Question

In Exercises 3–6, find (a) the maximum value of Q(x) subject to
the constraint x^Tx = 1, (b) a unit vector u where this maximum is
attained, and (c) the maximum of Q(x) subject to the constraints
x^Tx = 1 and x^Tu = 0. Q(x) = 3x²₁ + 9x²₂ + 8x₁x₂

Answer 1

Solution: The matrix of the quadratic form is given as

.

It is readily checked that

Solution(a) The maximum value of subject to the constraint is the greatest eigen value

of . In order to determine the eigenvalues of , we solve
  

Therefore the largest eigenvalue of is .

Solution(b) The maximum value of   subject to the constraint occurs at a unit eigenvector

corresponding to the greatest eigenvalue of .

Thus, we have to compute the eigenvector of corresponding to the eigenvalue .
Solving , we get

  

Let , then and hence the eigenvector of corresponding to the eigenvalue is
and so the unit vector .
Solution(c)   The maximum value of   subject to the constraint and is
the second-largest eigenvalue   of , which is