I am looking for analytic expressions for the eigenvalues of matrices of the form
$$A = \begin{bmatrix} 6 & -4 & 1 & 0 & 0 & 0 & 0 \\ -4 & 6 & -4 & 1 & 0 & 0 & 0 \\ 1 & -4 & 6 & -4 & 1 & 0 & 0 \\ 0 & 1 & -4 & 6 & -4 & 1 & 0 \\ 0 & 0 & 1 & -4 & 6 & -4 & 1 \\ 0 & 0 & 0 & 1 & -4 & 5 & -2 \\ 0 & 0 & 0 & 0 & 1 & -2 & 1 \\ \end{bmatrix}$$
for arbitrary dimensions. The displayed $(7 \times 7)$ dimension is chosen for expositional convenience only.
The matrix arises as follows. Let $U$ denote the lower triangular matrix with ones on and below the diagonal. In principle, I am interested in the eigenvalues of $B = (U^2)'U^2$. It seems easiest, however, to find the reciprocals as the eigenvalues of $A = B^{-1}$. Clearly, the matrix $A$ is almost equivalent to a symmetric pentadiagonal Toeplitz matrix, with the exception of the last two rows and columns. Perhaps it is possible to first find the eigenvalues of
$$A^* = \begin{bmatrix} 6 & -4 & 1 & 0 & 0 \\ -4 & 6 & -4 & 1 & 0 \\ 1 & -4 & 6 & -4 & 1 \\ 0 & 1 & -4 & 6 & -4 \\ 0 & 0 & 1 & -4 & 6 \\ \end{bmatrix}$$
although I am not sure how to extend these to $A$. Some background information that may be useful:
The $(n \times n)$ matrix $U'U$ (recall that we're interested in $(U^2)'U^2$) has eigenvalues whose reciprocals are given by
$$\lambda_i^{-1} = 4\sin^2\left(\frac{(2i-1)\pi}{2(2n+1)}\right),$$
as demonstrated in Akesson and Lehoczky (1998). One can numerically verify that
$$\lambda_i^{-2} = 16\sin^4\left(\frac{(2i-1)\pi}{2(2n+1)}\right)$$
seems to be a very accurate approximation of the reciprocal of the $i$-th eigenvalue of $(U^2)'U^2$, but it is not an exact solution. A circulant version of the matrix $A^*$ appears on page 3 of Wang and Long (2020), where it is stated that its eigenvalues are $16\sin^4\left(\frac{i\pi}{n}\right)$. Finally, theorem 2 in Elouafi (2011) provides a general characterization for the eigenvalues of symmetric pentadiagonal Toeplitz matrices in terms of the zeros of particular functions, but I struggle to obtain analytic expression for the eigenvalues of $A^*$ from this theorem (let alone for $A$).
I suspect that directly solving the system of equations $Av = \lambda v$ might correspond to solving a fourth order homogenous linear difference equation under suitable initial value conditions, although I am not sure this is a feasible route to follow.
References
Akesson and Lehoczky (1998) Discrete eigenfunction expansion of multi-dimensional Brownian motion and the Ornstein-Uhlenbeck process
Elouafi (2011) An eigenvalue localization theorem for pentadiagonal symmetric Toeplitz matrices
Wang and Long (2020) Multiple solutions of fourth-order functional difference equation with periodic boundary conditions
