THE SPECTRUM OF A GLUED MATRIX

摘要

A glued matrix can be obtained from a direct sum of p copies of an unreducedsymmetric tridiagonal matrix T by modifying the junctions by a glue 'y, in one of two ways, so that thenew tridiagonal matrix has no zero off-diagonal entries. Despite being unreduced, a glued matrix canhave some eigenvalues agreeing to hundreds of decimal places. This makes glued matrices practicallyuseful as test matrices for tridiagonal eigensolvers such as inverse iteration and the MRRR algorithm.However, the eigenvalue distribution of a glued matrix is a fascinating subject of theoretical interestin its own right. By means of secular equations, this paper studies how width and placement of theeigenvalue clusters of a glued matrix depend on T, on p, and on 7. Interlacing properties and thequestion of eigenvalue repetition between T and a glued matrix are also investigated.