Let X = ( X 1 , . . . , X n ) be a vector of n random variables. Consider the following maximization problem: max a , b C o v ( a ⋅ X , b ⋅ X ) under the constraint that ‖ a ‖ 2 = ‖ b ‖ 2 = 1.( a ⋅ X is the dot product between a and X). Would it be true that there is a solution to this maximization problem such that a = b?Thanks.

Question

Let   X  =  (      X    1    ,  .  .  .  ,      X    n    ) be a vector of   n random variables. Consider the following maximization problem:      max          a      ,      b              C    o    v    (  a  ⋅  X  ,  b  ⋅  X  ) under the constraint that   ‖  a      ‖    2    =  ‖  b      ‖    2    =  1.(  a  ⋅  X is the dot product between   a and   X). Would it be true that there is a solution to this maximization problem such that   a  =  b?Thanks.

reflam2kfnr · Accepted Answer

Since   C  ≥  0 and symmetric, we have  C  =  Q  L      Q    ′  where   L  =  diag  ⁡  (      λ    1    ,  .  .  .  ,      λ    n    ) and   Q is orthogonal. Optimization of       a    ′    C  b then reduces to the optimization of       a    ′    L  b  =      ∑          i      =      1        n              λ      i            a    i        b    i  . If at least one       λ    i    &amp;gt;  0 then for the optimal solution   a  =  b.

Let X = ( X 1 </msub> , . . . , X n </msu

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