Multiplicative convolution of real asymmetric and real anti-symmetric matrices

Mario Kieburg; Peter J. Forrester; Jesper R. Ipsen

doi:10.1515/apam-2018-0037

Published by De Gruyter January 20, 2019

Multiplicative convolution of real asymmetric and real anti-symmetric matrices

Mario Kieburg , Peter J. Forrester and Jesper R. Ipsen

From the journal Advances in Pure and Applied Mathematics

https://doi.org/10.1515/apam-2018-0037

Showing a limited preview of this publication:

Abstract

The singular values of products of standard complex Gaussian random matrices, or sub-blocks of Haar distributed unitary matrices, have the property that their probability distribution has an explicit, structured form referred to as a polynomial ensemble. It is furthermore the case that the corresponding bi-orthogonal system can be determined in terms of Meijer G-functions, and the correlation kernel given as an explicit double contour integral. It has recently been shown that the Hermitised product XM⁢⋯⁢X2⁢X1⁢A⁢X1T⁢X2T⁢⋯⁢XMT, where each Xi is a standard real Gaussian matrix and A is real anti-symmetric, exhibits analogous properties. Here we use the theory of spherical functions and transforms to present a theory which, for even dimensions, includes these properties of the latter product as a special case. As an example we show that the theory also allows for a treatment of this class of Hermitised product when the Xi are chosen as sub-blocks of Haar distributed real orthogonal matrices.

Keywords: Products of random matrices; spherical function; bi-orthogonal functions; matrix integrals

MSC 2010: 15A52; 42C05

Funding source: Australian Research Council

Award Identifier / Grant number: DP170102028

Award Identifier / Grant number: CRC 1283

Funding statement: We acknowledge support by the Australian Research Council through grant DP170102028 (PJF), the ARC Centre of Excellence for Mathematical and Statistical Frontiers (PJF,JRI,MK), and the German research council (DFG) via the CRC 1283: “Taming uncertainty and profiting from randomness and low regularity in analysis, stochastics and their applications”.

A Appendix

In this appendix we want to prove the explicit expression of the spherical function Φ as stated in Theorem 2.4. We do this in three steps. First we consider the action of a corank 2 projection on even-dimensional anti-symmetric matrices, see Section A.1. We use this projection to construct a recurrence relation in the dimension for the group integral in the denominator of equation (2.8), see Section A.2, which is solved in Section A.3.

A.1 Eigenvalue PDF for a corank 2 projection

For C,X∈H both 2⁢n×2⁢n real anti-symmetric matrices, we define the matrix-valued Fourier transform of X by

(A.1)ℱ⁢PH⁢(C)=∫exp⁡[ı2⁢Tr⁡X⁢C]⁢PH⁢(X)⁢𝑑X.

Let the singular values of C and X be denoted by c∈A and x∈A, respectively. Consider the circumstance that ℱ⁢PH⁢(C)=ℱ⁢PH⁢(k⁢C⁢kT) for all k∈K=O⁢(2⁢n) and write f^X⁢(c)=ℱ⁢PH⁢(ı⁢c⊗τ2). With pA⁢(x) denoting the jPDF of the singular values of X, by making use of the Harish-Chandra group integral (1.3) for K=O⁢(2⁢n), it was shown in [18, Proposition 3.4] (with an extra factor of 1/n! for the convention that the eigenvalues are not ordered) that

(A.2)pA⁢(x)=∏j=0n-1(-1)j⁢2π⁢(2⁢j)!⁢Δn⁢(x2)⁢1n!⁢∫0∞𝑑c1⁢⋯⁢∫0∞𝑑cn⁢f^X⁢(c)⁢Δn⁢(c2)⁢∏j=1ncos⁡(xj⁢cj).

We will use equation (A.2), with n↦n-1, to deduce the jPDF for the corank 2 projection given by the (2⁢n-2)×(2⁢n-2) random matrix

(A.3)X=Π2⁢n-2,2⁢n⁢k⁢(ı⁢a⊗τ2)⁢kT⁢Π2⁢n-2,2⁢nT,k∈K=O⁢(2⁢n),

where a=(a1,…,an)∈A is fixed.

Remark A.1.

