{"title":"A note on multiplicative property of the Smith normal form","authors":"M. Marcus, E. E. Underwood","doi":"10.6028/JRES.076B.016","DOIUrl":null,"url":null,"abstract":"In [2; p. 33]1 the following interesting result appears: If A and Bare n-square matrices over a principal ideal domain Rand g.c.d. (det (A), det (B»= 1 then S(AB)= S(A)S(B) where -SeA) is the Smith normal form of A. The purpose of this note is to present a simple proof of the result that uses elementary properties of compound matrices. LEMMA. Let Q=diag (q., ... , qn), P=diag (p., ... , Pn), qllqj' Pllpj,j=l, ... , nand g.c.d. (Pb qJ)= 1, i, j= 1, . .. , n. Let U be an n-square matrix with the property that g.c.d. (Ull' U2!, _ .. , Unl)= g.c.d. (Ull' U12, ... , Utn)= 1. Then the g.c.d. of all the entries in QUP is Plqt. PROOF. Obviously PlqlIQUP, i.e., Ptql divides every entry of QUP. Write QUP= PlqtD. Suppose that plD where p is a prime. It is simple to see that the first row and column of Dare respectively","PeriodicalId":166823,"journal":{"name":"Journal of Research of the National Bureau of Standards, Section B: Mathematical Sciences","volume":"170 1","pages":"0"},"PeriodicalIF":0.0000,"publicationDate":"1972-07-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"5","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Journal of Research of the National Bureau of Standards, Section B: Mathematical Sciences","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.6028/JRES.076B.016","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
引用次数: 5
Abstract
In [2; p. 33]1 the following interesting result appears: If A and Bare n-square matrices over a principal ideal domain Rand g.c.d. (det (A), det (B»= 1 then S(AB)= S(A)S(B) where -SeA) is the Smith normal form of A. The purpose of this note is to present a simple proof of the result that uses elementary properties of compound matrices. LEMMA. Let Q=diag (q., ... , qn), P=diag (p., ... , Pn), qllqj' Pllpj,j=l, ... , nand g.c.d. (Pb qJ)= 1, i, j= 1, . .. , n. Let U be an n-square matrix with the property that g.c.d. (Ull' U2!, _ .. , Unl)= g.c.d. (Ull' U12, ... , Utn)= 1. Then the g.c.d. of all the entries in QUP is Plqt. PROOF. Obviously PlqlIQUP, i.e., Ptql divides every entry of QUP. Write QUP= PlqtD. Suppose that plD where p is a prime. It is simple to see that the first row and column of Dare respectively