## Tunisian Journal of Mathematics

### Horn's problem and Fourier analysis

Jacques Faraut

#### Abstract

Let $A$ and $B$ be two $n×n$ Hermitian matrices. Assume that the eigenvalues $α1,…,αn$ of $A$ are known, as well as the eigenvalues $β1,…,βn$ of $B$. What can be said about the eigenvalues of the sum $C=A+B$? This is Horn’s problem. We revisit this question from a probabilistic viewpoint. The set of Hermitian matrices with spectrum ${α1,…,αn}$ is an orbit $Oα$ for the natural action of the unitary group $U(n)$ on the space of $n×n$ Hermitian matrices. Assume that the random Hermitian matrix $X$ is uniformly distributed on the orbit $Oα$ and, independently, the random Hermitian matrix $Y$ is uniformly distributed on $Oβ$. We establish a formula for the joint distribution of the eigenvalues of the sum $Z=X+Y$. The proof involves orbital measures with their Fourier transforms, and Heckman’s measures.

#### Article information

Source
Tunisian J. Math., Volume 1, Number 4 (2019), 585-606.

Dates
Revised: 5 September 2018
Accepted: 19 September 2018
First available in Project Euclid: 18 December 2018

https://projecteuclid.org/euclid.tunis/1545102024

Digital Object Identifier
doi:10.2140/tunis.2019.1.585

Mathematical Reviews number (MathSciNet)
MR3892253

Zentralblatt MATH identifier
07027467

#### Citation

Faraut, Jacques. Horn's problem and Fourier analysis. Tunisian J. Math. 1 (2019), no. 4, 585--606. doi:10.2140/tunis.2019.1.585. https://projecteuclid.org/euclid.tunis/1545102024

#### References

• Y. Baryshnikov, “GUEs and queues”, Probab. Theory Related Fields 119:2 (2001), 256–274.
• S. Ben Saïd and B. Ørsted, “Bessel functions for root systems via the trigonometric setting”, Int. Math. Res. Not. 2005:9 (2005), 551–585.
• F. A. Berezin and I. M. Gelfand, “Some remarks on the theory of spherical functions on symmetric Riemannian manifolds”, Amer. Math. Soc. Transl. $(2)$ 21 (1962), 193–238.
• R. Bhatia, Matrix analysis, Graduate Texts in Mathematics 169, Springer, 1997.
• R. Bhatia, “Linear algebra to quantum cohomology: the story of Alfred Horn's inequalities”, Amer. Math. Monthly 108:4 (2001), 289–318.
• R. Coquereaux and J.-B. Zuber, “The Horn problem for real symmetric and quaternonic self-dual matrices”, preprint, 2018.
• A. H. Dooley, J. Repka, and N. J. Wildberger, “Sums of adjoint orbits”, Linear and Multilinear Algebra 36:2 (1993), 79–101.
• M. Duflo, G. Heckman, and M. Vergne, “Projection d'orbites, formule de Kirillov et formule de Blattner”, Mém. Soc. Math. France $($N.S.$)$ 15 (1984), 65–128.
• J. Faraut, “Rayleigh theorem, projection of orbital measures and spline functions”, Adv. Pure Appl. Math. 6:4 (2015), 261–283.
• J. Faraut, “Projections of orbital measures for the actions of a pseudo-unitary group”, pp. 111–121 in 50th Seminar “Sophus Lie”, edited by A. Fialowski et al., Banach Center Publ. 113, Polish Acad. Sci. Inst. Math., Warsaw, 2017.
• P. Foth, “Eigenvalues of sums of pseudo-Hermitian matrices”, Electron. J. Linear Algebra 20 (2010), 115–125.
• A. Frumkin and A. Goldberger, “On the distribution of the spectrum of the sum of two Hermitian or real symmetric matrices”, Adv. in Appl. Math. 37:2 (2006), 268–286.
• W. Fulton, “Eigenvalues of sums of Hermitian matrices (after A. Klyachko)”, pp. [exposé] 845, pp. 255–269 in Séminaire Bourbaki, Vol. 1997/98, Astérisque 252, Société Mathématique de France, Paris, 1998.
• W. Fulton, “Eigenvalues, invariant factors, highest weights, and Schubert calculus”, Bull. Amer. Math. Soc. $($N.S.$)$ 37:3 (2000), 209–249.
• P. Graczyk and P. Sawyer, “The product formula for the spherical functions on symmetric spaces in the complex case”, Pacific J. Math. 204:2 (2002), 377–393.
• Harish-Chandra, “Differential operators on a semisimple Lie algebra”, Amer. J. Math. 79 (1957), 87–120.
• G. J. Heckman, “Projections of orbits and asymptotic behavior of multiplicities for compact connected Lie groups”, Invent. Math. 67:2 (1982), 333–356.
• A. Horn, “Doubly stochastic matrices and the diagonal of a rotation matrix”, Amer. J. Math. 76 (1954), 620–630.
• A. Horn, “Eigenvalues of sums of Hermitian matrices”, Pacific J. Math. 12 (1962), 225–241.
• R. A. Horn and C. R. Johnson, Matrix analysis, Cambridge University Press, 1985.
• C. Itzykson and J. B. Zuber, “The planar approximation, II”, J. Math. Phys. 21:3 (1980), 411–421.
• A. A. Klyachko, “Stable bundles, representation theory and Hermitian operators”, Selecta Math. $($N.S.$)$ 4:3 (1998), 419–445.
• A. Knutson and T. Tao, “The honeycomb model of ${\rm GL}_n({\bf C})$ tensor products, I: Proof of the saturation conjecture”, J. Amer. Math. Soc. 12:4 (1999), 1055–1090.
• A. Knutson, T. Tao, and C. Woodward, “The honeycomb model of ${\rm GL}_n(\mathbb C)$ tensor products, II: Puzzles determine facets of the Littlewood–Richardson cone”, J. Amer. Math. Soc. 17:1 (2004), 19–48.
• A. B. J. Kuijlaars and P. Román, “Spherical functions approach to sums of random Hermitian matrices”, preprint, 2016.
• V. B. Lidskii, “On the characteristic numbers of the sum and product of symmetric matrices”, Doklady Akad. Nauk SSSR $($N.S.$)$ 75 (1950), 769–772. In Russian.
• G. Olshanski, “Projections of orbital measures, Gelfand–Tsetlin polytopes, and splines”, J. Lie Theory 23:4 (2013), 1011–1022.
• M. Rösler, “A positive radial product formula for the Dunkl kernel”, Trans. Amer. Math. Soc. 355:6 (2003), 2413–2438.
• H. Weyl, “Das asymptotische Verteilungsgesetz der Eigenwerte linearer partieller Differentialgleichungen (mit einer Anwendung auf die Theorie der Hohlraumstrahlung)”, Math. Ann. 71:4 (1912), 441–479.
• J.-B. Zuber, “Horn's problem and Harish-Chandra's integrals: Probability density functions”, Ann. Inst. Henri Poincaré D 5:3 (2018), 309–338.
• D. I. Zubov, “Projections of orbital measures for classical Lie groups”, Funktsional. Anal. i Prilozhen. 50:3 (2016), 76–81.