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Focal power. (English) Zbl 1066.15015
Consider the square matrix $$M=\begin{pmatrix} A&C\\B&D\end{pmatrix}$$ with square diagonal blocks. We call $$M$$ “focused on the $$(1,1)$$ position” or $$(1,1)$$-focused, if $$(M^k)_{1,1}=(M)_{1,1}^k$$. Examples of this are block triangular matrices for which the block $$A$$ is $$1\times 1$$ (for such matrices one has either $$C=0$$ or $$B=0$$). This property can be characterized by the vanishing of all moments $$CD^kB$$. The authors derive several properties of focused matrices as related to convergence, graphs, invariant subspaces and chains.
##### MSC:
 15A24 Matrix equations and identities 15A09 Theory of matrix inversion and generalized inverses 15B57 Hermitian, skew-Hermitian, and related matrices
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