5.Special case of Hessian matrix


Remark

 In particular, a quadratic form

   

 for   is itself a function of two variables, and its frist partial derivatives are

   

   

 So, the Hessian of  is 

   .     Thus   is nonsigular if and only if .

 Since , it follows that the quadratic form  takes the local minimum(극소) at  if and only if

       for all  ,

  and   takes the local maximum(극대) at    if and only if

      for all .

  If  takes bothe positive and negative values, then  is a saddle point.

  In general(일반적으로), for

   , the matrix representation(행렬표현) of the quadratic form  can be rewritten as

     ( is the Hessian matrix of )

  Therefore, we can determine the extrema of  using the matrix .

Theorem 5

 Let  be a symmetric  matrix whose eigenvalues(고유값) are  in descending order.

 If  is constrained so that   relative to the Euclidean inner product on  , then

 1) ,

 2)    if  is eigenvectors(고유벡터) of  belonging to an eigenvalue .

 (참고 : Exercise Rqyleigh quatient http://math1.skku.ac.kr/home/pub/1661/ )

Proof)

1) Since  is symmetric(대칭), there is an orthonormal basis  for  cosisting of eigenvectors(고유벡터) of  belonging to the eigenvlues , respectively.

  If   denotes the Euclidean inner product , then any  in  may be expressed as

     

 Thus,

     

          .

   If  , then we obtain

       

                 

                 

                 ,

  since  is the largest eigenvalue. Simliarly, One can show .

  2)  If  is an eigenvector of  belonging to  and  , then

      .

 

Example) Find the maximum and minimum values of the quadratic form 

 

subject to the constraint  , and determine values of  and  at which the maximum and minimum occur.

Sol)

The matrix representation of the quadratic form is   .




The largest eigenvalue of  is   at   .




The smallest eigenvalue of  is   at    .




The unit vector of   is    , which maximum value of the form occur.




The unit vector of   is   , which minimum value of the form occur.







Example)

Find the maximum and minimum values of the quadratic form 

  

subject to the constraint   , and determine values of  and  at which the minimum and maximum occur.

 다음은 함수가 단위원 안에서 최대값과 최소값이 어느 점에서 일어나는 지를 보여주는 interact입니다.




Example)

Find the maximum and monimum of the following quadratic forms subject to the constraint  

and determine the values of  and  at which the maximum and minimum occur:

1) .

2) .

Sol)

1) The matrix representation of quadratic form is  .




The largest values of  is   at   .




The samllest value of  is   at   .







2) 다음은 함수가 단위구에서 최대값과 최소값이 어느 점에서 일어나는 지를 보여주는 interact입니다.




Reference:

Remark3, Theorem5, Example- Linear Algebra, Kwak&Hong, Birkhauser,p298~p300

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