--- /dev/null	Thu Jan 01 00:00:00 1970 +0000
+++ b/toolbox/invol.m	Wed May 06 14:56:53 2015 +0200
@@ -0,0 +1,21 @@
+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