{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,21]],"date-time":"2026-04-21T17:42:06Z","timestamp":1776793326156,"version":"3.51.2"},"reference-count":39,"publisher":"American Mathematical Society (AMS)","issue":"275","license":[{"start":{"date-parts":[[2011,12,13]],"date-time":"2011-12-13T00:00:00Z","timestamp":1323734400000},"content-version":"am","delay-in-days":365,"URL":"https:\/\/www.ams.org\/publications\/copyright-and-permissions"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Math. Comp."],"abstract":"

\n Let\n \n \n \n \n S<\/mml:mi>\n (<\/mml:mo>\n A<\/mml:mi>\n )<\/mml:mo>\n <\/mml:mrow>\n S(A)<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n denote the orbit of a complex or real matrix\n \n \n \n A<\/mml:mi>\n A<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n under a certain equivalence relation such as unitary similarity, unitary equivalence, unitary congruences etc. Efficient gradient-flow algorithms are constructed to determine the best approximation of a given matrix\n \n \n \n \n A<\/mml:mi>\n 0<\/mml:mn>\n <\/mml:msub>\n A_0<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n by the sum of matrices in\n \n \n \n \n S<\/mml:mi>\n (<\/mml:mo>\n \n A<\/mml:mi>\n 1<\/mml:mn>\n <\/mml:msub>\n )<\/mml:mo>\n ,<\/mml:mo>\n \n \u2026\n \n <\/mml:mo>\n ,<\/mml:mo>\n S<\/mml:mi>\n (<\/mml:mo>\n \n A<\/mml:mi>\n N<\/mml:mi>\n <\/mml:msub>\n )<\/mml:mo>\n <\/mml:mrow>\n S(A_1), \\dots , S(A_N)<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n in the sense of finding the Euclidean least-squares distance\n \n \\[\n \n \n \n min<\/mml:mo>\n \n \n {<\/mml:mo>\n <\/mml:mrow>\n <\/mml:mstyle>\n \n \n \u2016<\/mml:mo>\n <\/mml:mrow>\n <\/mml:mstyle>\n \n X<\/mml:mi>\n 1<\/mml:mn>\n <\/mml:msub>\n +<\/mml:mo>\n \n \u22ef\n \n <\/mml:mo>\n +<\/mml:mo>\n \n X<\/mml:mi>\n N<\/mml:mi>\n <\/mml:msub>\n \n \u2212\n \n <\/mml:mo>\n \n A<\/mml:mi>\n 0<\/mml:mn>\n <\/mml:msub>\n \n \n \u2016<\/mml:mo>\n <\/mml:mrow>\n <\/mml:mstyle>\n :<\/mml:mo>\n \n X<\/mml:mi>\n j<\/mml:mi>\n <\/mml:msub>\n \n \u2208\n \n <\/mml:mo>\n S<\/mml:mi>\n (<\/mml:mo>\n \n A<\/mml:mi>\n j<\/mml:mi>\n <\/mml:msub>\n )<\/mml:mo>\n ,<\/mml:mo>\n \u00a0<\/mml:mtext>\n j<\/mml:mi>\n =<\/mml:mo>\n 1<\/mml:mn>\n ,<\/mml:mo>\n \n \u2026\n \n <\/mml:mo>\n ,<\/mml:mo>\n N<\/mml:mi>\n \n \n }<\/mml:mo>\n <\/mml:mrow>\n <\/mml:mstyle>\n .<\/mml:mo>\n <\/mml:mrow>\n \\min \\Big \\{\\big \\|X_1+ \\cdots + X_N - A_0\\big \\|: X_j \\in S(A_j), \\ j = 1, \\dots , N\\Big \\}.<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n \\]\n <\/disp-formula>\n Connections of the results to different pure and applied areas are discussed.\n <\/p>","DOI":"10.1090\/s0025-5718-2010-02450-0","type":"journal-article","created":{"date-parts":[[2010,12,29]],"date-time":"2010-12-29T14:39:59Z","timestamp":1293633599000},"page":"1601-1621","source":"Crossref","is-referenced-by-count":4,"title":["Least-squares approximation by elements from matrix orbits achieved by gradient flows on compact lie groups"],"prefix":"10.1090","volume":"80","author":[{"given":"Chi-Kwong","family":"Li","sequence":"first","affiliation":[]},{"given":"Yiu-Tung","family":"Poon","sequence":"additional","affiliation":[]},{"given":"Thomas","family":"Schulte-Herbr\u00fcggen","sequence":"additional","affiliation":[]}],"member":"14","published-online":{"date-parts":[[2010,12,13]]},"reference":[{"key":"1","series-title":"Student Mathematical Library","isbn-type":"print","doi-asserted-by":"publisher","DOI":"10.1090\/stml\/022","volume-title":"An introduction to Lie groups and the geometry of homogeneous spaces","volume":"22","author":"Arvanitoyeorgos, Andreas","year":"2003","ISBN":"https:\/\/id.crossref.org\/isbn\/0821827782"},{"key":"2","series-title":"Fields Institute Communications","isbn-type":"print","doi-asserted-by":"publisher","DOI":"10.1090\/fic\/003","volume-title":"Hamiltonian and gradient flows, algorithms and control","volume":"3","year":"1994","ISBN":"https:\/\/id.crossref.org\/isbn\/0821802550"},{"key":"3","doi-asserted-by":"publisher","first-page":"79","DOI":"10.1016\/0024-3795(91)90021-N","article-title":"Dynamical systems that sort lists, diagonalize matrices, and solve linear programming problems","volume":"146","author":"Brockett, R. 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