Spectra of Certain Large Tridiagonal Matrices
Published In
Linear Algebra and its Application
Document Type
Citation
Publication Date
7-1-2018
Abstract
We characterize the eigenvalues and eigenvectors of a class of complex valued tridiagonal n by n matrices subject to arbitrary boundary conditions, i.e. with arbitrary elements on the first and last rows of the matrix. For large n, we show there are up to 4 eigenvalues, the so-called special eigenvalues, whose behavior depends sensitively on the boundary conditions. The other eigenvalues, the so-called regular eigenvalues vary very little as function of the boundary conditions. For large n, we determine the regular eigenvalues up to , O(n−2), and the special eigenvalues up to O ( κ n ) , for some κ ∈ ( 0 , 1 ) . . The components of the eigenvectors are determined up to O ( n − 1 ) . The matrices we study have important applications throughout the sciences. Among the most common ones are arrays of linear dynamical systems with nearest neighbor coupling, and discretizations of second order linear partial differential equations. In both cases, we give examples where specific choices of boundary conditions substantially influence leading eigenvalues, and therefore the global dynamics of the system.
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DOI
10.1016/j.laa.2018.03.005
Persistent Identifier
https://archives.pdx.edu/ds/psu/25411
Citation Details
Veerman, J. J. P., Hammond, D. K., & Baldivieso, P. E. (2018). Spectra of certain large tridiagonal matrices. Linear Algebra and its Applications, 548, 123-147.