## 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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