Notes on matrices with diagonally dominant properties
Let \(A \in \mathbb{C}^{n \times n}\) and \(a_{ij}\) denote its entries. For \(1 \leq k \leq n\), define \(r_k(A)\) by \(r_k(A) = \sum_{j=1,~j \neq k}^n~|a_{ij}|\) and the set \(J(A)\) by \(J(A)=\{i: 1 \leq i \leq n\) and \(|a_{ii}| > r_i(A)\}\). We say that \(A\) is diagonally dominant if \(|a_{jj}| \geq r_j(A)\) for all \(j\). \(A\) is said to be stricly diagonally dominant if \(J(A)=\{1,2,\dots,n\}\). \(A\) is called strictly generalized diagonally dominant if there exists a diagonal matrix \(Y\) such that \(AY\) is strictly diagonally dominant. For a nonempty proper subset \(S\) of \(\{1,2,\dots,n\}\), define \(r_k^S(A)\) by \(r_k^S(A) = \sum_{j \in S,~j \neq k}^n~|a_{kj}|\). A partition \(S_1,S_2\) of \(\{1,2,\dots,n\}\) is called a separation of \(\{1,2,\dots,n\}\), denoted by \((S_1,S_2)\). For a separation \((S_1,S_2)\), define \(f_A^{S_1}:S_1 \times S_2 \longrightarrow \mathbb{R}\) by \(f_A^{S_1}(i,j) = (|a_{ii}| - r_i^{S_1}(A)) (|a_{jj}| - r_j^{S_2}(A)) - r_i^{S_2}(A) r_j^{S_1}(A)\). \(A\) is called generalized doubly diagonally dominant if \(J(A)\) is nonempty and there exists a separation \((S_1,S_2)\) such that \(f_A^{S_1} \geq 0\). \(A\) is called strictly generalized doubly diagonally dominant if strict inequality holds in the inequality as above. As usual, the directed graph of \(A \in \mathbb{C}^{n \times n}\) is the directed graph on \(n\) distinct points called its vertices, denoted by \(v_1,\dots,v_n\), such that there is a directed arc from \(v_i\) to \(v_j\) if and only if \(a_{ij} \neq 0\). \(A\) is called cyclically diagonally dominant if for every cycle \(\varGamma\) in the graph of \(A\), the following inequality holds: \(\prod_{v_i \in \varGamma} |a_{ii}| \geq \prod_{v_i \in \varGamma} r_i(A)\). The author analyzes the relationship between certain classes of matrices with variants of the diagonal dominance property. The following are the main results of the paper: A sufficient condition for a generalized doubly diagonally dominant matrix to be invertible, sufficient conditions for a matrix to be strictly generalized diagonally dominant and a sufficient condition for the invertibility of a cyclically diagonally dominant matrix. In any of the results, the usual assumption on the irreducibility of the matrices is not made.
strictly generalized diagonally dominant matrix
generalized doubly diagonally dominant matrix
scientific article; zbMATH DE number 5943572
scientific article