Find the solutions of this linear equation: ( ⋆ ) X + ( X ⊤ C ) C = D where C = ( c 1 , … , c n ) ⊤ ∈ R n and D = ( d 1 , … , d n ) ⊤ ∈ R n are given and non-zero.I noticed that the equation ( ⋆ ) is equivalent to: ( ⋆ ⋆ ) C ~ X = D where C ~ ∈ M a t n ( R ) such that: C ~ i , j = { c j c i if i ≠ j 1 + c i 2 if i = j . To do so, I just said that the i-th component of the LHS of ( ⋆ ) is ( 1 + c i 2 ) x i + ∑ j ≠ i x j c j c i Is there another method to prove that ( ⋆ ) is equivalent to ( ⋆ ⋆ )?

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Find the solutions of this linear equation:                              (                  ⋆          )                            X        +        (                  X          ⊤                C        )        C        =        D            where   C  =  (      c      1    ,  …  ,      c      n        )    ⊤    ∈        R      n   and   D  =  (      d      1    ,  …  ,      d      n        )    ⊤    ∈        R      n   are given and non-zero.I noticed that the equation   (  ⋆  ) is equivalent to:                              (                      ⋆            ⋆                    )                                            C          ~                X        =        D            where       C    ~    ∈            M      a      t          n    (  R  ) such that:                         C        ~                    i        ,        j              =          {                                                  c                          j                                      c                          i                                            if             i            ≠            j                                                1            +                          c                          i                          2                                            if             i            =            j                                    . To do so, I just said that the   i-th component of the LHS of   (  ⋆  ) is       (    1    +          c          i          2        )          x          i        +          ∑              j        ≠        i                    x          j              c          j              c          i      Is there another method to prove that   (  ⋆  ) is equivalent to   (  ⋆  ⋆  )?

Camron Herrera · Accepted Answer

As   X and   C are real   n-vectors,                    (1)                    X        +        (                  X          ⊤                C        )        C                            =        X        +        C        (                  X          ⊤                C        )                            (2)                                  =        X        +        C        (                  X          ⊤                C                  )          ⊤                                                  =        X        +        C        (                  C          ⊤                X        )                                          =        (        I        +        C                  C          ⊤                )        X        .            Explanations:  (  1  ): if   u is a vector and   s is a scalar, we have   [  s  ]  u  =  u  [  s  ], where on the LHS we identify the   1  ×  1 matrix   [  s  ] with the scalar   s, so that   [  s  ]  u is the scalar multiplication   s  u.  (  2  ): the transpose of a real   1  ×  1 matrix is the matrix itself.

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