Tavakolipour, Hanieh; Shakeri, Fatemeh Asymptotics of the eigenvalues for exponentially parameterized pentadiagonal matrices. (English) Zbl 07286037 Numer. Linear Algebra Appl. 27, No. 6, e2330, 30 p. (2020). Summary: Let \(P(t)\) be an \(n\times n\) (complex) exponentially parameterized pentadiagonal matrix. In this article, using a theorem of Akian, Bapat, and Gaubert, we present explicit formulas for asymptotics of the moduli of the eigenvalues of \(P(t)\) as \(t\to\infty\). Our approach is based on exploiting the relation with tropical algebra and the weighted digraphs of matrices. We prove that this asymptotics tends to a unique limit or two limits. Also, for \(n-2\) largest magnitude eigenvalues of \(P(t)\) we compute the asymptotics as \(n\to\infty\), in addition to \(t\). When \(P(t)\) is also symmetric, these formulas allow us to compute the asymptotics of the \(2\)-norm condition number. The number of arithmetic operations involved, does not depend on \(n\). We illustrate our results by some numerical tests. MSC: 15A18 Eigenvalues, singular values, and eigenvectors 15A80 Max-plus and related algebras Keywords:asymptotics of eigenvalues; pentadiagonal matrix; tropical algebra; weighted digraph PDFBibTeX XMLCite \textit{H. Tavakolipour} and \textit{F. Shakeri}, Numer. Linear Algebra Appl. 27, No. 6, e2330, 30 p. (2020; Zbl 07286037) Full Text: DOI