{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,8,22]],"date-time":"2026-08-22T07:10:56Z","timestamp":1787382656214,"version":"3.56.0"},"reference-count":22,"publisher":"Society for Industrial & Applied Mathematics (SIAM)","issue":"1","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["SIAM J. Matrix Anal. Appl."],"published-print":{"date-parts":[[1996,1]]},"abstract":"<jats:p>Let A be an $n \\times n$ unitary matrix, and let the columns of an $n \\times l\\,( l &lt; n )$ matrix $\\tilde X_1 $ form an orthonormal basis for an approximate eigenspace $\\tilde{\\mathcal{X}}_1 $ of A. Then there are two problems: How near is $\\tilde{\\mathcal{X}}_1 $ to an eigenspace of A? How can we make use of the $l \\times l$ matrix $\\tilde X_1^H A\\tilde X_1 $ to get l approximate eigenvalues of A? This paper gives solutions to these problems. In particular, this paper reveals such a fact: One can use the eigenvalues of the unitary polar factor of $\\tilde X_1^H A\\tilde X_1 $ (or the eigenvalues of the matrix $\\tilde X_1^H A\\tilde X_1 $) as l approximate eigenvalues of A, and the precision of the eigenvalues of the unitary polar factor of $\\tilde X_1^H A\\tilde X_1 $ (or the eigenvalues of $\\tilde X_1^H A\\tilde X_1 $) as l approximate eigenvalues of A is higher than that of $\\tilde{\\mathcal{X}}_1 $ as an approximate eigenspace of A.<\/jats:p>","DOI":"10.1137\/s0895479894270014","type":"journal-article","created":{"date-parts":[[2005,2,27]],"date-time":"2005-02-27T07:15:07Z","timestamp":1109488507000},"page":"69-82","source":"Crossref","is-referenced-by-count":1,"title":["Residual Bounds on Approximate Solutions for the Unitary Eigenproblem"],"prefix":"10.1137","volume":"17","author":[{"given":"Ji-Guang","family":"Sun","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"351","published-online":{"date-parts":[[2012,2,17]]},"reference":[{"key":"R1","doi-asserted-by":"publisher","DOI":"10.1080\/03081088408817578"},{"key":"R2","doi-asserted-by":"publisher","DOI":"10.1007\/BF02227441"},{"key":"R3","doi-asserted-by":"publisher","DOI":"10.1016\/0024-3795(91)90402-I"},{"key":"R4","doi-asserted-by":"publisher","DOI":"10.1016\/0024-3795(93)90469-5"},{"key":"R5","doi-asserted-by":"publisher","DOI":"10.1073\/pnas.37.11.760"},{"key":"R6","doi-asserted-by":"publisher","DOI":"10.1090\/S0002-9939-1955-0067841-7"},{"key":"R7","volume-title":"Matrix computations","author":"Golub G.","year":"1989"},{"key":"R8","doi-asserted-by":"publisher","DOI":"10.1016\/0377-0427(86)90169-X"},{"key":"R9","doi-asserted-by":"publisher","DOI":"10.1007\/BF01386438"},{"key":"R10","doi-asserted-by":"publisher","DOI":"10.1137\/0907079"},{"key":"R11","doi-asserted-by":"publisher","DOI":"10.1215\/S0012-7094-53-02004-3"},{"key":"R12","doi-asserted-by":"publisher","DOI":"10.1017\/CBO9780511810817"},{"key":"R13","doi-asserted-by":"publisher","DOI":"10.1137\/0719030"},{"key":"R14","volume-title":"The symmetric eigenvalue problem","author":"Parlett Beresford N.","year":"1980"},{"key":"R15","doi-asserted-by":"publisher","DOI":"10.1137\/1015095"},{"key":"R16","doi-asserted-by":"publisher","DOI":"10.1137\/1019104"},{"key":"R17","volume-title":"Matrix perturbation theory","author":"Stewart G. W.","year":"1990"},{"key":"R18","doi-asserted-by":"publisher","DOI":"10.1007\/BF01385798"},{"key":"R19","doi-asserted-by":"publisher","DOI":"10.1007\/BF01385757"},{"key":"R20","unstructured":"J.G. Sun,  On the variation of the spectrum of a normal matrix, Report, UMINF-94.06, ISSN-0348-0542, Department of Computing Science, Ume\u00e5 University,  1994"},{"key":"R21","first-page":"286","volume":"1","author":"von Neumann J.","year":"1937","journal-title":"Tomsk Univ. Rev."},{"key":"R22","volume-title":"The algebraic eigenvalue problem","author":"Wilkinson J. H.","year":"1965"}],"container-title":["SIAM Journal on Matrix Analysis and Applications"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/epubs.siam.org\/doi\/pdf\/10.1137\/S0895479894270014","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,8,21]],"date-time":"2026-08-21T13:34:40Z","timestamp":1787319280000},"score":1,"resource":{"primary":{"URL":"https:\/\/epubs.siam.org\/doi\/10.1137\/S0895479894270014"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[1996,1]]},"references-count":22,"journal-issue":{"issue":"1","published-print":{"date-parts":[[1996,1]]}},"alternative-id":["10.1137\/S0895479894270014"],"URL":"https:\/\/doi.org\/10.1137\/s0895479894270014","relation":{},"ISSN":["0895-4798","1095-7162"],"issn-type":[{"value":"0895-4798","type":"print"},{"value":"1095-7162","type":"electronic"}],"subject":[],"published":{"date-parts":[[1996,1]]}}}