Geodesic
Sum of Hermitian Matrices with Given Eigenvalues: Inertia, Rank, and Multiple Eigenvalues
Canadian journal of mathematics, Tome 62 (2010) no. 1, pp. 109-132

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Let $A$ and $B$ be $n\,\times \,n$ complex Hermitian (or real symmetric) matrices with eigenvalues ${{a}_{1}}\,\ge \,\cdots \,\ge \,{{a}_{n}}$ and ${{b}_{1}}\,\ge \,\cdots \,\ge \,{{b}_{n}}$ . All possible inertia values, ranks, and multiple eigenvalues of $A\,+\,B$ are determined. Extension of the results to the sum of $k$ matrices with $k\,>\,2$ and connections of the results to other subjects such as algebraic combinatorics are also discussed.
DOI : 10.4153/CJM-2010-007-2
Mots-clés : complex Hermitian matrices, real symmetric matrices, inertia, rank, multiple eigenvalues 
Li, Chi-Kwong; Poon, Yiu-Tung. Sum of Hermitian Matrices with Given Eigenvalues: Inertia, Rank, and Multiple Eigenvalues. Canadian journal of mathematics, Tome 62 (2010) no. 1, pp. 109-132. doi: 10.4153/CJM-2010-007-2
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[1] [1] H., Bercovici, Li, W. S. and D., Timotin, The Horn conjecture for sums of compact selfadjoint operators. http://arXiv.org/abs/0709.1088. Google Scholar

[2] [2] A. S., Buch, Eigenvalues of Hermitian matrices with positive sum of bounded rank. Linear Algebra Appl. 418(2006), no. 2-3, 480-488. doi:10.1016/j.laa.2006.02.024 Google Scholar

[3] [3] M. D., Choi and P. Y., Wu, Convex combinations of projections. Linear Algebra Appl. 136(1990), 25-42. doi:10.1016/0024-3795(90)90019-9 Google Scholar

[4] [4] M. D., Choi and P. Y., Wu, Finite-rank perturbations of positive operators and isometries. Studia Math. 173(2006), no. 1, 73-79. doi:10.4064/sm173-1-5 Google Scholar

[5] [5] J., Day, W., So, and R. C., Thompson, The spectrum of a Hermitian matrix sum. Linear Algebra Appl. 280(1998), no. 2-3, 289-332. doi:10.1016/S0024-3795(98)10019-8 Google Scholar

[6] [6] K., Fan and G., Pall, Imbedding conditions for Hermitian and normal matrices. Canad. J. Math. 9(1957), 298-304. Google Scholar

[7] [7] W., Fulton, Eigenvalues, invariant factors, highest weights, and Schubert calculus. Bull. Amer.Math. Soc. 37(2000), no. 3, 209-249. doi:10.1090/S0273-0979-00-00865-X Google Scholar

[8] [8] A., Horn, Eigenvalues of sums of Hermitian matrices. Pacific J. Math 12(1962), 225-241. Google Scholar

[9] [9] A. A., Klyachko, Stable bundles, representation theory and Hermitian operators. Selecta Math. 4(1998), no. 2, 419-445. doi:10.1007/s000290050037 Google Scholar

[10] [10] A., Knutson and T., Tao, The honeycomb model of GLn(c) tensor products. I. Proof of the saturation conjecture. J. Amer. Math. Soc. 12(1999), no. 4, 1055-1090. doi:10.1090/S0894-0347-99-00299-4 Google Scholar

[11] [11] R. C., Thompson and L. J., Freede, On the eigenvalues of sums of Hermitian matrices. Linear Algebra and Appl. 4(1971), 369-376. doi:10.1016/0024-3795(71)90007-3 Google Scholar

[12] [12] R. C., Thompson and L. J., Freede, On the eigenvalues of sums of Hermitian matrices. II. Aequationes Math 5(1970), 103-115. doi:10.1007/BF01819276 Google Scholar

[13] [13] Weyl, H., Das asymtotische Verteilungsgesetz der Eigenwerte lineare partieller Differentialgleichungen. Math. Ann. 71(1912), no. 4, 441-479 doi:10.1007/BF01456804 Google Scholar

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