Eigenvalues and eigenvectors of Kac–Murdock–Szegő Toeplitz matrices (interactive visualization)

This program illustrates one example from our article (J. M. Bogoya Ramírez, A. Böttcher, S. Grudsky, E. A. Maximenko).
Some of the formulas and facts used in this program were previously proved by William F. Trench.

Kac–Murdock–Szegő Toeplitz matrices are real symmetric Toeplitz matrices generated by the symbol g(x) = (1 − ρ2) / (1 − 2 ρ cos(x) + ρ2).
The first column of the KMS matrix of order n is formed by the numbers 1, ρρ2, …, ρn−1.

Choose the value of the parameter ρ and the order n of the matrix:

     ρ = 0.5

     n = 8

Next picture shows the graph of the generating function g.
The ordinates of the yellow points are the eigenvalues of the corresponding n×n matrix.

1 g(π) g(0) π 2π θn,j λn,j

The eigenvalues λn,1, …, λn,n are numbered in the descending order.
For each j = 1, …, n, the eigenvalue λn,j is computed as g(θn,j), where θn,j is the unique solution of the equation
     (n + 1) x + η(x) = j π,
and the function η is defined by
     η(x) = 2 atan(ρ sin(x) / (1 − ρ cos(x))).



Next figure shows the components of an eigenvector vn,j associated to λn,j.
Recall that n = 8 (the value of n can be changed above).
Choose the index j of the eigenvalue/eigenvector pair:

     j = 1

For each p = 1, …, n, the p-st component of vn,j is computed as
     vn,j,p = sin(p j π / (n + 1) +  (1 / 2 − p / (n + 1)) η(θn,j)).
The thin gray curve is the graph of the smooth function defined by the same formula, with a continuous variable p.

The vector is ,
and the number of sign changes is ,
in accordance with results by William Trench.