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Subdirect sums of doubly diagonally dominant matrices. (English) Zbl 1151.15025

The paper deals with problem of when the \(k\)-subdirect sum of matrices belongs to the same class. The authors discuss this for doubly diagonally dominant matrices and for \(S\)-strictly diagonally dominant matrices.
Let \(A\) and \(B\) be two square matrices of order \(n_1\) and \(n_2\), respectively, and let \(k\) be an integer such that \(1 \leq k \leq \min{(n_1,n_2)}\). Let \(A\) and \(B\) be partitioned as follows,
\[ A= \left( \begin{matrix} A_{11} & A_{12} \\ A_{21} & A_{22} \end{matrix} \right), \qquad B= \left( \begin{matrix} B_{11} & B_{12} \\ B_{21} & B_{22} \end{matrix} \right), \]
where \(A_{22}\) and \(B_{11}\) are square matrices of order \(k\). The square matrix of order \(n=n_1+n_2-k\) given by
\[ C=\left( \begin{matrix} A_{11} & A_{12} & 0 \\ A_{21} & A_{22}+B_{11}& B_{12} \\ 0 & B_{21} & B_{22} \end{matrix} \right) \]
is called the \(k\)-subdirect sum of \(A\) and \(B\) and denoted by \(C=A \oplus_k B\) A matrix \(A=(a_{ij}) \in\mathbb C^{n,n}\) it said to be doubly diagonally dominant (DDD) if
\[ | a_{ii}| | a_{jj}| \geq R_i(A)R_j(A), \quad i,j=1,2,\ldots,n, \;i \neq j, \]
where \(R_i(A)= \sum_{j=1, j \neq i}^n | a_{ij}| \). Given a nonempty subset \(S\) of \(N=\{ 1,2, \ldots,n\}\), matrix \(A\) it said to be \(S\)-strictly diagonally dominant (\(S\)-SDD) if
\[ \begin{alignedat}{2} | a_{ii}| &> R_i^S(A), & \forall i &\in S, \tag{i}\\ (| a_{ii}| -R_i^S(A))(| a_{jj}| -R_j^{\overline{S}}(A)) &> R_i^{\overline{S}}(A)R_j^S(A), \quad&\forall i &\in S,\;\forall j \in \overline{S} \tag{ii}\end{alignedat} \]
where \(R_i^S(A)= \sum_{j \in S, j \neq i} | a_{ij}| \) and \(\overline{S}\) the complement of \(S\) in \(N\).
The authors give sufficient conditions in order to guarantee that the \(k\)-subdirect sum of DDD matrices is also a DDD matrix. The same situation is analyzed for \(S\)-SDD matrices and new sufficient conditions are given.

MSC:

15B57 Hermitian, skew-Hermitian, and related matrices
65F10 Iterative numerical methods for linear systems
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