The `Clement' matrix is a tridiagonal
matrix with zero diagonal entries
and known eigenvalues. The symmetric case is diagonally similar to the unsymmetric case.

If the order is odd, the matrix is singular, otherwise the inverse is known
and may be computed here.

Relevance

A test case for computations of matrix inverses and eigenvalues.
The eigenvalues
are ±(N-1), ±(N-3), ±(N-5), ..., (±1 or 0).
About 64% of the entries of the inverse (for even order) are zero.

P.A. Clement, A class of triple-diagonal matrices for test
purposes, SIAM Review, 1 (1959), pp. 50-52.

A. Edelman and E. Kostlan, The road from Kac's matrix to Kac's
random polynomials. In John~G. Lewis, editor, Proceedings of
the Fifth SIAM Conference on Applied Linear Algebra Society
for Industrial and Applied Mathematics, Philadelphia, 1994,
pp. 503-507.

O. Taussky and J. Todd, Another look at a matrix of Mark Kac,
Linear Algebra and Appl., 150 (1991), pp. 341-360.