Explicit inverse of tridiagonal matrix with applications in autoregressive modelling

Abstract

Abstract 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 is 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 modelling. 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.