Asymptotics of the eigenvalues for exponentially parameterized pentadiagonal matrices

Summary Let P ( t ) be an n  ×  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  →  ∞ . 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  →  ∞ , 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.