We want to underline that to be fully rigorous we have to introduce a regularizing function in equation (A.2) like a Gaussian or a function similar to equation (2.15) with a vanishing parameter ϵ→0 to guarantee the absolute integrability, since f^X⁢(c)⁢Δn⁢(c2) is not necessarily an L1-function. For general PH, it can grow like polynomial on ℝn. The particular problem to be overcome is the interchange of the inverse Laplace transform and the expectation value (A.1). In the present case, where we want to compute the distribution of the projection (A.3) with a fixed a, the function f^X⁢(c)⁢Δn⁢(c2) is an L1-function on ℝ, see below. Thus, we can neglect these regularizing terms.

Proposition A.2.

Let x={xj}j=1n-1∈A denote the singular values of the random matrix (A.3), and let pA(x|a) denote the corresponding jPDF. We have

pA(x|a)=(2⁢n-2)!(n-1)!Δn-1⁢(x2)Δn⁢(a2)det[1 ⋯ 1(ak-xj)⁢Θ⁢(ak-xj)]j=1,…,n-1k=1,…,n,

where Θ denotes the Heaviside step function.

Proof.

According to definition (A.1), with X given by equation (A.3), upon use of the cyclic property of the trace and the definition of Π2⁢n-2,2⁢n, we have

(A.4)ℱ⁢PH⁢(C)=∫Kexp⁡[-12⁢Tr⁡k⁢(a⊗τ2)⁢kT⁢[C⊕diag⁡(0,0)]]⁢d*⁢k.

The variables c=(c1,…,cn-1)∈A are the singular values of C. The matrix integral corresponding to the average (A.4) is an example of the Harish-Chandra group integral (1.3) for O⁢(2⁢n), where C⊕diag⁡(0,0) is to be thought of as C⊕[0ϵ-ϵ0] with ϵ→0. As such, it has the evaluation

(A.5)f^X⁢(c)=ℱ⁢PH⁢(ı⁢c⊗τ2)=∏j=0n-1(2⁢j)!⁢(-1)n⁢(n-1)/2Δn⁢(a2)⁢Δn-1⁢(c2)⁢∏l=1n-1cl2⁢det⁡[1 ⋯ 1cos⁡(cj⁢ak)]j=1,…,n-1k=1,…,n.

Substituting equation (A.5) in equation (A.2), the latter with n↦n-1, we obtain

pA(x|a)=(2⁢n-2)!(n-1)!Δn-1⁢(x2)Δn⁢(a2)(∏j=1n-1-2π)∫Adcdet[1 ⋯ 1cos⁡(cj⁢ak)]j=1,…,n-1k=1,…,n∏j=1n-1cos⁡xj⁢cjcj2.

In preparation for further simplification, we subtract the first row of the determinant from each of the subsequent rows. The integrations can then be carried out row-by-row to give

pA(x|a)=(2⁢n-2)!(n-1)!Δn-1⁢(x2)Δn⁢(a2)det[1 ⋯ 12π⁢∫0∞cos⁡(xj⁢c)⁢(1-cos⁡(ak⁢c))c2⁢𝑑c]j=1,…,n-1k=1,…,n.

Evaluating the integral with the help of the residue theorem gives

2π⁢∫0∞cos⁡xj⁢c⁢(1-cos⁡ak⁢c)c2⁢𝑑c=(ak-xj)⁢Θ⁢(ak-xj),

which concludes the proof. ∎

Remark A.3.

(a) Let us order a and x as a1<a2<⋯<an and x1<x2<⋯<xn-1. Examining the determinant, we see that the intervals [0,a1] and [an-1,an] can be maximally occupied by a single singular value, namely, x1 and xn-1, respectively. More singular values would yield that either the first three rows or last two rows become linearly dependent. Furthermore, in each interval [aj,aj+1] we cannot have more than two singular values xj because of the same reason (three or more rows become linearly dependent). The same also applies to the intervals [xj,xj+1], which can maximally comprise two singular values of a (three or more columns become linearly dependent when violated). A detailed analysis tells us that those orders, where the determinant does not vanish, create a matrix which is upper triangular with maximally one additional lower diagonal. Once a combined order of a and x is given as 0<α1<α2<⋯<α2⁢n-2<α2⁢n-1, with α a permutation of (a,x), the determinant evaluates to ∏j=1n-1(α2⁢j+1-α2⁢j), which directly follows from the particular form of the matrix. The reason why α1 does not appear in the product follows from the following case discussion. We have either x1<a1, which means that we can subtract the first row times x1 with the second row canceling x1, or a1<x1, where a1 immediately drops out because of the Heaviside step-function, which vanishes in this case.

