Explicit inverse of tridiagonal matrix with applications in autoregressive modeling
Provides a theoretical result for a specific matrix class used in econometrics and spatial modeling, but the contribution is incremental as it extends known inversion techniques.
The paper derives an explicit inverse for a class of symmetric tridiagonal matrices that are almost Toeplitz, with applications in autoregressive modeling and cubic spline interpolation. The inverse is expressed via Chebyshev polynomials, and properties like row sum bounds and trace limits are provided.
We present the explicit inverse of a class of symmetric tridiagonal matrices which is almost Toeplitz, except that the first and last diagonal elements are different from the rest. This class of tridiagonal matrices are of special interest in complex statistical models which uses the first order autoregression to induce dependence in the covariance structure, for instance, in econometrics or spatial modeling. They also arise in interpolation problems using the cubic spline. We show that the inverse can be expressed as a linear combination of Chebyshev polynomials of the second kind and present results on the properties of the inverse, such as bounds on the row sums, the trace of the inverse and its square, and their limits as the order of the matrix increases.