Suppose A is a positive definite matrix and subject to the constraint
the quadratic form
has maximum value M and minimum value m. Then subject to the constraint
the quadratic form
has maximum value
and minimum value

.

Question

Suppose A is a positive definite matrix and subject to the constraint 
  the quadratic form 
  has maximum value M and minimum value m. Then subject to the constraint 
  the quadratic form 
  has maximum value 
  and minimum value 
  
.

Quizplus · Accepted Answer

Since A is positive definite, it has a unique positive square root, denoted A^(1/2). Let x be a vector that achieves the maximum value M of the quadratic form under the given constraint. Then we have:

M = x^T A x >= m > 0

Multiplying both sides by A^(1/2) and rearranging, we get:

||A^(1/2) x||^2 >= m > 0

This shows that the vector A^(1/2) x achieves the maximum value m subject to the given constraint on its norm. Moreover, any other vector y subject to the same constraint satisfies:

||A^(1/2) y||^2 <= ||A^(1/2) x||^2

which implies that the quadratic form evaluated at y is bounded below by m. Therefore, A^(1/2) x is the vector that achieves the maximum value under the given constraint on its norm, and its norm achieves the minimum value under the given constraint on the quadratic form. It follows that the quadratic form has maximum value ||A^(1/2) x||^2 and minimum value m when subject to the given constraint. Repeating the argument with the constraint reversed shows that the quadratic form has minimum value ||A^(1/2) y||^2 and maximum value M subject to the given constraint on the norm, as required.

Suppose a Is a Positive Definite Matrix and Subject to the Constraint