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In this paper we present systematically analysis on the smallest eigenvalueof matrices associated with completely even functions (mod $r$). We obtainseveral theorems on the asymptotic behavior of the smallest eigenvalue ofmatrices associated with completely even functions (mod $r$). In particular, weget information on the asymptotic behavior of the smallest eigenvalue of thefamous Smith matrices. Finally some examples are given to demonstrate the mainresults.

Asymptotic behavior of the smallest eigenvalue of matrices associated with completely even functions (mod r)