The Singular Values of the \mathrm{GUE} (Less is More)

The Singular Values of the (Less is More)

Alan Edelman Department of Mathematics, Massachusetts Institute of Technology edelman@mit.edu  and  Michael LaCroix Department of Mathematics, Massachusetts Institute of Technology malacroi@mit.edu
Abstract.

Some properties that nominally involve the eigenvalues of Gaussian Unitary Ensemble () can instead be phrased in terms of singular values. By discarding the signs of the eigenvalues, we gain access to a surprising decomposition: the singular values of the are distributed as the union of the singular values of two independent ensembles of Laguerre type. This independence is remarkable given the well known phenomenon of eigenvalue repulsion.

The structure of this decomposition reveals that several existing observations about large limits of the are in fact manifestations of phenomena that are already present for finite random matrices. We relate the semicircle law to the quarter-circle law by connecting Hermite polynomials to generalized Laguerre polynomials with parameter . Similarly, we write the absolute value of the determinant of the as a product independent random variables to gain new insight into its asymptotic log-normality. The decomposition also provides a description of the distribution of the smallest singular value of the , which in turn permits the study of the leading order behavior of the condition number of matrices.

The study is motivated by questions involving the enumeration of orientable maps, and is related to questions involving powers of complex Ginibre matrices. The inescapable conclusion of this work is that the singular values of the GUE play an unpredictably important role that had gone unnoticed for decades even though, in hindsight, so many clues had been around.

Key words and phrases:
random matrices, GUE, anti-GUE, LUE, singular values, condition number, semicircle law, quarter-circle law
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1. Introduction

This paper highlights some surprising interrelationships between problems that involve singular values of random matrices. By discarding the signs of eigenvalues, we gain access to additional structure, since despite the pairwise repulsion of its eigenvalues, the singular values of the can be decomposed as the union of two independent sets. The decomposition is equivalent to a result of Jackson and Visentin [23] from enumerative combinatorics, and was previously reported by Forrester in [13, Sec. 2.2]. Our contribution is to consider the decomposition as a complete result about singular values instead of a specialized result about eigenvalues, and to note that this single decomposition underlies several diverse phenomena. From this perspective, we can translate results about asymptotically large matrices to the finite setting, and we can capitalize on the independence to describe the determinant and extreme singular values of the .

Several results, that we find individually surprising, are in fact hidden facets of the same phenomenon. Our aim is to expose these surprises and the interconnections between them.

  1. It is possible to partition the singular values of the into two statistically independent sets (stated in [13] in terms of eigenvalues). This stands in striking contrast, almost in contradiction with, the familiar fact that eigenvalues repel.

  2. The logarithm of the absolute value of the determinant of the can be written as a sum of independent random variables (speculated as impossible by Tao and Vu in [35]).

  3. Matrices of nominal half-integer size play a key role.

  4. The decomposition is equivalent to a result from enumerative combinatorics that relates the cardinalities of two classes of orientable maps on surface of positive genus ([23]).

  5. A bi-diagonal model for singular values gives all the moments of the determinant.

  6. The bulk-scaling limit of the behaves as a superposition of two hard edges. The first author has long since argued for the relatively obvious importance of the singular value view for Laguerre (Wishart) ensembles, and the less well known, but easy to recognize, generalized singular value view for Jacobi (MANOVA) ensembles (see the first author’s course notes for course 18.337 at MIT). The importance of a singular value view for the , however, is far more astonishing.

Our approach is analogous to replacing a semicircle with a pair of quarter-circles. These curves occur as famous limiting distributions. In particular, Wigner’s semicircle law is the limiting distribution of the eigenvalues of the (). The Marchenko-Pastur distribution similarly describes the limiting distribution for the singular values of large rectangular random matrices. In particular, Laguerre ensemble singular values satisfy the quarter-circle law. There is an obvious geometric relationship between these distributions; a semicircle is the union of two quarter-circles (Figure 1–top). The semicircle and quarter-circles also have a less obvious relationship: the semicircle is symmetric about the -axis, and its restriction to the first quadrant is the average of two quarter-circles. This second relationship generalizes to matrices of finite size (Figure 1–right), with the quarter-circles replaced by the distributions of singular values of rectangular matrices of nominal half-integer size (Figure 1–bottom). Variations of this second relationship form the basis for this paper.

