The spectrum of a glued matrix

Abstract

In this paper, we analyze the distribution of the eigenvalues of glued tridiagonal matrices. Such matrices provide a useful class of test matrix because, despite being unreduced, a glued matrix can have some eigenvalues agreeing to hundreds of decimal places. A glued matrix can be obtained from a direct sum of p copies of an unreduced symmetric tridiagonal matrix T by modifying the junctions, in one of two ways, so that the new matrix has no zero off-diagonal entries. We exhibit, in gradually increasing detail, how width and placement of the eigenvalue clusters of a glued matrix depend on T, on p, and on the strength of the glue γ.