An Eigenvector Test for Inflation Matrices and ZME-Matrices
Jeffrey L. Stuart · SIAM Journal on Matrix Analysis and Applications · 1989
It is shown that a matrix A is of the form $A = B \times \times U + \rho G(V)$, where U is an inflator and $ \times \times $ is the inflation product, if and only if A has a row and a column eigenvector for some eigenvalue such that the eigenvectors satisfy a simple restriction on their supports. This test is extended to recover the inflation sequence for a ZME-matrix. These results imply that the maximal eigenvalue (and hence spectral radius) of a ZME-matrix is the maximum of the maximal eigenvalues of its $2 \times 2$ principal submatrices. Additionally, it is shown that for every inflation sequence, there exists an equivalent normalized inflation sequence.