Horn's problem – to find the support of the spectrum of eigenvalues of the sum of two by Hermitian matrices whose eigenvalues are known – has been solved by Klyachko and by Knutson and Tao. Here the probability distribution function (PDF) of the eigenvalues of is explicitly computed for low values of , for and uniformly and independently distributed on their orbit, and confronted to numerical experiments. Similar considerations apply to skew-symmetric and symmetric real matrices under the action of the orthogonal group. In the latter case, where no analytic formula is known in general and we rely on numerical experiments, curious patterns of enhancement appear.
Accepté le :
Publié le :
DOI : 10.4171/aihpd/56
Publié le :
DOI : 10.4171/aihpd/56
Classification :
15-XX, 60-XX
Mots-clés : Horn problem, Harish-Chandra integrals
Mots-clés : Horn problem, Harish-Chandra integrals
@article{AIHPD_2018__5_3_309_0, author = {Zuber, Jean-Bernard}, title = {Horn's problem and {Harish-Chandra's} integrals. {Probability} density functions}, journal = {Annales de l{\textquoteright}Institut Henri Poincar\'e D}, pages = {309--338}, volume = {5}, number = {3}, year = {2018}, doi = {10.4171/aihpd/56}, mrnumber = {3835548}, zbl = {1397.15008}, language = {en}, url = {http://archive.numdam.org/articles/10.4171/aihpd/56/} }
TY - JOUR AU - Zuber, Jean-Bernard TI - Horn's problem and Harish-Chandra's integrals. Probability density functions JO - Annales de l’Institut Henri Poincaré D PY - 2018 SP - 309 EP - 338 VL - 5 IS - 3 UR - http://archive.numdam.org/articles/10.4171/aihpd/56/ DO - 10.4171/aihpd/56 LA - en ID - AIHPD_2018__5_3_309_0 ER -
Zuber, Jean-Bernard. Horn's problem and Harish-Chandra's integrals. Probability density functions. Annales de l’Institut Henri Poincaré D, Tome 5 (2018) no. 3, pp. 309-338. doi : 10.4171/aihpd/56. http://archive.numdam.org/articles/10.4171/aihpd/56/
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