function A = invol(n)
%INVOL   An involutory matrix.
%        A = INVOL(N) is an N-by-N involutory (A*A = EYE(N)) and
%        ill-conditioned matrix.
%        It is a diagonally scaled version of HILB(N).
%        NB: B = (EYE(N)-A)/2 and B = (EYE(N)+A)/2 are idempotent (B*B = B).

%        Reference:
%        A.S. Householder and J.A. Carpenter, The singular values
%        of involutory and of idempotent matrices, Numer. Math. 5 (1963),
%        pp. 234-237.

A = hilb(n);

d = -n;
A(:, 1) = d*A(:, 1);

for i = 1:n-1
    d = -(n+i)*(n-i)*d/(i*i);
    A(i+1, :) = d*A(i+1, :);
end