Figure 1. A semicircle describes the limiting density of the eigenvalues of the . It decomposes as two quarter-circles (top), related to the limiting densities of the singular values of rectangular matrices. For finite matrices, the density of a random eigenvalue of the (bottom left) is still described by a weighted average (right) of the densities of of the positive square roots of the eigenvalues from two s (bottom center).

Much of random matrix theory involves the behavior of eigenvalues of asymptotically large matrices. It is not always clear how such phenomena correspond to finite matrices. In this paper, we connect the infinite to the finite by phrasing phenomena in terms of singular values. For Hermitian matrices, this amounts to considering the magnitudes of eigenvalues and discarding their signs. One might assume that discarding signs limits the scope of possible conclusions, but in practice several problems that are nominally about eigenvalues are better analyzed in terms of singular values. One could even argue that existing results about the extreme eigenvalues of Laguerre and Jacobi ensembles are elegant precisely because they are essentially about singular values.

The change of setting becomes advantageous when we observe that the singular values of the exhibit an unexpected decomposition: Theorem 1 shows that they are distributed identically to the union of the distinct non-zero singular values of two independent anti- ensembles (an anti- matrix consists of purely imaginary Gaussian entries that are independently distributed subject to skew-symmetry) one of order , the other of order . An equivalent result was previously observed by Forrester in [13, Sec. 2.2] where it was stated explicitly for the case that is even. Since the eigenvalues of the are readily seen to be pairwise dependent, the existence of such a decomposition is itself somewhat surprising.

The decomposition allows us to analyze several statistics of the , including the physically significant gap probability, in terms of the anti-. Ironically, most of the relevant facts about the anti- can be found in Mehta’s physically motivated text, [28, Ch. 13], where his description is asserts that such matrices have “no immediate physical interest”. After a change of variables, the positive eigenvalues of the anti- are seen to have distributions of Laguerre-type (Section 2), corresponding to complex matrices with a half-integral dimension and Laguerre parameter (this is the case of a more general analysis presented by Dumitriu and Forrester in [8]). It is thus possible to draw conclusions about the from an understanding of corresponding facts about Laguerre ensembles. Physically significant existing results about level densities, the absolute value of the determinant, the distributions of the largest singular value, and the bulk-scaling limit can all be analyzed using this framework.

As an unexpected consequence, we obtain the square of the determinant of the as a product of independent random variables (Theorem 2). This is a direct analogue to the result of Goodman for Wishart matrices [18], and precisely the form that Tao and Vu speculated did not exist when discussing the log-normality of the absolute value of the determinant of the in [35].

In addition to providing a common framework for understanding existing results about the , the decomposition permits a study of the distribution of the smallest singular value of a matrix from the ensemble. This quantity may initially appear somewhat unnatural, but for some applications it is an appropriate analog for the smallest eigenvalue of Laguerre and Jacobi ensembles, in some ways behaving as though governed by the existence of a virtual hard-edge. The distribution of the smallest singular value is also closely related to the distribution of conditions numbers, and has implications for the analysis of numerical stability of operations involving random matrices.

The decomposition was first identified by the authors as part of an attempt to find a combinatorial derivation for a functional identity, given by Jackson and Visentin in [20], between generating series for two classes of orientable maps. Physical implications of their identity involve matrix models of 2-dimensional gravity, and are discussed in [19]. They later generalized the identity, in [23], to a stronger form that is essentially equivalent to the existence of our decomposition. Their generating series are effectively cumulant generating series for suitably scaled ensembles of matrix eigenvalues, but Jackson and Visentin appear to have been unaware of the random matrix interpretation of one of the series, possibly because its direct interpretation involves a half-integer evaluation of a parameter that nominally represents one of the dimensions of a rectangular matrix of complex Gaussians. While their work required subtle manipulation of characters of the symmetric group, we believe that the present proof is elementary and enlightening from the perspective of random matrix theory, although a combinatorial interpretation still remains elusive.

