The Spectral Structure of TN Matrices

This chapter highlights the spectral structure of TN matrices. By “spectral structure” this chapter refers to facts about the eigenvalues and eigenvectors of matrices in a particular class. In view of the well-known Perron-Frobenius theory that describes the spectral structure of general entrywise nonnegative matrices, it is not surprising that TN matrices have an important and interesting special spectral structure. Nonetheless, that spectral structure is remarkably strong, identifying TN matrices as a very special class among nonnegative matrices. All eigenvalues are nonnegative real numbers, and the sign patterns of the entries of the eigenvectors are quite special as well. This spectral structure is most apparent in the case of IITN matrices (that is, the classical oscillatory case originally described by Gantmacher and Krein).

THE GENERALIZED GOODWIN–STATON INTEGRAL

Some properties of the generalized Goodwin–Staton integral are derived. Explicit error bounds for the asymptotic expansion are determined. In addition, results are obtained for the oscillatory case and when logarithmic factors are present.