On elementary divisors perturbation of nonnegative matrices
A theorem from \textit{W. Guo} [ibid. 266, 261--270 (1997; Zbl 0903.15003)] concerning the spectra of nonnegative matrices is extended to elementary divisors. One main result states that if \(\Lambda=\{\lambda_1,\lambda_2,\dots,\lambda_n\}\) is the spectrum of an \(n\times n\) nonnegative matrix \(A\) with constant row sums, with real \(\{\lambda_1>\lambda_2\geq\dots\geq\lambda_p\}\), \(2\leq p\leq n\), and elementary divisors \(\lambda-\lambda_1\), \((\lambda-\lambda_2)^{n_2},\dots,(\lambda-\lambda_k)^{n_k}\), \(n_2+\cdots+n_k=n-1\) then, for any \(t>0\), \(\Lambda_t=\{\lambda_1+t,\lambda_2+t,\dots,\lambda_n\}\) is also the spectrum of a nonnegative matrix \(B\) with constant row sums and elementary divisors \(\lambda-\lambda_1-t,\;\lambda-\lambda_2-t,\;(\lambda-\lambda_2)^{n_2-1},\;(\lambda-\lambda_3)^{n_3}, \dots,(\lambda-\lambda_k)^{n_k}\). A similar result is provided with \(\lambda-\lambda_2+t\) instead \(\lambda-\lambda_2-t\). The matrix \(B\) is constructed by means of two rank one perturbations. Two examples illustrates the results.
scientific article; zbMATH DE number 5649994
scientific article