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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
##### Keywords:
Nonnegative matrices; Spectral radius; Block matrices
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##### References:
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