{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,8,21]],"date-time":"2026-08-21T18:28:03Z","timestamp":1787336883954,"version":"build-2736575974"},"reference-count":19,"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":[[2004,1]]},"abstract":"<jats:p>\n                    A polynomial p (with real coefficients) in noncommutative variables is matrix convex provided \\[ p(tX+(1-t)Y) \\le tp(X)+(1-t)p(Y) \\] for all $0 \\le t \\le 1$ and for all tuples X=(X\n                    <jats:sub>1<\/jats:sub>\n                    ,. . .,X\n                    <jats:sub>g<\/jats:sub>\n                    and Y=(Y\n                    <jats:sub>1<\/jats:sub>\n                    ,. . .,Y\n                    <jats:sub>g<\/jats:sub>\n                    ) of symmetric matrices on a common finite dimensional vector space of a sufficiently large dimension (depending upon p). The main result of this paper is that every matrix convex polynomial has degree two or less. More generally, the polynomial p has degree at most two if convexity holds only for all matrices X and Y in an \"open set.\" An analogous result for nonsymmetric variables is also obtained.\n                  <\/jats:p>\n                  <jats:p>Matrix convexity is an important consideration in engineering system theory. This motivated our work, and our results suggest that matrix convexity in conjunction with a type of \"system scalability\" produces surprisingly heavy constraints.<\/jats:p>","DOI":"10.1137\/s0895479803421999","type":"journal-article","created":{"date-parts":[[2004,7,15]],"date-time":"2004-07-15T21:03:14Z","timestamp":1089925394000},"page":"1124-1139","source":"Crossref","is-referenced-by-count":25,"title":["Convex Noncommutative Polynomials Have Degree Two or Less"],"prefix":"10.1137","volume":"25","author":[{"given":"J. William","family":"Helton","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Scott","family":"McCullough","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"351","published-online":{"date-parts":[[2006,7,31]]},"reference":[{"key":"R1","doi-asserted-by":"publisher","DOI":"10.1016\/0024-3795(79)90179-4"},{"key":"R2","doi-asserted-by":"publisher","DOI":"10.1090\/S0002-9947-1955-0082655-4"},{"key":"R3","doi-asserted-by":"publisher","DOI":"10.1007\/s00020-001-1147-7"},{"key":"R4","doi-asserted-by":"publisher","DOI":"10.1137\/1.9780898719833"},{"key":"R5","doi-asserted-by":"publisher","DOI":"10.2307\/3597203"},{"key":"R6","unstructured":"J. Helton, Manipulating matrix inequalities automatically, IMA Vol. Math. Appl., Vol. 134, Springer, New York, 2003, 237\u20132562005a:15019"},{"key":"R7","unstructured":"J. W. Helton and O. Merino,\n                      Sufficient conditions for optimization of matrix functions\n                      , in IEEE Conference on Decision and Control, Tampa, FL, IEEE Control Systems Society, Piscataway, NJ, 1998, pp. 1\u20135."},{"key":"R8","unstructured":"J. W. Helton, S. McCullough, and M. Putinar,\n                      Matrix representations for positive non\u2010commutative polynomials\n                      , Positivity, to appear;"},{"key":"R8","unstructured":"also available from http:\/\/www.math.ufl.edu\/\u223csam\/xandh.dvi(ps)(pdf)."},{"key":"R9","doi-asserted-by":"publisher","DOI":"10.1007\/BF01180403"},{"key":"R10","doi-asserted-by":"publisher","DOI":"10.1137\/S0036144599337290"},{"key":"R11","doi-asserted-by":"publisher","DOI":"10.1007\/BF01170633"},{"key":"R12","doi-asserted-by":"publisher","DOI":"10.1016\/S0024-3795(00)00285-8"},{"key":"R13","unstructured":"S. McCullough and M. Putinar,\n                      Non\u2010commutative sums of squares\n                      , Pacific J. Math., to appear;"},{"key":"R13","unstructured":"also available from http:\/\/www.math.ufl.edu\/\u223csam\/sos.dvi(ps)(pdf)."},{"key":"R14","doi-asserted-by":"publisher","DOI":"10.2307\/121083"},{"key":"R15","doi-asserted-by":"publisher","DOI":"10.1090\/conm\/253"},{"key":"R16","doi-asserted-by":"publisher","DOI":"10.2307\/3315013"},{"key":"R17","volume-title":"A unified algebraic approach to linear control design","author":"Skelton Robert","year":"1998"}],"container-title":["SIAM Journal on Matrix Analysis and Applications"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/epubs.siam.org\/doi\/pdf\/10.1137\/S0895479803421999","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,8,21]],"date-time":"2026-08-21T17:35:02Z","timestamp":1787333702000},"score":1,"resource":{"primary":{"URL":"https:\/\/epubs.siam.org\/doi\/10.1137\/S0895479803421999"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2004,1]]},"references-count":19,"journal-issue":{"issue":"4","published-print":{"date-parts":[[2004,1]]}},"alternative-id":["10.1137\/S0895479803421999"],"URL":"https:\/\/doi.org\/10.1137\/s0895479803421999","relation":{},"ISSN":["0895-4798","1095-7162"],"issn-type":[{"value":"0895-4798","type":"print"},{"value":"1095-7162","type":"electronic"}],"subject":[],"published":{"date-parts":[[2004,1]]}}}