Abstract
In the 1950s J. H. Wilkinson introduced two families of symmetric tridiagonal integer matrices. Most of the eigenvalues are close to diagonal entries. We develop the structure of their eigenvectors...
Highlights
The diagonal entries are given by diag(W2±m+1) = m m − 1 . . . 1 0 ±1 ±2 . . . ±m
W0(k) is the w0 vector pushed so that its central value 1 is on the same row as is the diagonal entry k in W∞−, namely row k in the new notation
This doubly infinite matrix obtained from W2+m+1 by letting m → ∞ retains its persymmetry and so each eigenpair of V∞ determines an eigenvector of W∞+ that is anti-symmetric about the 0 index and which has the same eigenvalue as that pair
Summary
Each matrix illustrated subtle difficulties in the automatic computation of eigenvectors and eigenvalues as described These matrices did not come from applications. Our analysis begins with the triangular factorization LDLt of Vm, the leading principal m × m submatrix of both families It is the rapid convergence, as m → ∞, of the pivots in D that drives the eigenvalues of all three matrices Vm, W2−m+1 and W2+m+1 towards integer values. The final section, the big picture, lets m reach infinity and reveals that W∞− has the integers Z for its spectrum and all eigenvectors are displaced versions of each other.
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Free) 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a call