(b) Let b=diag⁡(b1,…,bn)∈ℝn, u∈K=U⁢(n), and consider the corank 1 projection given by the (n-1)×(n-1) random matrix Πn-1,n⁢u⁢b⁢u†⁢Πn-1,nT. The method used for proving Proposition A.2, with the role of the Harish-Chandra group integral for O⁢(2⁢n) now played by its unitary (U⁢(n)) counterpart, can be adapted to show that the jPDF of the eigenvalues of this random matrix is equal to

Δn-1⁢(x)Δn⁢(b)⁢det⁡[1 ⋯ 1Θ⁢(bk-xj)]j=1,…,n-1k=1,…,n=Δn-1⁢(x)Δn⁢(b)⁢χb1<x1<b2<⋯<xn-1<bn,

where χJ is the indicator function for the condition J. This result is well known [7].

A.2 A recurrence relation

We denote the numerator in equation (2.8) by

fn⁢(s,x)=∫K=O⁢(2⁢n)d*⁢k⁢∏j=1n[det⁡Π2⁢j,2⁢n⁢k⁢x⁢kT⁢Π2⁢j,2⁢nT](sj-sj+1)/2-1.

We first restrict to s∈ℂn, with Re⁡(sj-sj+1)≥2 for all j=1,…,n-1, to have a bounded integrand, and then analytically continue at the end. Our present interest is in establishing a recurrence in the matrix dimension n.

Lemma A.4 (Recursion of fn).

Let n>2, let a∈A=R+n be the singular values of x∈H=o⁢(2⁢n), assumed non-degenerate, and let s∈Cn with Re⁡(sj-sj+1)≥2 for all j=1,…,n-1. We have

fn⁢(s,ı⁢a⊗τ2)=(2⁢n-2)!(n-1)!⁢(det⁡a)sn+n-1⁢∫ℝ+n-1𝑑a~⁢Δn-1⁢(a~2)Δn⁢(a2)⁢det⁡[1 ⋯ 1(ac-a~b)⁢Θ⁢(ac-a~b)]b=1,…,n-1c=1,…,n

(A.6)×fn-1⁢(diag⁡(s1-sn-n,…,sn-1-sn-n),ı⁢diag⁡(a~1,…,a~n-1)⊗τ2).

Proof.

Since the function fn is K-invariant, i.e., fn⁢(s,x)=fn⁢(s,k⁢x⁢kT) for all k∈K and x∈H, we can replace x by ı⁢a⊗τ2, with a∈A its singular values. Then the j=n term in the product of the integrand evaluates to

(det⁡Π2⁢n,2⁢n⁢k⁢x⁢kT⁢Π2⁢n,2⁢nT)(sn+n+1)/2-1=(det⁡a)sn+n-1.

Being independent of the orthogonal matrix k, this term can be taken outside of the integral. Since Π2⁢j,2⁢n=Π2⁢j,2⁢(n-1)⁢Π2⁢(n-1),2⁢n for each j=1,…,n-1, for the remaining terms, we can write

∏j=1n-1(det⁡Π2⁢j,2⁢n⁢k⁢x⁢kT⁢Π2⁢j,2⁢nT)(sj-sj+1)/2-1

=∏j=1n-1(det⁡Π2⁢j,2⁢(n-1)⁢Π2⁢(n-1),2⁢n⁢k⁢x⁢kT⁢Π2⁢(n-1),2⁢nT⁢Π2⁢j,2⁢(n-1)T)(sj-sj+1)/2-1.

