar Hermitian matrix, then and the numerical measure is absolutely continuous with respect to the (one-dimensional) Lebesgue measure on . The numerical density is smooth outside the spectrum .
As is explained in the introduction, Proposition 2.4 covers all interesting cases. Indeed, if the numerical range has empty interior, then either is reduced to a single point, in which case is a scalar matrix and is just a Dirac mass, or is a line segment of nonzero length, in which case can be reduced to a nonscalar Hermitian matrix by a simple affine transformation.
1) is an algebraic submanifold of of codimension at least . By Sard’s lemma, this is the case if and only if has nonempty interior. In that situation, since we already know that has a smooth density outside the critical set , we only need to show that . Given , let , where “” denotes here the geodesic distance on the unit sphere. We decompose
where . Since , we have as . Moreover, the proof of Lemma 2.3 shows is a codimension two submanifold of , so that for any . Using the definition of the numerical measure, we conclude that .
2) . This is the case if and only if has empty interior, and without loss of generality we can then assume that the matrix is Hermitian. Since , it is more natural here to consider as a map from into . If we do that, then repeating the proofs of Lemmas 2.2 and 2.3 we easily find that the critical points of are exactly the eigenvectors of . Moreover, the numerical measure has a smooth density on , and is absolutely continuous with respect to the Lebesgue measure if is not a scalar matrix. We skip the details here, because the Hermitian case will be treated in full details in Section 3 below.
To conclude this section, we show that the numerical density is strictly positive in the interior of . A little care is needed in the formulation of that result, because as we shall see in Section 4.2 the numerical density need not be a continuous function.
If is an interior point of , then
Proof. If is an interior point of , it is shown in [17, Proposition 2.11] that the preimage contains at least one regular point . Let be an open geodesic ball centerd at whose closure does not intersect . Proceeding as in the proof of Lemma 2.3, we find
If is sufficiently small, the integral inside the curly brackets is a smooth and positive function of , and (9) follows.
2.2 The direct sum formula
Let and . Given and , the direct orthogonal sum of and is the matrix defined by
In this situation, we have a formula for the numerical measure in terms of and .
For any , we have
where is Euler’s beta function
Proof. Any unit vector can be written as
where , , and . Up to negligible sets, the map defines a diffeomorphism from onto . With this parametrization it is not difficult to verify that the Euclidean measure on has the following expression
Equivalently, since , the normalized Euclidean measure satisfies
Thus, using definition (1) and the fact that , we easily obtain
which is the desired result.
As an application, if we choose in Proposition 2.7, we obtain the following relation between the Fourier transforms of the measures , and :
The formula given in Proposition 2.7 can be generalized in a straightforward way to a direct sum with an arbitrary number of terms. Assume that where , so that with . Setting , we denote
where and denotes the -dimensional simplex
Using (10) and proceeding by induction over , we easily obtain the general formula
where it is understood again that .
3 The numerical density of a normal matrix
If is a normal matrix, the numerical measure is entirely determined by the spectrum . Indeed, we know that is unitarily equivalent to the diagonal matrix , and that a unitary conjugation does not affect the numerical measure. Using this observation and the direct sum formula of Section 2.2, we shall prove that the numerical density of is a piecewise polynomial function, which can be characterized as a multivariate -spline whose knots are the eigenvalues of . We begin with the important particular case where all eigenvalues of are colinear.
3.1 The Hermitian case
If is a Hermitian matrix, then and the numerical measure is therefore supported on the real axis. Assuming that is not a multiple of the identity matrix, we show in this section that is absolutely continuous with respect to Lebesgue’s measure on , and we give a simple characterization of the numerical density . The result is:
If is a nonscalar Hermitian matrix, the numerical density is the normalized -spline of degree whose knots are the eigenvalues of .
To make the statement clear, we briefly recall the definition and some elementary properties of the classical -splines . If are pairwise distinct, the divided difference of a continuous function at the points is the quantity
This is the leading coefficient of the unique polynomial of degree at most which agrees with at the points . It is easy to verify that the right-hand side of (14) is a completely symmetric function of the variables . If , the divided difference can be extended by continuity to arbitrary (not necessarily distinct) values of , and we have the integral formula:
where is the -dimensional simplex defined in (12) and . In what follows, we shall always assume that the set is not reduced to a single point, so that the divided difference is well-defined as soon as is of class in a neighborhood of .
With these notations, the normalized -spline of degree with knots is the function defined by
where denotes the map . If , it is not difficult to show that vanishes identically outside , and coincides with a polynomial of degree at most on each nonempty interval . Moreover, if denotes the multiplicity of in , one can verify that is continuous at together with its derivatives up to order , provided . If , then is discontinuous at . Finally, we shall see below that is positive on and that .
Proof of Proposition 3.1. Let be a normal matrix with eigenvalues . To compute the numerical density, we can assume without loss of generality that is diagonal, namely