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Journal Articles SIAM Journal on Matrix Analysis and Applications Year : 2019

Schur's Lemma for Coupled Reducibility and Coupled Normality

Abstract

Let $\mathcal A = \{A_{ij} \}_{i, j \in \mathcal I}$, where $\mathcal I$ is an index set, be a doubly indexed family of matrices, where $A_{ij}$ is $n_i \times n_j$. For each $i \in \mathcal I$, let $\mathcal V_i$ be an $n_i$-dimensional vector space. We say $\mathcal A$ is {\em reducible in the coupled sense} if there exist subspaces, $\mathcal U_i \subseteq \mathcal V_i$, with $\mathcal U_i \neq \{0\}$ for at least one $i \in \mathcal I$, and $\mathcal U_i \neq \mathcal V_i$ for at least one $i$, such that $A_{ij} (\mathcal U_j) \subseteq \mathcal U_i$ for all~$i, j$. Let $\mathcal B = \{B_{ij} \}_{i, j \in \mathcal I}$ also be a doubly indexed family of matrices, where $B_{ij}$ is $m_i \times m_j$. For each $i \in \mathcal I$, let $X_i$ be a matrix of size $n_i \times m_i$. Suppose $A_{ij} X_j = X_i B_{ij}$ 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.
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Dates and versions

hal-02305142 , version 1 (03-10-2019)

Identifiers

Cite

Dana Lahat, Christian Jutten, Helene Shapiro. Schur's Lemma for Coupled Reducibility and Coupled Normality. SIAM Journal on Matrix Analysis and Applications, 2019, 40 (3), pp.998-1021. ⟨10.1137/18M1232462⟩. ⟨hal-02305142⟩
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