It should be noted that while the decomposition discussed here has many superficial parallels with the ideas of superposition and decimation superposition explored by Forrester and Rains ([14, 15]), the concepts are distinct, although it is not difficult to imagine a more general setting in which both their result and ours exist as special cases.

Outline

The remainder of the paper has the following structure:

  • Section 2 describes the matrix ensembles we need to formulate the decomposition.

  • Section 3 uses the level density of the as a warm-up exercise.

  • Section 4 demonstrates the decomposition. We also describes its equivalence to an identity of Jackson and Visentin, and discuses how the decomposition can be observed experimentally.

  • Section 5 applies the decomposition to provide a unified explanation to existing results.

  • Section 6 relates the decomposition to properties of the complex Ginibre ensemble, and draws parallels to an earlier investigation by Rains of powers of compact Lie groups [31, 32].

  • Finally, in Section 7 we discuss some related questions for future work.

2. The Ensembles

Gaussian Unitary Ensembles

The Gaussian Unitary Ensemble of order , (), consists of Hermitian matrices invariant after conjugation by any unitary matrix, and with entries that are normal, and independently distributed, subject to Hermitian symmetry. The ensemble is completely defined by specifying the variance of the diagonal entries, and we choose a normalization with diagonal entries standard normal. As a consequence, the real and imaginary parts of the off-diagonal entries are independently normal with mean and variance . The ensemble can be sampled as , where the real and imaginary parts of the entries of the matrix are independently standard normal, and denotes the Hermitian conjugate of .

Remark.

It is also common to work with a normalization where real and imaginary parts of the off-diagonal entries are standard normal, as in [28, 29], or where the variance depends on (when the primary concern is taking large- limits). Our choice is motivated by combinatorial considerations from the map enumeration setting studied by Jackson and Visentin ([23]), and provides the property that for every partition , the moment is a polynomial in with non-negative integer coefficients depending only on . A convenient consequence of this normalization is that is a product of odd integers for every and (see Theorem 2).

An element of the has real eigenvalues, so the distribution on the matrices induces a distribution on -tuples of eigenvalues. The joint density function for this distribution on , is

(1)

where is such that the density defines a probability measure. A thorough discussion of the is given by Mehta in [28], though with a different choice of normalization. It is convenient to consider the density as consisting of two factors: the Vandermonde squared factor, , occurs because the ensemble is unitarily invariant, while the second factor, , is associated to the Hermite weight in the study of orthogonal polynomials (explaining the use of ‘’ in our notation), and occurs because the density of a matrix is proportional to .

Remark.

It is also common to consider the in terms of a density on sets of eigenvalues, and thus use a density that is supported only on . For the present purposes, we prefer to have a density function that is invariant under permutation of its arguments, and so we consider a density on -tuples constructed by randomly permuting the eigenvalues. The two approaches are not substantially different, but would manifest as a factor of if were to be stated explicitly. In Section 4 we will consider an alternate density on -tuples that induces the same density on sets.

Laguerre Unitary Ensembles

The Laguerre Unitary Ensembles () are a two-parameter family of distributions on positive definite Hermitian matrices. The parameter corresponds to the order of the matrix, while the parameter determines the shape of the distribution. In contrast to the which traditionally found applications arising in physics, the are more closely associated with statistics where the relevant matrices are often referred to as Wishart matrices. Many statistical applications of the , an analogous ensemble based on real instead of complex matrix entries, can be found in [30], and much of the commentary there applies to the with minor modifications. There are two related models of the : one model applies when is a positive integer, while a second model applies when is a positive real number.

When is a positive integer, the ensemble can be sampled as , where is an matrix of independent complex Gaussian entries. The spectrum of is completely determined by , and it is often more natural to work with the singular values of than with the eigenvalues of . By convention we will consider centered matrix entries, with equally distributed real and imaginary parts chosen such that , so that the real and imaginary parts of each entry are independent standard normal. With this normalization, the joint density for the eigenvalues of the on , is

(2)

In fact (2) continues to define a probability density for non-integral , and the densities are realized when is bi-diagonal with its non-zero entries independently -distributed according to

(3)

The correctness of this model for integer values of is verified by considering the effect of applying Householder reflections to a matrix of complex Gaussians, and can be seen to extend to non-integral via the fact that the moments of (2) must depend polynomially on . A complete derivation of the bi-diagonal model for Laguerre ensembles is given in a more general setting in [7]. In the present paper, we will be primarily interested in Laguerre ensembles for which and their relationship to anti- matrices of even and odd order, although ensembles corresponding to arbitrary values of are closely related to the combinatorics in [23] that motivated the present study.

