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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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