{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,21]],"date-time":"2026-04-21T10:54:35Z","timestamp":1776768875521,"version":"3.51.2"},"reference-count":22,"publisher":"American Mathematical Society (AMS)","issue":"214","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Math. Comp."],"abstract":"

\n A well-known result on spectral variation of a Hermitian matrix due to Mirsky is the following:\n \n Let\n \n \n \n A<\/mml:mi>\n A<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n and\n \n \n \n \n \n A<\/mml:mi>\n \n ~\n \n <\/mml:mo>\n <\/mml:mover>\n <\/mml:mrow>\n \\widetilde A<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n be two\n \n \n \n \n n<\/mml:mi>\n \n \u00d7\n \n <\/mml:mo>\n n<\/mml:mi>\n <\/mml:mrow>\n n\\times n<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n Hermitian matrices, and let\n \n \n \n \n \n \u03bb\n \n <\/mml:mi>\n 1<\/mml:mn>\n <\/mml:msub>\n \\lambda _1<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n , \u2026,\n \n \n \n \n \n \u03bb\n \n <\/mml:mi>\n \n n<\/mml:mi>\n <\/mml:mrow>\n <\/mml:msub>\n \\lambda _{n}<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n and\n \n \n \n \n \n \n \n \u03bb\n \n <\/mml:mi>\n \n ~\n \n <\/mml:mo>\n <\/mml:mover>\n <\/mml:mrow>\n 1<\/mml:mn>\n <\/mml:msub>\n \\widetilde \\lambda _1<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n , \u2026,\n \n \n \n \n \n \n \n \u03bb\n \n <\/mml:mi>\n \n ~\n \n <\/mml:mo>\n <\/mml:mover>\n <\/mml:mrow>\n \n n<\/mml:mi>\n <\/mml:mrow>\n <\/mml:msub>\n \\widetilde \\lambda _{n}<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n be their eigenvalues arranged in ascending order. Then\n \n \n \n \n \n \u2980<\/mml:mo>\n diag<\/mml:mi>\n \n \u2061\n \n <\/mml:mo>\n (<\/mml:mo>\n \n \n \u03bb\n \n <\/mml:mi>\n 1<\/mml:mn>\n <\/mml:msub>\n \n \u2212\n \n <\/mml:mo>\n \n \n \n \n \u03bb\n \n <\/mml:mi>\n \n ~\n \n <\/mml:mo>\n <\/mml:mover>\n <\/mml:mrow>\n 1<\/mml:mn>\n <\/mml:msub>\n ,<\/mml:mo>\n \n \u2026\n \n <\/mml:mo>\n ,<\/mml:mo>\n \n \n \u03bb\n \n <\/mml:mi>\n n<\/mml:mi>\n <\/mml:msub>\n \n \u2212\n \n <\/mml:mo>\n \n \n \n \n \u03bb\n \n <\/mml:mi>\n \n ~\n \n <\/mml:mo>\n <\/mml:mover>\n <\/mml:mrow>\n n<\/mml:mi>\n <\/mml:msub>\n )<\/mml:mo>\n \u2980<\/mml:mo>\n <\/mml:mrow>\n \n \u2264\n \n <\/mml:mo>\n \n \u2980<\/mml:mo>\n A<\/mml:mi>\n \n \u2212\n \n <\/mml:mo>\n \n \n A<\/mml:mi>\n \n ~\n \n <\/mml:mo>\n <\/mml:mover>\n <\/mml:mrow>\n \u2980<\/mml:mo>\n <\/mml:mrow>\n <\/mml:mrow>\n \\left \\Vvert \\operatorname {diag} (\\lambda _1- \\widetilde \\lambda _1,\\ldots ,\\lambda _n- \\widetilde \\lambda _n) \\right \\Vvert \\le \\left \\Vvert A-\\widetilde A \\right \\Vvert<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n for any unitarily invariant norm\n \n \n \n \n \u2980<\/mml:mo>\n \n \u22c5\n \n <\/mml:mo>\n \u2980<\/mml:mo>\n <\/mml:mrow>\n \\Vvert \\cdot \\Vvert<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n .\n <\/italic>\n In this paper, we generalize this to the perturbation theory for diagonalizable matrix pencils with real spectra. The much studied case of definite pencils is included in this.\n <\/p>","DOI":"10.1090\/s0025-5718-96-00699-0","type":"journal-article","created":{"date-parts":[[2002,7,26]],"date-time":"2002-07-26T18:14:28Z","timestamp":1027707268000},"page":"637-645","source":"Crossref","is-referenced-by-count":12,"title":["On perturbations of matrix pencils with real spectra. 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