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New lower bounds on eigenvalue of the Hadamard product of an \(M\)-matrix and its inverse. (English) Zbl 1163.15019

The main results which improve some earlier ones are the following. Let \(A= (a_{ij}) \in\mathbb R^{n \times n} \) be an \(M\)-matrix and \(\tau (A \circ A^{-1})\) be the minimum eigenvalue of the Hadamard product of \(A\) and \(A^{-1}\). Then \(\tau (A \circ A^{-1}) \geqslant \min _i\{{1-\frac{1} {{a_{ii}}}\sum_{j \neq i} {|{a_{ji} }|m_{ji}}}\}\), where \(m_{ji}= \frac{{|{a_{ji}}|+ \sum_{k \neq j,i} {|{a_{jk}}|r_i }}} {{a_{jj}}}\), \(j \neq i\), \(j\in\mathbb N\); \(r_i= \max _{l \neq i} \frac{{|{a_{li}}|}} {{|{a_{ll}}|- \sum_{k\neq l,i} {|{a_{lk}}|}}}\). In addition let \(A^{-1}\) be doubly stochastic. Then \(\tau(A\circ A^{-1}) \geqslant \min _i \{{\frac{{a_{ii}-m_i \sum_{k\neq i} {|{a_{ik}}|}}} {{1+ \sum_{j \neq i} {m_{ji}}}}}\}\).

MSC:

15A42 Inequalities involving eigenvalues and eigenvectors
15A15 Determinants, permanents, traces, other special matrix functions
15B48 Positive matrices and their generalizations; cones of matrices
15B51 Stochastic matrices
15A09 Theory of matrix inversion and generalized inverses
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References:

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