I am given the following problem: A = [ 4 1 1 1 2 3 1 3 2 ] Find max x | ( A x , x ) | ( x , x ) where ( . , . ) is a dot product of vectors and the maximization is performed over all x = [ x 1 x 2 x 3 ] T ∈ R 3 , such that ∑ i = 1 3 x i = 0 I have found the eigenvectors for A and they happen to match the sum criterion: E ( λ 1 ) = span ( [ − 2 1 1 ] T )for λ 1 = 3 and E ( λ 2 ) = span ( [ 0 − 1 1 ] T )for λ 2 = − 1.(For λ 3 = 6 there are no eigenvectors).Can the above eigenvectors and eigenvalues be used for solving this maximization problem?

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I am given the following problem:  A  =      [                            4                          1                          1                                      1                          2                          3                                      1                          3                          2                      ]  Find      max    x                      |            (      A      x      ,      x      )              |                    (      x      ,      x      )      where   (  .  ,  .  ) is a dot product of vectors and the maximization is performed over all   x  =            [                                                  x              1                                                          x              2                                                          x              3                                          ]        T    ∈            R        3  , such that             ∑              i        =        1                    3                    x      i        =    0  I have found the eigenvectors for   A and they happen to match the sum criterion:  E  (      λ    1    )  =  span  (            [                                    −            2                                1                                1                              ]        T    )for       λ    1    =  3 and  E  (      λ    2    )  =  span  (            [                                    0                                −            1                                1                              ]        T    )for       λ    2    =  −  1.(For       λ    3    =  6 there are no eigenvectors).Can the above eigenvectors and eigenvalues be used for solving this maximization problem?

Xzavier Shelton · Accepted Answer

You wrote &quot;for       λ    3    =  6 there are no eigenvectors&quot;. This is not true !We have, since   A is symmetric, that       max    x                      |            (      A      x      ,      x      )              |                    (      x      ,      x      )        =  max  {  λ  :  λ  ∈  σ  (  A  )  }, where   σ  (  A  ) denotes the set of eigenvalues of   A.Hence      max    x                      |            (      A      x      ,      x      )              |                    (      x      ,      x      )        =  6.

I am given the following problem: A = [ <mtable rowspacing="4pt" columnspaci

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