We then introduce an orthogonal matrix k~∈O⁢(2⁢n-2) by the multiplicative shift k→diag⁡(k~,𝟙2)⁢k and integrate k~ via the Haar measure on O⁢(2⁢n-2) to obtain

fn⁢(s,ı⁢a⊗τ2)=(det⁡a)sn+n-1⁢∫K=O⁢(2⁢n)d*⁢k⁢∫K=O⁢(2⁢n-2)d*⁢k~

×∏j=1n-1[det⁡Π2⁢j,2⁢(n-1)⁢k~⁢Π2⁢(n-1),2⁢n⁢k⁢(ı⁢a⊗τ2)⁢kT⁢Π2⁢(n-1),2⁢nT⁢k~T⁢Π2⁢j,2⁢(n-1)T](sj-sj+1)/2-1

=(det⁡a)sn+n-1⁢∫K=O⁢(2⁢n)d*⁢k

×fn-1⁢(diag⁡(s1-sn-n,…,sn-1-sn-n),Π2⁢(n-1),2⁢n⁢k⁢(ı⁢a⊗τ2)⁢kT⁢Π2⁢(n-1),2⁢nT).

Since also fn-1 is O⁢(2⁢n-2)-invariant we can replace x~=Π2⁢(n-1),2⁢n⁢k⁢(ı⁢a⊗τ2)⁢kT⁢Π2⁢(n-1),2⁢nT by ı⁢a~⊗τ2, with a~∈ℝ+n-1 the singular values of the [2⁢(n-1)×2⁢(n-1)]-dimensional real anti-symmetric random matrix Π2⁢(n-1),2⁢n⁢k⁢(ı⁢a⊗τ2)⁢kT⁢Π2⁢(n-1),2⁢nT. The jPDF of a~ is given in Proposition A.2, which yields the recursion (A.6), completing the proof. ∎

A.3 Proof of Theorem 2.4

Let a∈A=ℝ+n be the singular values of x∈H=o⁢(2⁢n), assumed non-degenerate, and let s∈ℂn with Re⁡(sj-sj+1)≥2 for all j=1,…,n-1. The spherical function (2.8) is given in terms of fn according to

Φ⁢(s;x)=fn⁢(s,x)fn⁢(s,ı⁢𝟙n⊗τ2),

and so the explicit knowledge of fn implies the explicit form of Φ.

Regarding the latter, we will use complete induction to show

(A.7)fn⁢(s;ı⁢a⊗τ2)=cn⁢(s)⁢det⁡[acsb+n-1]b,c=1,…,nΔn⁢(a2),

where

(A.8)cn⁢(s)=(-1)n⁢(n-1)/2⁢∏j=0n-1(2⁢j)!Δn⁢(s)⁢∏1≤k<l≤n(sk-sl-1).

For n=1, we have f1⁢(s1;ı⁢a1⁢τ2)=a1s1, agreeing with ansatz (A.7) with c1⁢(s1)=1 as is consistent with equation (A.8).

For the induction step we substitute ansatz (A.7) for fn-1 in the recursion (A.6) to obtain

fn⁢(s,ı⁢a⊗τ2)=cn-1⁢(diag⁡(s1-sn-n,…,sn-1-sn-n))⁢(2⁢n-2)!(n-1)!⁢(det⁡a)sn+n-1

×∫ℝ+n-1𝑑a~⁢Δn-1⁢(a~2)Δn⁢(a2)⁢det⁡[1 ⋯ 1(ac-a~b)⁢Θ⁢(ac-a~b)]b=1,…,n-1c=1,…,n⁢det⁡[a~csb-sn-2]b,c=1,…,n-1Δn-1⁢(a~2)

=cn-1⁢(diag⁡(s1-sn-n,…,sn-1-sn-n))⁢(2⁢n-2)!⁢(det⁡a)sn+n-1Δn⁢(a2)

×det⁡[1 ⋯ 1∫0ac𝑑t⁢(ac-t)⁢tsb-sn-2]b=1,…,n-1c=1,…,n

=(-1)n-1⁢(2⁢n-2)!∏l=1n-1(sl-sn-1)⁢(sl-sn)⁢cn-1⁢(diag⁡(s1-sn-n,…,sn-1-sn-n))

(A.9)×det⁡[acsb+n-1]b,c=1,…,nΔn⁢(a2).

