Estimates for bounds of some numerical characters of matrices (Q5926396)

Let \(A=(a_{ij})\) be an \(n\times n\) matrix and \(\alpha,\beta\subset\{1,\dots,n\}\), \(\alpha'= \{1,\dots,n\}\setminus\alpha\). Denote by \(A[\alpha|\beta]\) the submatrix of \(A\), whose rows and columns are indexed by \(\alpha\) and \(\beta\), respectively. Let \(A_\alpha=(a^\prime_{ij})\) be determined by the relations \(A_{\alpha}[\alpha|\alpha]=A[\alpha|\alpha]\), \(A_{\alpha}[\alpha'|\alpha'] =A[\alpha'|\alpha']\), \(A_{\alpha}[\alpha|\alpha^\prime]= (\|A[\alpha^\prime|\alpha]\|/ \|A[\alpha|\alpha^\prime]\|)^{1/2} A[\alpha,\alpha^\prime]\), \(A_{\alpha}[\alpha^\prime|\alpha]= (\|A[\alpha|\alpha^\prime]\|/ \|A[\alpha^\prime|\alpha]\|)^{1/2} A[\alpha^\prime|\alpha]\). The main result of this paper is the inequality NEWLINE\[NEWLINE \sum_{i=1}^{n}|\lambda_{i}|^2 \leq \sum_{i=1}^{n} \biggl(\sum_{j=1}^n|a'_{ij}|^2\Bigr)^{1/2} \Bigl(\sum_{j=1}^n|a'_{ji}|^2\biggr)^{1/2}. NEWLINE\]NEWLINE Here \((\lambda_1,\lambda_2,\dots,\lambda_n)\) denote the eigenvalues of \(A\) and \(\|\cdot \|\) is the Euclidean norm. NEWLINENEWLINENEWLINEIt is shown that this estimate is better than the classical Schur inequality (\(\sum|\lambda_{i}|\leq\|A\|^2\)) and some of its known improvements [e.g. \textit{P. Novosad} and \textit{R. Tovar}, Linear Algebra Appl. 31, 179-197 (1980; Zbl 0431.15019), \textit{R. Kreß, H. L. de Vries} and \textit{R. Wegmann}, Linear Algebra Appl. 8, 109-120 (1974; Zbl 0212.05301)]. NEWLINENEWLINENEWLINEThe main result is applied to obtain a new estimate of \({\text{rank}}(A)\) and to construct an algorithm that solves the following problem: For given irreducible \(A=(a_{ij})\), \(a_{ij}\geq 0\), find a diagonal matrix NEWLINE\[NEWLINED_{x}={\text{diag}}(x_1,x_2,\dots,x_{n}),\quad x_i>0,NEWLINE\]NEWLINE such that for \((b_{ij})=D_{x}^{-1}AD_{x}\) the equality \(\sum_{j}b_{ij}=\sum_{j}b_{ji}\) holds for each \(i=1,\dots,n\).