Remark.

As with the , moments of the can be interpreted combinatorially. Taking , the moments are each polynomials in and with non-negative integer coefficients and are symmetric in and . These coefficients are related to the enumeration of hypermaps and associated with the generating series discussed in [23], though a direct interpretation of the combinatorial results there requires the alternate normalization .

Anti-GUE

The anti- consists of anti-symmetric Hermitian matrices with independent (subject to anti-symmetric) normal entries. Such matrices were identified by Mehta as having a particularly elegant theory, with no immediate applications to physics [28, Ch. 13]. Every such matrix is of the form , where is a real skew-symmetric matrix. Such a matrix is unitarily diagonalizable, so its singular values are the absolute values of its eigenvalues. Since the characteristic polynomial of has real coefficients, its eigenvalues occur in complex conjugate pairs, and it follows that the eigenvalues of occur in plus/minus pairs, so each non-zero singular values occurs with even multiplicity. If is for with , then except on a set of measure zero, has distinct non-zero singular values, which we can denote by . When the imaginary parts of the entries of are distributed as independent standard Gaussians (up to Hermitian symmetry), the joint probability density function for the distinct singular values of (in this case also the positive eigenvalues), supported on , is given by

(4)

an expression that combines the two cases described in [28, Section 3.4] or [11, Ex 1.3 q.5] after accounting for the differing choice of normalization. Key to the existence of the decomposition in Theorem 1 is that the final factor, , which is common to both this density and the density, is a symmetric product of even functions.

For completeness, we will outline how this density can be derived from the Laguerre density, (2), by establishing the existence of a bi-diagonal model for the singular values. This follows closely one of the approaches used by Dumitriu and Forrester in [8], where several other derivations are also presented. By applying a sequence of orthogonal Householder transformation to , it is seen to have the same eigenvalue distribution as the tri-diagonal anti-symmetric matrix

which by simultaneously permuting rows and columns is orthogonally similar to a matrix of the form , where depending on the parity of ,

Despite the differing form for even and odd , for many purposes these bi-diagonal should be considered as comprising a one-parameter family. Their moments, for example, can be seen to linked, and to depend polynomially on . Figure 2 emphasizes the uniformity by illustrating how each bi-diagonal matrix is obtained from one of lower order by adding a single additional non-zero matrix element. Notice that when is odd, the matrix is not square.

Figure 2. The bi-diagonal models for the positive eigenvalues of the anti- form a one-parameter family, where each is obtained from the previous by adding a single non-zero matrix element.

When is even, the matrix is the transpose of the Laguerre form from (3), with . For odd values of , the singular values of can also be seen to be Laguerre distributed, in this case with , by noting that

have identically distributed singular values. Dumitriu and Forrester [8, Claim 6.5] demonstrated this equivalence by noting that describes the distribution of the Cholesky factor of . The following lemma can be used to establish the same claim while working directly with and , potentially avoiding numerical pitfalls associated with constructing .

Lemma 1.

If has independent entries with , , and , and is the reflection matrix , then has independent entries distributed as , , and .

Proof.

This is equivalent to the more familiar fact that if , , and are independent with , then , , and are also independent and distributed as . This is established by a change of variables in appropriate joint probability density functions. ∎

By iteratively applying the lemma, a matrix distributed as can be orthogonally transformed into one distributed as via a sequence of orthogonal matrices that act on two columns at a time. Subsequently dropping the column of zeros does not alter the singular values. In particular, the lemma gives a constructive method for sampling from a sample of . Figure 3 illustrates the equivalence schematically for .

Figure 3. Four orthogonal transformations (gray arrows) act on two columns at a time to transform a matrix distributed as into one distributed as .
Remark.

