2018/11/20 by Dana Lahat, Christian Jutten, Lahat, Dana +3
Mathematics · #15A04 #15A21 #15A24 #15A99 #20C99 #FOS: Mathematics #Rings and Algebras (math.RA) #math.RA #msc:15A04 #msc:15A21 #msc:15A24 #msc:15A99 #msc:20C99
paper · pdf · doi:10.48550/arxiv.1811.08467
35 pages. Second version corrects some typos in the original submission and makes some changes in MSC classification numbers
arxiv created 2018/11/29 · arxiv updated 2018/12/03
Let \mathcal A = \Aij \i, j ∈ \mathcal I, where \mathcal I is an index set, be a doubly indexed family of matrices, where Aij is ni × nj. For each i ∈ \mathcal I, let \mathcal Vi be an ni-dimensional vector space. We say \mathcal A is reducible in the coupled sense if there exist subspaces, \mathcal Ui ⊆ \mathcal Vi, with \mathcal Ui ≠ \0\ for at least one i ∈ \mathcal I, and \mathcal Ui ≠ \mathcal Vi for at least one i, such that Aij (\mathcal Uj) ⊆ \mathcal Ui for all i, j. Let \mathcal B = \Bij \i, j ∈ \mathcal I also be a doubly indexed family of matrices, where Bij is mi × mj. For each i ∈ \mathcal 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 \mathcal A, \mathcal B satisfying coupled irreducibility conditions. We also consider a refinement of Schur's Lemma for sets of normal matrices and prove corresponding versions for \mathcal A, \mathcal B satisfying coupled normality and coupled irreducibility conditions.