{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,10,2]],"date-time":"2026-10-02T21:17:34Z","timestamp":1790975854615,"version":"4.1.0"},"reference-count":14,"publisher":"Society for Industrial & Applied Mathematics (SIAM)","issue":"4","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["SIAM J. Matrix Anal. Appl."],"published-print":{"date-parts":[[1995,10]]},"abstract":"<jats:p>In this paper we introduce a new representation of orthogonal matrices. We show that any orthogonal matrix can be represented in the form $Q = I - YSY^T $, which we call the basis-kernel representation of Q. In particular, we point out that the kernel S can be chosen to be triangular and that a familiar representation of an orthogonal matrix as a product of Householder matrices can be readily deduced from a basis-kernel representation with triangular kernel. We also show that there exists, in some sense, a minimal orthogonal transformation between two subspaces of same dimension, an important application of which is on block elimination problems. We explore how the basis Y determines the subspaces that Q acts on in a nontrivial fashion, and how S determines the way Q acts on this subspace. Especially, there is a canonical representation that explicitly shows that Q partitions ${\\bf R}^n $ into three invariant subspaces in which it acts as the identity, a reflector, and a rotator, respectively. We also present a generalized Cayley representation for arbitrary orthogonal matrices, which illuminates the degrees of freedom we have in choosing orthogonal matrices acting on a predetermined subspace.<\/jats:p>","DOI":"10.1137\/s0895479894276369","type":"journal-article","created":{"date-parts":[[2005,2,27]],"date-time":"2005-02-27T07:14:44Z","timestamp":1109488484000},"page":"1184-1196","source":"Crossref","is-referenced-by-count":19,"title":["A Basis-Kernel Representation of Orthogonal Matrices"],"prefix":"10.1137","volume":"16","author":[{"given":"Xiaobai","family":"Sun","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Christian","family":"Bischof","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"351","published-online":{"date-parts":[[2006,7,17]]},"reference":[{"key":"R1","volume-title":"LAPACK User's Guide","author":"Anderson E.","year":"1992"},{"key":"R2","unstructured":"C. Bischof, X. Sun,  On orthogonal block elimination, Tech. Report, MCS-P450-0794, Mathematics and Computer Science Division, Argonne National Laboratory, Argonne, IL,  1994"},{"key":"R3","doi-asserted-by":"publisher","DOI":"10.1137\/0912073"},{"key":"R4","doi-asserted-by":"publisher","DOI":"10.1145\/355984.355990"},{"key":"R5","doi-asserted-by":"publisher","DOI":"10.1137\/1.9781611971811"},{"key":"R6","volume-title":"Applications of the theory of matrices","author":"Gantmacher F. R.","year":"1959"},{"key":"R7","volume-title":"Matrix computations","author":"Golub Gene H.","year":"1989"},{"key":"R8","doi-asserted-by":"publisher","DOI":"10.1137\/0913042"},{"key":"R9","doi-asserted-by":"publisher","DOI":"10.1137\/0725014"},{"key":"R10","doi-asserted-by":"publisher","DOI":"10.1137\/0910005"},{"key":"R11","volume-title":"Introduction to matrix computations","author":"Stewart G. W.","year":"1973"},{"key":"R12","doi-asserted-by":"publisher","DOI":"10.1137\/0717034"},{"key":"R13","volume-title":"Matrix perturbation theory","author":"Stewart G. W.","year":"1990"},{"key":"R14","doi-asserted-by":"publisher","DOI":"10.1137\/0909010"}],"container-title":["SIAM Journal on Matrix Analysis and Applications"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/epubs.siam.org\/doi\/pdf\/10.1137\/S0895479894276369","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,8,21]],"date-time":"2026-08-21T13:39:02Z","timestamp":1787319542000},"score":1,"resource":{"primary":{"URL":"https:\/\/epubs.siam.org\/doi\/10.1137\/S0895479894276369"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[1995,10]]},"references-count":14,"journal-issue":{"issue":"4","published-print":{"date-parts":[[1995,10]]}},"alternative-id":["10.1137\/S0895479894276369"],"URL":"https:\/\/doi.org\/10.1137\/s0895479894276369","relation":{},"ISSN":["0895-4798","1095-7162"],"issn-type":[{"value":"0895-4798","type":"print"},{"value":"1095-7162","type":"electronic"}],"subject":[],"published":{"date-parts":[[1995,10]]}}}