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An estimation of the spectral radius of a product of block matrices. (English) Zbl 1043.15012
Let \(C(r)= \lfloor C_{ij}\rfloor\), \(r= 1,2,\dots, R\), be block \(m\times m\) matrices, where \(C_{ij}(r)\) are nonnegative \(N_i\times N_j\) matrices for \(i,j= 1,2,\dots, m\). For each \(r\) let \(B(r)= \lfloor\| C_{ij}(r)\|\rfloor\) be the \(m\times m\) matrix, where \(\|\cdot\|\) denotes a consistent norm. The authors, among other results, give an estimation of the spectral radius of a product of block matrices. They show that \[ \rho\Biggl(\prod^R_{r=1} C(r)\Biggr)\leq \rho\Biggl(\prod^R_{r=1} B(r)\Biggr), \] where \(\rho(A)\) denotes the spectral radius of the matrix \(A\).

MSC:
15A42 Inequalities involving eigenvalues and eigenvectors
15B48 Positive matrices and their generalizations; cones of matrices
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