function A = clement(n, k)
%CLEMENT   Clement matrix - tridiagonal with zero diagonal entries.
%          CLEMENT(N, K) is a tridiagonal matrix with zero diagonal entries
%          and known eigenvalues.  It is singular if N is odd.  About 64
%          percent of the entries of the inverse are zero.  The eigenvalues
%          are plus and minus the numbers N-1, N-3, N-5, ..., (1 or 0).
%          For K = 0 (the default) the matrix is unsymmetric, while for
%          K = 1 it is symmetric.
%          CLEMENT(N, 1) is diagonally similar to CLEMENT(N).

%          Similar properties hold for TRIDIAG(X,Y,Z) where Y = ZEROS(N,1).
%          The eigenvalues still come in plus/minus pairs but they are not
%          known explicitly.
%
%          References:
%          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.

if nargin == 1, k = 0; end

n = n-1;

x = n:-1:1;
z = 1:n;

if k == 0
   A = diag(x, -1) + diag(z, 1);
else
   y = sqrt(x.*z);
   A = diag(y, -1) + diag(y, 1);
end