SCHUR'S LEMMA FOR COUPLED REDUCIBILITY AND COUPLED NORMALITY.

Let 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 reducible in the coupled sense if there exist subspaces, Ui ⊆ Vi, with Ui ≠{0} for at least one i∈I, and Ui ≠ 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 times mj. For each i∈I, let Xi be a matrix of size ni times mi. Suppose Aij} Xj = Xi Bij for all i, j. We prove versions of Schur's lemma for 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 A, mathcal B satisfying coupled normality and coupled irreducibility conditions. [ABSTRACT FROM AUTHOR]