{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,8,21]],"date-time":"2026-08-21T14:19:49Z","timestamp":1787321989448,"version":"3.56.0"},"reference-count":7,"publisher":"Society for Industrial & Applied Mathematics (SIAM)","issue":"2","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["SIAM J. Matrix Anal. Appl."],"published-print":{"date-parts":[[1996,4]]},"abstract":"<jats:p>The polar decomposition of a square matrix is a major step toward the singular value decomposition, and is an important device in its own right. Singular values of a matrix A, being the eigenvalues for a matrix closely related to A, generally cannot be computed by a finite process if only arithmetic operations and radicals are allowed. However, this consideration alone does not prove that the polar decomposition is not finitely computable.<\/jats:p>\n                  <jats:p>The problem of finite computability of the polar decomposition is not settled in this paper, but we do show it to be equivalent to the following simpler-looking problem. Suppose f is a real polynomial of degree $n &gt; 4$, and all the roots of f are distinct positive numbers. Denote by g a polynomial of the same degree whose zeros are the positive square roots of the zeros of f. Can this polynomial g always be computed finitely for a given polynomial f? In the Appendix we discuss one nontrivial situation where the polar decomposition can indeed be computed finitely.<\/jats:p>","DOI":"10.1137\/s0895479894274541","type":"journal-article","created":{"date-parts":[[2005,2,27]],"date-time":"2005-02-27T07:15:07Z","timestamp":1109488507000},"page":"348-354","source":"Crossref","is-referenced-by-count":4,"title":["Is the Polar Decomposition Finitely Computable?"],"prefix":"10.1137","volume":"17","author":[{"given":"Alan","family":"George","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Kh.","family":"Ikramov","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"351","published-online":{"date-parts":[[2012,2,17]]},"reference":[{"key":"R1","doi-asserted-by":"publisher","DOI":"10.1016\/0024-3795(90)90323-5"},{"key":"R2","volume-title":"The Theory of Matrices","author":"Gantmacher F. R.","year":"1974"},{"key":"R3","doi-asserted-by":"publisher","DOI":"10.1137\/0907079"},{"key":"R4","doi-asserted-by":"publisher","DOI":"10.1137\/0613044"},{"key":"R5","volume-title":"Solving least squares problems","author":"Lawson Charles L.","year":"1974"},{"key":"R6","volume-title":"The symmetric eigenvalue problem","author":"Parlett Beresford N.","year":"1980"},{"key":"R7","doi-asserted-by":"publisher","DOI":"10.1016\/0024-3795(93)90268-S"}],"container-title":["SIAM Journal on Matrix Analysis and Applications"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/epubs.siam.org\/doi\/pdf\/10.1137\/S0895479894274541","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,8,21]],"date-time":"2026-08-21T13:16:54Z","timestamp":1787318214000},"score":1,"resource":{"primary":{"URL":"https:\/\/epubs.siam.org\/doi\/10.1137\/S0895479894274541"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[1996,4]]},"references-count":7,"journal-issue":{"issue":"2","published-print":{"date-parts":[[1996,4]]}},"alternative-id":["10.1137\/S0895479894274541"],"URL":"https:\/\/doi.org\/10.1137\/s0895479894274541","relation":{},"ISSN":["0895-4798","1095-7162"],"issn-type":[{"value":"0895-4798","type":"print"},{"value":"1095-7162","type":"electronic"}],"subject":[],"published":{"date-parts":[[1996,4]]}}}