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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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