Bounds for the infinity norm of the inverse for certain \(M\)- and \(H\)-matrices
Let \(\bigcup_{i=1}^n M_i\) be a partition of the index set \(\{1,\dots, n\}\) into disjoint nonempty subsets. For an \(m\times m\) complex matrix \(A\), let \(r_i(A)\) be the \(i\)th row sum of the entries of \(A\), denote \(A_{ij}=A(M_i,M_j)\), for \(i,j=1,\dots,n\), the matrix that lies in the rows of \(A\) indexed by \(M_i\) and the columns indexed by \(M_j\), and consider the \(n\times n\) matrix \(A^{(k_1,k_2,\dots,k_n)}=(r_{k_i}(A_{ij}))\). The author shows that if \(A\) is a partitioned \(M\)-matrix, then it is nonsingular, and \(A^{-1}\) verifies the inequalities \[ \min_{k_1,\dots,k_n}\| (A^{(k_1,\dots,k_n)})^{-1}\| _\infty \leqslant \| A^{-1}\| _\infty \leqslant \max_{k_1,\dots,k_n}\| (A^{(k_1,\dots,k_n)})^{-1}\| _\infty . \] A condition for the equality case is presented. An analogous result is established for an upper bound for partitioned \(H\)-matrices and the main result is compared with others already known.
scientific article; zbMATH DE number 5382410
scientific article