The spectral decomposition of near-Toeplitz tridiagonal matrices

Some properties of near-Toeplitz tridiagonal matrices with specific perturbations in the first and last main diagonal entries are considered. Applying the relation between the determinant and Chebyshev polynomial of the second kind, we first give the explicit expressions of determinant and characteristic polynomial, then eigenvalues are shown by finding the roots of the characteristic polynomial, which is due to the zeros of Chebyshev polynomial of the first kind, and the eigenvectors are obtained by solving symmetric tridiagonal linear systems in terms of Chebyshev polynomial of the third kind or the fourth kind. By constructing the inverse of the transformation matrices, we give the spectral decomposition of this kind of tridiagonal matrices. Furthermore, the inverse (if the matrix is invertible), powers and a square root are also determined. Keywords—Tridiagonal matrices, Spectral decomposition, Powers, Inverses, Chebyshev polynomials