I have a problem with finding: max a ∈ R p a ′ C a a ′ B a ,where C and B are p × p symmetric matrices.After differentiation I get the following result: 2 a ′ C ( a ′ B a ) − 2 a ′ B ( a ′ C a ) ( a ′ B a ) ( a ′ B ′ a ) = 0I can’t solve the above equation for a. How to find it?

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I have a problem with finding:      max          a      ∈                        R                p                                a        ′            C      a                      a        ′            B      a        ,where   C and   B are   p  ×  p symmetric matrices.After differentiation I get the following result:            2              a        ′            C      (              a        ′            B      a      )      −      2              a        ′            B      (              a        ′            C      a      )              (              a        ′            B      a      )      (              a        ′                    B        ′            a      )        =  0I can’t solve the above equation for   a. How to find it?

Abigail Palmer · Accepted Answer

The maximum won&#039;t generally exist. It can be guaranteed to exist, however, if   B is positive definite.In this case, suppose that   B  =      M    ′    M for some invertible matrix   M (we can compute   M via a Cholesky decomposition for instance). We then have      max    a                      a        ′            C      a                      a        ′            B      a        =      max    a                      a        ′            C      a                      a        ′                    M        ′            M      a        =      max    a                      a        ′            C      a              (      M      a              )        ′            (      M      a      )          =      max    b              (              M                  −          1                    b              )        ′            C      (              M                  −          1                    b      )                      b        ′            b        =      max    b                      b        ′            (      [              M        ′                    ]                  −          1                    C      M      )      b                      b        ′            b        .By the Rayleigh-Ritz theorem, this maximum is the largest eigenvalue of   [      M    ′        ]          −      1        C  M.

vittorecostao1 · Accepted Answer

Denote the scalar which you wish to optimize as  λ  =                    a        T            C      a                      a        T            B      a      Take your gradient result, transpose it, multiply it by                       a        T            B      a        2    ,  , and write it as  C  a  =  λ  B  aThis is a generalized eigenvalue problem and you want the eigenvector corresponding to the largest such eigenvalue.If         B          −      1       exists, then this can be reduced to an ordinary eigenvalue problem for   L  =      B          −      1        C.  L  a  =  λ  aIf         C          −      1       exists, then it becomes an eigenvalue problem for the reciprocal eigenvalues             (        μ  =      1    λ              )       of the matrix   M  =      C          −      1        B.  M  a  =  μ  a

I have a problem with finding: <munder> <mo movablelimits="true" form="prefix">max

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