In the second equality we used a variant of Andréief’s integration identity [29, Appendix C.1], and in the third we integrated by parts using the fact that the boundary terms vanish for b=1,…,n-1 because Re⁡(sb-sn)≥2⁢(n-b). Comparison of this result with ansatz (A.7) yields the recursion in the constants

cn⁢(s)=(-1)n-1⁢(2⁢n-2)!∏l=1n-1(sl-sn-1)⁢(sl-sn)⁢cn-1⁢(diag⁡(s1-sn-n,…,sn-1-sn-n)),

which, with the initial condition c1⁢(s)=1, implies equation (A.8).

For a=𝟙n, with the help of l’Hôpital’s rule, we obtain

(A.10)fn⁢(s;ı⁢𝟙n⊗τ2)=∏j=0n-1(2⁢j)!/j!∏1≤k<l≤n2⁢(sk-sl-1).

Dividing equation (A.9) by equation (A.10) yields equation (2.10).

The result (A.9) can be analytically continued in the complex variables zj=sj-sj+1-2 with j=1,…,n-1 when dividing the spherical function Φ⁢(s,x) by maxj=1,…,n⁡{aj∑l=1nsl}. This division is equivalent with restricting all aj to the interval [0,1]. Then the singular values of each matrix Π2⁢j,2⁢n⁢k⁢x⁢kT⁢Π2⁢j,2⁢nT are also restricted to the interval [0,1] and fn⁢(s,ı⁢a⊗τ2)⁢∏1≤k<l≤n(sk-sl-1)⁢(sl-sk) is a bounded analytic function in each zj on the complex half-hyperplanes Re⁡zj≥0. Carlson’s theorem can be applied to equation (A.9) for each zj to uniquely analytically continue to the whole complex plane, which concludes the proof of Theorem 2.4.

Acknowledgements

The first author wants to thank the University of Melbourne for its hospitality where this project was carried out.

References

[1] G. Akemann, T. Checinski and M. Kieburg, Spectral correlation functions of the sum of two independent complex Wishart matrices with unequal covariances, J. Phys. A 49 (2016), no. 31, Article ID 315201. 10.1088/1751-8113/49/31/315201Search in Google Scholar

[2] G. Akemann and J. R. Ipsen, Recent exact and asymptotic results for products of independent random matrices, Acta Phys. Polon. B 46 (2015), no. 9, 1747–1784. 10.5506/APhysPolB.46.1747Search in Google Scholar

[3] G. Akemann, J. R. Ipsen and M. Kieburg, Products of rectangular random matrices: Singular values and progressive scattering, Phys. Rev. E (3) 88 (2013), Article ID 052118. 10.1103/PhysRevE.88.052118Search in Google Scholar

[4] G. Akemann, M. Kieburg and L. Wei, Singular value correlation functions for products of Wishart random matrices, J. Phys. A 46 (2013), no. 27, Article ID 275205. 10.1088/1751-8113/46/27/275205Search in Google Scholar

[5] M. C. Andréief, Note sur une relation les intégrales définies des produits des fonctions, Brod. Mém. (3) 2 (1886), 1–14. Search in Google Scholar

[6] W. N. Bailey, Generalized Hypergeometric Series, Cambridge Tracts in Math. Math. Phys. 32, Stechert–Hafner, New York, 1964. Search in Google Scholar

[7] Y. Baryshnikov, GUEs and queues, Probab. Theory Related Fields 119 (2001), no. 2, 256–274. 10.1007/PL00008760Search in Google Scholar

[8] A. Borodin, Biorthogonal ensembles, Nuclear Phys. B 536 (1998), no. 3, 704–732. 10.1016/S0550-3213(98)00642-7Search in Google Scholar

[9] M. Defosseux, Orbit measures, random matrix theory and interlaced determinantal processes, Ann. Inst. Henri Poincaré Probab. Stat. 46 (2010), no. 1, 209–249. 10.1214/09-AIHP314Search in Google Scholar

[10] A. Edelman, The probability that a random real Gaussian matrix has k real eigenvalues, related distributions, and the circular law, J. Multivariate Anal. 60 (1997), no. 2, 203–232. 10.1006/jmva.1996.1653Search in Google Scholar

[11] A. Edelman and N. Raj Rao, Random matrix theory, Acta Numer. 14 (2005), 233–297. 10.1017/S0962492904000236Search in Google Scholar

