{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,7,24]],"date-time":"2026-07-24T14:54:07Z","timestamp":1784904847507,"version":"3.55.0"},"reference-count":0,"publisher":"Walter de Gruyter GmbH","issue":"3","license":[{"start":{"date-parts":[[2011,1,1]],"date-time":"2011-01-01T00:00:00Z","timestamp":1293840000000},"content-version":"unspecified","delay-in-days":0,"URL":"http:\/\/creativecommons.org\/licenses\/by-nc-nd\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Comput. Methods Appl. Math."],"published-print":{"date-parts":[[2011]]},"abstract":"<jats:title>Abstract<\/jats:title>\n                  <jats:p>We show that the recent tensor-train (TT) decompositions of matrices come up from its \n\t\t\trecursive Kronecker-product representations with a systematic use of common bases. The names \n\t\t\tTTM and QTT used in this case stress the relation with multilevel matrices or quantization \n\t\t\tthat increases artificially the number of levels. Then we investigate how the tensor-train \n\t\t\tranks of a matrix can be related to those of its inverse. In the case of a banded Toeplitz \n\t\t\tmatrix, we prove that the tensor-train ranks of its inverse are bounded above by 1+(l+u)^2, \n\t\t\twhere l and u are the bandwidths in the lower and upper parts of the matrix without the main \n\t\t\tdiagonal.<\/jats:p>","DOI":"10.2478\/cmam-2011-0022","type":"journal-article","created":{"date-parts":[[2020,6,4]],"date-time":"2020-06-04T07:49:59Z","timestamp":1591256999000},"page":"394-403","source":"Crossref","is-referenced-by-count":21,"title":["Tensor-Train Ranks for Matrices and Their Inverses"],"prefix":"10.2478","volume":"11","author":[{"given":"Ivan","family":"Oseledets","sequence":"first","affiliation":[{"name":"Institute of Numerical Mathematics, Russian Academy of Sciences, 8 Gubkin Street, Moscow, 119333, Russia."}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Eugene","family":"Tyrtyshnikov","sequence":"additional","affiliation":[{"name":"Institute of Numerical Mathematics, Russian Academy of Sciences, 8 Gubkin Street, Moscow, 119333, Russia."}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Nickolai","family":"Zamarashkin","sequence":"additional","affiliation":[{"name":"Institute of Numerical Mathematics, Russian Academy of Sciences, 8 Gubkin Street, Moscow, 119333, Russia."}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"374","published-online":{"date-parts":[[2011]]},"container-title":["Computational Methods in Applied Mathematics"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.degruyter.com\/view\/journals\/cmam\/11\/3\/article-p394.xml","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.degruyterbrill.com\/document\/doi\/10.2478\/cmam-2011-0022\/xml","content-type":"application\/xml","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.degruyterbrill.com\/document\/doi\/10.2478\/cmam-2011-0022\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,6,16]],"date-time":"2026-06-16T21:55:31Z","timestamp":1781646931000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.degruyterbrill.com\/document\/doi\/10.2478\/cmam-2011-0022\/html"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2011]]},"references-count":0,"journal-issue":{"issue":"3","published-online":{"date-parts":[[2011]]},"published-print":{"date-parts":[[2011]]}},"alternative-id":["10.2478\/cmam-2011-0022"],"URL":"https:\/\/doi.org\/10.2478\/cmam-2011-0022","relation":{},"ISSN":["1609-9389","1609-4840"],"issn-type":[{"value":"1609-9389","type":"electronic"},{"value":"1609-4840","type":"print"}],"subject":[],"published":{"date-parts":[[2011]]}}}