On a result of J.J. Sylvester

For any algebraically closed field F and any two square matrices A,B over F, Sylvester (1884) [8] and Cecioni (1910) [1] showed that AX=XB implies X=0 if and only if A and B have no common eigenvalue. It is proved that a third equivalent statement is that, for any given polynomials f, g in F[t], the...

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Bibliographic Details
Published inLinear algebra and its applications Vol. 505; pp. 361 - 366
Main Author Drazin, Michael P.
Format Journal Article
LanguageEnglish
Published Elsevier Inc 15.09.2016
Subjects
Associative algebras
Associative rings
Core-nilpotent decomposition
Eigenvalues
Fitting's lemma
Generalized inverses
Matrix polynomials
Semigroups
Fitting's lemma
Core-nilpotent decomposition
Associative rings
Generalized inverses
15A09
16–XX
15A18
15A24
20Mxx
Eigenvalues
Associative algebras
Semigroups
Matrix polynomials
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Summary:For any algebraically closed field F and any two square matrices A,B over F, Sylvester (1884) [8] and Cecioni (1910) [1] showed that AX=XB implies X=0 if and only if A and B have no common eigenvalue. It is proved that a third equivalent statement is that, for any given polynomials f, g in F[t], there exists h in F[t] such that f(A)=h(A) and g(B)=h(B). Corresponding results hold also for any finite set of square matrices over F, and these lead to a new property of all associative rings and algebras (even over arbitrary fields) with 1.
ISSN:0024-3795
1873-1856
DOI:10.1016/j.laa.2016.05.007