Geršgorin variations. I: On a theme of Pupkov and Solov'ev
Let \(A\) be a square complex matrix. New variations by \textit{V. A. Pupkov} and \textit{V. N. Solov'ev} of the classic Gershgorin theorem appeared in [Zh. Vychisl. Mat. Mat. Fiz. 23, No. 6, 1304-1313 (1983; Zbl 0569.65030) and Izv. Akad. Nauk SSSR, Ser. Mat. 47, No. 6, 1285-1302 (1983; Zbl 0551.65017)], respectively. The present author generalizes their results and provides still more variations. For fixed indices \(i\) and \(k\), define \(R_i= \sum_{j\neq i}|a_{ij}|\) and \(C_{i,k}= \max_{|S|= k-1} \{\sum_{j\in S,j\neq i}|a_{ji}|\}\). Then \(A\) is nonsingular if (1) For each set \(S\) of indices, \(\sum_{i\in S}|a_{ii}|> \min\{\sum_{i\in S} R_i, \sum_{i\in S} C_{i,k}\}\), and (2) There exists an index \(k\) such that \(|S|= k\) implies \(\sum_{i\in S}|a_{ii}|> \sum_{i\in S} R_i\). Another result is: \(A\) is nonsingular if condition (1) holds and in addition. \((2')\) There exists an index \(k\) such that \(|S|= k\) implies \(\sum_{i\in S}|a_{ii}|\geq \sum_{i\in S} R_i\), and (3) For some \(i\), \(|a_{ii}|< R_i\). Condition (3) is curious, the opposite of a diagonal dominance type hypothesis. As a corollary, if for some \(\alpha\), \(0<\alpha< 1\), one has \(|a_{ii}|> \alpha R_i+(1- \alpha)C_{i,k}\) for all \(i\), and if conditions (2), or \((2')\) and (3), hold, then \(A\) is nonsingular. A similar theorem and corollary are given for irreducible \(A\).
scientific article; zbMATH DE number 1420039