×

Horn’s problem and Fourier analysis. (English) Zbl 1406.15012

Summary: Let \(A\) and \(B\) be two \(n\times n\) Hermitian matrices. Assume that the eigenvalues \(\alpha _1,\dots ,\alpha _n\) of \(A\) are known, as well as the eigenvalues \(\beta _1,\dots ,\beta _n\) 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 \(\{\alpha _1,\ldots ,\alpha _n\}\) is an orbit \(\mathcal{O}_{\alpha }\) for the natural action of the unitary group \(U(n)\) on the space of \(n\times n\) Hermitian matrices. Assume that the random Hermitian matrix \(X\) is uniformly distributed on the orbit \(\mathcal{O}_{\alpha }\) and, independently, the random Hermitian matrix \(Y\) is uniformly distributed on \(\mathcal{O}_{\beta }\). 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.

MSC:

15A18 Eigenvalues, singular values, and eigenvectors
15A42 Inequalities involving eigenvalues and eigenvectors
22E15 General properties and structure of real Lie groups
42B37 Harmonic analysis and PDEs
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] 10.1007/PL00008760 · doi:10.1007/PL00008760
[2] 10.1155/IMRN.2005.551 · Zbl 1065.22010 · doi:10.1155/IMRN.2005.551
[3] ; Berezin, Amer. Math. Soc. Transl. (2), 21, 193 (1962)
[4] 10.1007/978-1-4612-0653-8 · doi:10.1007/978-1-4612-0653-8
[5] 10.2307/2695237 · Zbl 1016.15014 · doi:10.2307/2695237
[6] 10.1080/03081089308818278 · Zbl 0797.15010 · doi:10.1080/03081089308818278
[7] ; Duflo, Mém. Soc. Math. France (N.S.), 65 (1984)
[8] 10.1515/apam-2015-5012 · Zbl 1326.15058 · doi:10.1515/apam-2015-5012
[9] ; Faraut, 50th Seminar “Sophus Lie”. Banach Center Publ., 113, 111 (2017)
[10] ; Foth, Electron. J. Linear Algebra, 20, 115 (2010)
[11] 10.1016/j.aam.2005.12.007 · Zbl 1115.22010 · doi:10.1016/j.aam.2005.12.007
[12] ; Fulton, Séminaire Bourbaki, Vol. 1997/98. Astérisque, 252 (1998)
[13] 10.1090/S0273-0979-00-00865-X · Zbl 0994.15021 · doi:10.1090/S0273-0979-00-00865-X
[14] 10.2140/pjm.2002.204.377 · Zbl 1049.43005 · doi:10.2140/pjm.2002.204.377
[15] 10.2307/2372387 · Zbl 0072.01901 · doi:10.2307/2372387
[16] 10.1007/BF01393821 · Zbl 0497.22006 · doi:10.1007/BF01393821
[17] 10.2307/2372705 · Zbl 0055.24601 · doi:10.2307/2372705
[18] 10.2140/pjm.1962.12.225 · Zbl 0112.01501 · doi:10.2140/pjm.1962.12.225
[19] 10.1017/CBO9780511810817 · Zbl 0576.15001 · doi:10.1017/CBO9780511810817
[20] 10.1063/1.524438 · Zbl 0997.81549 · doi:10.1063/1.524438
[21] 10.1007/s000290050037 · Zbl 0915.14010 · doi:10.1007/s000290050037
[22] 10.1090/S0894-0347-99-00299-4 · Zbl 0944.05097 · doi:10.1090/S0894-0347-99-00299-4
[23] 10.1090/S0894-0347-03-00441-7 · Zbl 1043.05111 · doi:10.1090/S0894-0347-03-00441-7
[24] ; Lidskii, Doklady Akad. Nauk SSSR (N.S.), 75, 769 (1950)
[25] ; Olshanski, J. Lie Theory, 23, 1011 (2013)
[26] 10.1090/S0002-9947-03-03235-5 · Zbl 1015.33010 · doi:10.1090/S0002-9947-03-03235-5
[27] 10.1007/BF01456804 · JFM 43.0436.01 · doi:10.1007/BF01456804
[28] 10.4171/AIHPD/56 · Zbl 1397.15008 · doi:10.4171/AIHPD/56
[29] 10.1007/s10688-016-0151-2 · Zbl 1360.22025 · doi:10.1007/s10688-016-0151-2
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.