[12] J. Faraut and A. Korányi, Analysis on Symmetric Cones, Oxford Math. Monogr., Clarendon Press, New York, 1994. Search in Google Scholar

[13] A. P. Ferrer, B. Eynard, P. Di Francesco and J.-B. Zuber, Correlation functions of Harish-Chandra integrals over the orthogonal and the symplectic groups, J. Stat. Phys. 129 (2007), no. 5–6, 885–935. 10.1007/s10955-007-9350-9Search in Google Scholar

[14] J. A. Fischmann, Eigenvalue distributions on a single ring, Ph.D. thesis, Queen Mary University of London, 2012. Search in Google Scholar

[15] J. A. Fischmann, W. Bruzda, B. A. Khoruzhenko, H.-J. Sommers and K. Życzkowski, Induced Ginibre ensemble of random matrices and quantum operations, J. Phys. A 45 (2012), no. 7, Article ID 075203. 10.1088/1751-8113/45/7/075203Search in Google Scholar

[16] P. J. Forrester, Log-gases and Random Matrices, London Math. Soc. Monogr. Ser. 34, Princeton University, Princeton, 2010. 10.1515/9781400835416Search in Google Scholar

[17] P. J. Forrester and J. R. Ipsen, Selberg integral theory and Muttalib–Borodin ensembles, Adv. in Appl. Math. 95 (2018), 152–176. 10.1016/j.aam.2017.11.004Search in Google Scholar

[18] P. J. Forrester, J. R. Ipsen and D.-Z. Liu, Matrix product ensembles of Hermite type and the hyperbolic Harish–Chandra–Itzykson–Zuber integral, Ann. Henri Poincaré 19 (2018), no. 5, 1307–1348. 10.1007/s00023-018-0654-xSearch in Google Scholar

[19] P. J. Forrester and D. Wang, Muttalib–Borodin ensembles in random matrix theory—realisations and correlation functions, Electron. J. Probab. 22 (2017), Paper No. 54. 10.1214/17-EJP62Search in Google Scholar

[20] Y.-P. Förster, M. Kieburg and H. Kösters, Polynomial ensembles and Pólya frequency functions, preprint (2017), https://arxiv.org/abs/1710.08794. 10.1007/s10959-020-01030-zSearch in Google Scholar

[21] I. M. Gelfand and M. A. Naĭmark, Unitäre Darstellungen der klassischen Gruppen, Akademie Verlag, Berlin, 1957. Search in Google Scholar

[22] Harish-Chandra, Differential operators on a semisimple Lie algebra, Amer. J. Math. 79 (1957), 87–120. 10.1007/978-1-4899-7407-5_44Search in Google Scholar

[23] S. Helgason, Groups and Geometric Analysis, Math. Surveys Monogr. 83, American Mathematical Society, Providence, 2000. 10.1090/surv/083Search in Google Scholar

[24] J. R. Ipsen and M. Kieburg, Weak commutation relations and eigenvalue statistics for products of rectangular random matrices, Phys. Rev. E (3) 89 (2014), Article ID 032106. 10.1103/PhysRevE.89.032106Search in Google Scholar PubMed

[25] C. Itzykson and J. B. Zuber, The planar approximation. II, J. Math. Phys. 21 (1980), no. 3, 411–421. 10.1063/1.524438Search in Google Scholar

[26] J. Jorgenson and S. Lang, Spherical Inversion on SLn⁢(𝐑), Springer Monogr. Math., Springer, New York, 2001. 10.1007/978-1-4684-9302-3Search in Google Scholar

[27] B. A. Khoruzhenko, H.-J. Sommers and K. Życzkowski, Truncations of random orthogonal matrices, Phys. Rev. E (3) 82 (2010), no. 4, Article ID 040106. 10.1103/PhysRevE.82.040106Search in Google Scholar PubMed

[28] M. Kieburg, Additive matrix convolutions of Pólya ensembles and polynomial ensembles, preprint (2017), https://arxiv.org/abs/1710.09481. 10.1142/S2010326321500027Search in Google Scholar

[29] M. Kieburg and T. Guhr, Derivation of determinantal structures for random matrix ensembles in a new way, J. Phys. A 43 (2010), no. 7, Article ID 075201. 10.1088/1751-8113/43/7/075201Search in Google Scholar

