Given matrix A ∈ R k × n , define scalar field F : R n ∖ { 0 } → R by F ( x ) := | A x | 2 | x | 2 and find ∇ F

Question

Given matrix   A  ∈            R              k      ×      n      , define scalar field   F  :            R              n           ∖  {  0  }  →      R   by  F  (  x  )  :=                    |            A      x                        |                2                            |            x                        |                2            and find   ∇  F

acilschoincg8 · Accepted Answer

You can think of it like function compositions/operations:Notice first that since for any   h  ∈            R        n  ,   A  (  x  +  h  )  =  A  x  +  A  h  =  A  x  +  A  h  +  o  (  h  ), the differential of   A  :  x  ↦  A  x is given by dA(x)=A and so      ∇    A  =      A    T  In particular,       ∇    x  =      I    n   is the identity. Then, since       |    x            |        2    =  x  ·  x, by the product rule       ∇    (      |    x            |        2    )  =  2    x. By the rule for taking differentials of composition of functions you get   ∇  (      |    A  x            |        2    )  =  2        A    T    A    x. Now we are ready to compute the result.  ∇      (                            |                A        x                              |                    2                                      |                x                              |                    2                      )    =            2                    |            x                        |                2                          A        T            A            x      −      2                    |            A      x                        |                2                  x                      |            x                        |                4              =      2                  |            x                        |                2                    A    T    A    x  −            2                    |            A      x                        |                2                            |            x                        |                4                x

Damon Cowan · Accepted Answer

Define the scalar functions (and their differentials)                    β                    =        ‖        A        x                  ‖          2                                    =                          x          T                B        x                                    ⟹                        d        β        =        2        B        x        :        d        x        ,                            B        =                  A          T                A                            γ                    =        ‖        x                  ‖          2                                    =                          x          T                I        x                                    ⟹                        d        γ        =        2        I        x        :        d        x        ,                            I        =                  I          d          e          n          t          i          t          y                    Use these to rewrite your own scalar function, then calculate its differential and gradient.                    λ                    =                  γ                      −            1                          β                            d        λ                    =                  γ                      −            1                          d        β                −                β                  γ                      −            2                          d        γ                                  =                  γ                      −            1                                                (                          2        B        x        :        d        x                              )                          −        β                  γ                      −            2                                                (                          2        I        x        :        d        x                              )                                                    =        2                  γ                      −            1                                                (                          B        x        −                  γ                      −            1                          β        x                              )                          :        d        x                                                  ∂            λ                                ∂            x                                      =        2                  γ                      −            1                                                (                          B        x        −        λ        x                              )                              which can be rewritten in terms of the original variables                                          ∂            F                                ∂            x                                      =                  2                      ‖            x                          ‖              2                                                (                      A            T                    A          x          −                      (                                          ‖                A                x                                  ‖                  2                                                            ‖                x                                  ‖                  2                                                      )                    x          )

Given matrix A in RR^(k xx n), define scalar field F:RR^n ∖{0} -> RR by F(x):=(|Ax|^2)/(|x|^2) and find grad F

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