Some New Lower Bounds for the Spread of a Nonnegative Matrix with a Zero Diagonal Element
Ram Asrey Rajput *
Department of Mathematics, Bundelkhand College Jhansi, Affiliated to Bundelkhand University, Jhansi, UP, 284001, India.
*Author to whom correspondence should be addressed.
Abstract
Let \(\mathbb{N}\)\(_n\) (with n ≥ 2) be the family of all nonnegative n × n matrices A = [aij], where a11 = 0 and the remaining entries aij ∈ [0, 1) with a spectral radius ρ(A) = 1. We can establish a lower bound for the additional spread s(A) ≥ \(\frac{k}{n-1}\), where k is the count of zero diagonal elements in matrix A. Furthermore, if matrix A possesses only two distinct eigenvalues, then it follows that s(A) ≥ \(\frac{n−2}{n−1}\). Additionally, we derived a few other lower bounds under a special family of matrices.
Keywords: Nonnegative matrix, spectral radius;, zero diagonal elements, lower bounds