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

The eigenvalues λ_{n,1}, …, λ_{n,n} are numbered in the descending order.
For each j = 1, …, n, the eigenvalue λ_{n,j} is computed as
g(θ_{n,j}),
where θ_{n,j} is the unique solution of the equation
(n + 1) x + η(x) = jπ,
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 eigenvectorv_{n,j} associated to λ_{n,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 v_{n,j} is computed as v_{n,j,p} = sin(pjπ / (n + 1) +
(1 / 2 − p / (n + 1)) η(θ_{n,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.

Next picture shows a family portrait of the eigenvectors.
The color of the cell (p, j) corresponds to the value of the component p of a normalized eigenvector associated to λ_{n,j}.
For example, the column corresponds to the eigenvector shown above (after normalization).