'Society for Industrial & Applied Mathematics (SIAM)'
Publication date
29/11/2018
Field of study
International audienceLet A={Aij​}i,j∈I​, where I is an index set, be a doubly indexed family of matrices, where Aij​ is ni​×nj​. For each i∈I, let Vi​ be an ni​-dimensional vector space. We say A is {\em reducible in the coupled sense} if there exist subspaces, Ui​⊆Vi​, with Uiâ€‹î€ ={0} for at least one i∈I, and Uiâ€‹î€ =Vi​ for at least one i, such that Aij​(Uj​)⊆Ui​ for all~i,j. Let B={Bij​}i,j∈I​ also be a doubly indexed family of matrices, where Bij​ is mi​×mj​. For each i∈I, let Xi​ be a matrix of size ni​×mi​. Suppose Aij​Xj​=Xi​Bij​ for all~i,j. We prove versions of Schur's Lemma for A,B satisfying coupled irreducibility conditions. We also consider a refinement of Schur's Lemma for sets of normal matrices and prove corresponding versions for A,B satisfying coupled normality and coupled irreducibility conditions