Spectral Properties of Tridiagonal k-Toeplitz Matrices

We derive the spectral properties of tridiagonal k-Toeplitz matrices in generality i.e. with non symmetric complex entries and any periodicity k. Previous work has highlighted some special spectral properties of real symmetric tridiagonal k-Toeplitz matrices and note that all square matrices have similarity transformation to tridiagonal form. Toeplitz matrices are used in convolution, discrete transforms and lumped physical systems, and it can be shown that every matrix is a product of Toeplitz matrices. We begin with numerical results of spectra of some special k-Toeplitz matrices as a motivation. This is followed by a derivation of spectral properties of a general tridiagonal k-Toeplitz matrix using three term recurrence relations and C - R,C - I kth order polynomial mappings. These relations establish a support for the limiting eigenvalue distribution of a tridiagonal Toeplitz matrix which has dimensions much larger than k. Numerical examples are used to graphically demonstrate theorems. As an addendum, we derive expressions for O(k) computation of the determinant of tridiagonal k-Toeplitz matrices of any dimension.