I`m trying to solve the optimization problem on x and y max ( x ⊗ y ) T A ( x ⊗ y ) s . t ‖ x ‖ 2 = 1 ‖ y ‖ 2 = 1 , where x ∈ R m and y ∈ R n . The matrix A is a rank-one symmetric matrix given by A = v v T , where v ∈ R m n .

Question

I`m trying to solve the optimization problem on       x   and       y                      max                (                  x                ⊗                  y                          )          T                A        (                  x                ⊗                  y                )                            s        .        t                ‖                  x                          ‖          2                =        1                                    ‖                  y                          ‖          2                =        1        ,            where       x    ∈            R              m       and       y    ∈            R              n        . The matrix   A is a rank-one symmetric matrix given by   A  =      v              v              T        , where       v    ∈            R              m      n        .

Carmelo Payne · Accepted Answer

First, note that for any two equal-size vectors   v,  w,       w    T    (  v      v    T    )  w  =  (      w    T    v      )    2  . Subdivide   v into   n  ×  1 blocks       v          (      i      )      ,   i  =  1  ,  2  ,  …  ,  m. Then  (  x  ⊗  y      )    T    v  =      ∑          i      =      1        m        x    i        y    T        v          (      i      )        =      y    T        ∑          i      =      1        m        x    i        v          (      i      )        =      y    T    V  xwhere   V  ∈            R              n      ×      m       collects the vectors       v          (      i      )       into columns. The problem becomes      max          ‖      x              ‖        2            =      1      ,      ‖      y              ‖        2            =      1        (  x  ⊗  y      )    T    A  (  x  ⊗  y  )  =      max          ‖      x              ‖        2            =      1      ,      ‖      y              ‖        2            =      1                  (              y        T            V      x      )        2      (  ∗  )But       max          ‖      y              ‖        2            =      1        (      y    T    w      )    2    =  ‖  w      ‖    2    2  , so      max          ‖      x              ‖        2            =      1      ,      ‖      y              ‖        2            =      1        (  x  ⊗  y      )    T    A  (  x  ⊗  y  )  =      max          ‖      x              ‖        2            =      1                  ‖      V      x      ‖        2    2    =      σ    1    (  V      )    2  where       σ    1    (  V  ) is the largest singular value of   V. The maximizing x is any right-singular vector associated with       σ    1    (  V  ), and the corresponding maximizing   y is   y  =  ±  (      σ    1    (  V  )      )          −      1        V  x, which is a corresponding left singular vector of   V. In hindsight, I could have jumped right to the answer after (*).

I`m trying to solve the optimization problem on <mrow class="MJX-TeXAtom-ORD"> <mi mathvari

Answered question

Answer & Explanation

New Questions in High school geometry