[30] M. Kieburg and H. Kösters, Exact relation between singular value and eigenvalue statistics, Random Matrices Theory Appl. 5 (2016), no. 4, Article ID 1650015. 10.1142/S2010326316500155Search in Google Scholar

[31] M. Kieburg and H. Kösters, Products of random matrices from polynomial ensembles, Ann. Inst. Henri Poincaré Probab. Stat., to appear. 10.1214/17-AIHP877Search in Google Scholar

[32] M. Kieburg, A. B. J. Kuijlaars and D. Stivigny, Singular value statistics of matrix products with truncated unitary matrices, Int. Math. Res. Not. IMRN 2016 (2016), no. 11, 3392–3424. 10.1093/imrn/rnv242Search in Google Scholar

[33] A. B. J. Kuijlaars, Transformations of polynomial ensembles, Modern Trends in Constructive Function Theory, Contemp. Math. 661, American Mathematical Society, Providence (2016), 253–268. 10.1090/conm/661/13286Search in Google Scholar

[34] A. B. J. Kuijlaars and P. Román, Spherical functions approach to sums of random Hermitian matrices, Int. Math. Res. Not. IMRN (2017), https://doi.org/10.1093/imrn/rnx146. 10.1093/imrn/rnx146Search in Google Scholar

[35] A. B. J. Kuijlaars and D. Stivigny, Singular values of products of random matrices and polynomial ensembles, Random Matrices Theory Appl. 3 (2014), no. 3, Article ID 1450011. 10.1142/S2010326314500117Search in Google Scholar

[36] A. B. J. Kuijlaars and L. Zhang, Singular values of products of Ginibre random matrices, multiple orthogonal polynomials and hard edge scaling limits, Comm. Math. Phys. 332 (2014), no. 2, 759–781. 10.1007/s00220-014-2064-3Search in Google Scholar

[37] T. Lueck, H.-J. Sommers and M. R. Zirnbauer, Energy correlations for a random matrix model of disordered bosons, J. Math. Phys. 47 (2006), no. 10, Article ID 103304. 10.1063/1.2356798Search in Google Scholar

[38] M. L. Mehta, Random Matrices, 3rd ed., Pure Appl. Math. (Amsterdam) 142, Elsevier, Amsterdam, 2004. Search in Google Scholar

[39] K. A. Muttalib, Random matrix models with additional interactions, J. Phys. A 28 (1995), no. 5, L159–L164. 10.1088/0305-4470/28/5/003Search in Google Scholar

[40] G. Pólya, Über Annäherung durch Polynome mit lauter reellen Wurzeln, Palermo Rend. 36 (1913), 279–295. 10.1007/BF03016033Search in Google Scholar

[41] G. Pólya, Algebraische Untersuchungen über ganze Funktionen vom Geschlechte Null und Eins, J. Reine Angew. Math. 145 (1915), 224–249. 10.1515/crll.1915.145.224Search in Google Scholar

[42] I. J. Schoenberg, On Pólya frequency functions. I. The totally positive functions and their Laplace transforms, J. Anal. Math. 1 (1951), 331–374. 10.1007/BF02790092Search in Google Scholar

[43] A. Terras, Harmonic Analysis on Symmetric Spaces and Applications. II, Springer, Berlin, 1988. 10.1007/978-1-4612-3820-1Search in Google Scholar

[44] E. C. Titchmarsh, Introduction to the Theory of Fourier Integrals, 3rd ed., Chelsea, New York, 1986. Search in Google Scholar

Received: 2018-03-01

Revised: 2018-11-19

Accepted: 2018-12-08

Published Online: 2019-01-20

Published in Print: 2019-10-01

Multiplicative convolution of real asymmetric and real anti-symmetric matrices

Abstract

A Appendix

A.1 Eigenvalue PDF for a corank 2 projection

Remark A.1.

Proposition A.2.

Proof.

Remark A.3.

A.2 A recurrence relation

Lemma A.4 (Recursion of fn).

Proof.

A.3 Proof of Theorem 2.4

Acknowledgements

References

Journal and Issue

Articles in the same Issue