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The Perron root of a weighted geometric mean of nonnegative matrices. (English) Zbl 0684.15007
For a nonnegative square matrix $$A=(\alpha_{ij}),$$ let $$\rho$$ (A) and M(A) denote the spectral radius of A and the matrix with diagonal entries $$\alpha_{ii}$$ and off-diagonal entries $$-\alpha_{ij}$$, respectively. Let $$\circ$$ denote the entry-wise Hadamard product of matrices and for $$\alpha >0,$$ let $$A^{(\alpha)}=(\alpha^{\alpha}_{ij}).$$ The authors prove that for k nonnegative n-by-n matrices $$A_ 1,...,A_ k$$ and for any k positive numbers $$\alpha_ 1,...,\alpha_ k$$ such that $$\alpha_ 1+...+\alpha_ k\geq 1$$ the inequality $$\rho (A_ 1^{(\alpha_ 1)}\circ...\circ A_ k^{(\alpha_ k)})\leq \rho (A_ 1)^{\alpha_ 1}...\rho (A_ k)^{\alpha_ k}$$ holds. They discuss in detail the nontrivial equality case and convexity of the function $$(\alpha_ 1,...,\alpha_ k)\to \rho (A_ 1^{(\alpha_ 1)}\circ...\circ A_ k^{(\alpha_ k)}).$$ These results generalize the earlier results of A. J. Schwenk [Linear Algebra Appl. 75, 257-265 (1986; Zbl 0654.15011)] and S. Karlin and F. Ost [ibid. 68, 47-65 (1985; Zbl 0575.15006)]. They obtained also the dual inequality fo the least eigenvalues of M(A) and some other interesting inequalities or new proofs of known results.
Reviewer: Z.Dostal

##### MSC:
 15B48 Positive matrices and their generalizations; cones of matrices 15A45 Miscellaneous inequalities involving matrices
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##### References:
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