### The inverse of the positive definite symmetric tridiagonal Toeplitz matrix, interactive visualization

This program is written by Egor Maximenko and Gabino Sánchez Arzate.
The formulas used in this program are based on the article by William F. Trench (1985): Explicit inversion formulas for Toeplitz band matrices. DOI: 10.1137/0606054

On this page we construct and visualize the inverse matrix T−1, where T is the tridiagonal real symmetric Toeplitz matrix of order n, with generating symbol g(x) = 4 (sin(x/2))2 + α.
We suppose that α > 0. In this case the matrix T is positive definite.

The first column of T is 2 + α, −1, 0, …, 0.

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

α = 0.5

n = 8

In this example T−1 = S − H, where S is a Toeplitz matrix and H is a Hankel matrix.
Next picture shows T−1, S, and H. All their entries are positive. Black color corresponds to the maximum values.

 rect.densityrect { fill: yellow; stroke: black; stroke-width: 1; } = rect.densityrect { fill: yellow; stroke: black; stroke-width: 1; } − rect.densityrect { fill: yellow; stroke: black; stroke-width: 1; }

All these matrices are symmetric and persymmetric. Next picture shows the first column of S: 