function A = minij(n)
%MINIJ   Symmetric positive definite matrix MIN(i,j).
%        A = MINIJ(N) is the N-by-N symmetric positive definite matrix with
%        A(i,j) = MIN(i,j).
%        Properties, variations:
%        A has eigenvalues .25*sec^2(r*PI/(2*N+1)), r=1:N, and the eigenvectors
%                    are also known explicitly.
%        INV(A) is tridiagonal: it is minus the second difference matrix
%                    except its (N,N) element is 1.
%        2*A-ONES(N) (Givens' matrix) has tridiagonal inverse and
%                    eigenvalues .5*sec^2((2r-1)PI/4N), r=1:N.
%        (N+1)*ONES(N)-A also has a tridiagonal inverse.
%        FLIPUD(TRIW(N,1)) is a square root of A.

%        References:
%        J. Fortiana and C. M. Cuadras, A family of matrices, the discretized
%           Brownian bridge, and distance-based regression, Linear Algebra
%           Appl., 264 (1997), 173-188.  (For the eigensystem of A.)
%        J. Todd, Basic Numerical Mathematics, Vol. 2: Numerical Algebra,
%           Birkhauser, Basel, and Academic Press, New York, 1977, p. 158.
%        D.E. Rutherford, Some continuant determinants arising in physics and
%           chemistry---II, Proc. Royal Soc. Edin., 63, A (1952), pp. 232-241.
%           (For the eigenvalues of Givens' matrix.)

A = min( ones(n,1)*(1:n), (1:n)'*ones(1,n) );
