>> % Demonstration of the benefit of picking v1 = a1 + sign(a1(1))*norm(a1)*e1,
>> % rather than v1 = a1 - norm(a1)*e1 for a 2-by-2 matrix.

>> A = [1 1; 1e-9 1]
A =
   1.00000000000000   1.00000000000000
   0.00000000100000   1.00000000000000

>> a1 = A(:,1)
a1 =
   1.00000000000000
   0.00000000100000

>> % First, pick v1 = a1 - norm(a1)*e1

>> v1 = a1-norm(a1)*[1;0]
v1 =
   1.0e-09 *
                  0
   1.00000000000000

>> v1'*v1
ans =
     1.000000000000000e-18

>> v1*v1'
ans =
   1.0e-18 *
                  0                  0
                  0   1.00000000000000

>> v1*v1'/(v1'*v1)
ans =
     0     0
     0     1

>> Hv1 = eye(2)-2*v1*v1'/(v1'*v1)
Hv1 =
     1     0
     0    -1

>> Hv1*A
ans =
   1.00000000000000   1.00000000000000
  -0.00000000100000  -1.00000000000000

>> % Note:  Hv1*A should be upper triangular:  it is not!

>> % Now, pick v1 = a1 + norm(a1)*e1 (i.e., v1 = v1 + sign(a1(1))*norm(a1)*e1

>> v1 = a1 + norm(a1)*[1;0]
v1 =
   2.00000000000000
   0.00000000100000

>> v1*v1'
ans =
   4.00000000000000   0.00000000200000
   0.00000000200000   0.00000000000000

>> v1'*v1
ans =
     4

>> v1*v1'/(v1'*v1)
ans =
   1.00000000000000   0.00000000050000
   0.00000000050000   0.00000000000000

>> Hv1 = eye(2)-2*v1*v1'/(v1'*v1)
Hv1 =
  -1.00000000000000  -0.00000000100000
  -0.00000000100000   1.00000000000000

>> Hv1*A
ans =
  -1.00000000000000  -1.00000000100000
                  0   0.99999999900000

>> % Note: Hv1*A is now upper triangular