Heuristically, the equivalence between the singular value distributions of and can be anticipated by considering the effect of applying Householder reflections to bi-diagonalize a hypothetical complex random matrix with fractional size, namely . Beginning the process by reducing the first column and then alternating between rows and columns produces the first distribution, while starting with the first row produces the second distribution.

In both the cases of even and odd , the singular values of an anti- matrix are the singular values of a bi-diagonal matrix of Laguerre type (Figure 4), and the probability density function (4) follows from (2) after a change of variable, taking and thus , with additional factors of absorbed into .

Figure 4. The positive eigenvalues of even and odd order anti- matrices are modeled by two different families of square bi-diagonal matrices of Laguerre type.
Remark.

It can also be advantageous to view the equivalence between the anti- and Laguerre ensembles from the opposite perspective. In particular, the relationship formalizes a sense in which the ensembles and are naturally part of a single one-parameter family. In particular, the moments of and share the same polynomial dependence on , with each evaluated at half-integers relative to the other. This matches our intuition that for the purpose of considering singular values, the dimensions of a rectangular matrix should be interchangeable, so that both and a hypothetical should involve the singular values of a nominal matrix.

3. Warm-up: the level densities of the and the Semicircle Law

Before proceeding to the general setting, we examine more closely the motivating problem. How is the semicircle from the related to the quarter-circles describing singular values of bi-diagonal matrices of Laguerre type? What is the analogous relationship for matrices of finite size? By dropping limits, and using orthogonal polynomials to represent relevant probability densities associated with finite random matrices, we see that the semicircle associated with the emerges from an average of two quarter-circles.

For a distribution on -sets, the -point correlation function, describes the induced distribution on uniformly selected subsets of size . By convention, is not a probability distribution, but is instead normalized such that

Conceptually, when the underlying random process generates a single unordered -set, it can be thought of as producing corresponding ordered -tuples. We will be interested primarily in , which describes the pdf of a uniformly selected -set. When the distribution on the -sets takes the form

as with the () and the (), the -point correlation function, , is given by an determinant

(5)

where , and are orthonormal polynomials associated with the weight such that has degree and . This result is based on the fact that the Vandermonde matrix can be expanded in terms of any monic polynomials, and the resulting integrals can be evaluated column by column, and can be conceptualized as a generalization of the Cauchy-Binet formula to matrices of continuous dimension. A more complete discussion can be found, for example, in [4, Sec. 5.4] or [28, Ch. 5].

For the , , and the functions are related to probabilists’ Hermite polynomials described by the initial conditions and , and by the -term recurrence for . It follows from the evaluation

that the level density, describing the probability density function for the distribution of a single eigenvalue selected uniformly from the eigenvalues of the order is given by

(6)

Applying the Christoffel-Darboux formula to the sum provides the compact representation

from which the eponymous semicircle law can then be recovered using asymptotic properties of Hermite polynomials, as in [28, Appendix A.9]. Figure 5 shows the level density for the as an approximation of a semicircle. The relationship to s and the quarter-circle law will be observed by considering the even and odd terms of the summand in (6) separately.

Figure 5. A semi-ellipse (red), with equation , is approximated by the probability density function for the distributions of a uniformly selected eigenvalue of the (blue) as given by .

For the , , and the relevant can be expressed in terms of , the generalized Laguerre polynomials of parameter , which satisfy

Although not required in the present context, it is convenient to note that the Laguerre polynomials are given explicitly by . The weight function for the Laguerre polynomials differs by a factor of in the exponential from our normalization of the , and this is the source of the rescaled parameters in the subsequent formulae. When is a half-integer, we can write in terms of factorials, and obtain

corresponding to and , with both functions supported on the positive real axis. The pdf for the distribution of a uniformly selected singular values is thus given by (the extra factor of is because the density is associated with an implicit differential, so the change of variable also induces the substitution ). By the Marchenko-Pastur law, both and converge to quarter-ellipses as .

To see the relationship between the semicircle law and the quarter-circle law, we note that the left-right symmetry of is a consequence of the fact that the matrix entries of the are distributed symmetrically about the origin, so that the density of a matrix is identical to the density of its negation. The semicircle law is thus an example of a property that nominally involves eigenvalues, but can instead be analyzed in terms of singular values: it is sufficient to show a relationship between