zbMATH — the first resource for mathematics

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 \(A\).

15A42 Inequalities involving eigenvalues and eigenvectors
15B48 Positive matrices and their generalizations; cones of matrices
Full Text: DOI
[1] Bose, T; Chen, M.-Q; Joo, K.S; Xu, G.-F, Stability of two-dimensional discrete systems with periodic coefficients, IEEE trans. circuits and systems, part II: analog and digital signal processing, 45, 7, 839-847, (1998) · Zbl 0989.94015
[2] Horn, R; Johnson, C, Matrix analysis, (1985), Cambridge University Press Cambridge · Zbl 0576.15001
[3] Varga, R, Matrix iterative analysis, (2000), Springer Berlin · Zbl 0998.65505
[4] You, Z.-Y, Block estimation of the spectral radius of a matrix, Numerical math. J. Chinese univ., 1, 1, 129-130, (1979) · Zbl 0432.65021
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.