Rewriting a quadratic Matrix equation as a quadratic vector equationConsider the set of N × N matrices { W i } i = 1 i = L , set of N × 1 vectors { g i } i = 1 i = L and { h i } i = 1 i = L . Now consider the following sum S = ∑ i ∑ j g i H W i h i h j H W j H g j where the summation runs through all L for all i,j. Clearly, this equation is quadratic in the matrix variables { W i } i = 1 i = L . Now define the column vector w = [ vec ⁡ ( W 1 ) vec ⁡ ( W 2 ) ⋮ vec ⁡ ( W L ) ] where for a matrix A, vec(A) denotes the column vector containing the columns of A starting from the first column. The question is, can we write S = w H Q w where Q is a matrix which is a function of { g i } i = 1 i = L and { h i } i = 1 i = L . If so, what is the structure of Q?

Question

Rewriting a quadratic Matrix equation as a quadratic vector equationConsider the set of   N  ×  N matrices   {      W    i        }          i      =      1              i      =      L      , set of   N  ×  1 vectors   {      g    i        }          i      =      1              i      =      L       and   {      h    i        }          i      =      1              i      =      L      . Now consider the following sum                    S        =                  ∑                      i                                    ∑                      j                                    g          i          H                          W          i                          h          i                          h          j                      H                                    W          j                      H                                    g          j                    where the summation runs through all L for all i,j. Clearly, this equation is quadratic in the matrix variables   {      W    i        }          i      =      1              i      =      L      . Now define the column vector                    w        =                  [                                                    vec                ⁡                (                                  W                  1                                )                                                                    vec                ⁡                (                                  W                  2                                )                                                                    ⋮                                                                    vec                ⁡                (                                  W                  L                                )                                              ]                    where for a matrix A, vec(A) denotes the column vector containing the columns of A starting from the first column. The question is, can we write                    S        =                  w                      H                          Q        w            where Q is a matrix which is a function of   {      g    i        }          i      =      1              i      =      L       and   {      h    i        }          i      =      1              i      =      L      . If so, what is the structure of Q?

Faith Wise · Accepted Answer

Step 1                    S                            =                  ∑                      i                                    ∑                      j                                    g          i          H                          W          i                          h          i                          h          j          H                          W          j          H                          g          j                                                  =                  ∑                      i                                    ∑                      j                          trace        ⁡        (                  g          j                          g          i          H                          W          i                          h          i                          h          j          H                          W          j          H                )                                          =                  ∑                      i                                    ∑                      j                          vec        ⁡        (                  W          j                          )          H                vec        ⁡        (                  g          j                          g          i          H                          W          i                          h          i                          h          j          H                )                                                                  {        ∵        trace        ⁡        (        A                  B                      H                          )        =        trace        ⁡        (                  A                      H                          B        )        =        vec        ⁡        (        A                  )                      H                          vec        ⁡        (        B        )        }                                                              =                  ∑                      i                                    ∑                      j                          vec        ⁡        (                  W          j                          )          H                          (          (                      h            i                                h            j            H                                )            T                    ⊗          (                      g            j                                g            i            H                    )          )                vec        ⁡        (                  W          i                )                                                                  {        ∵        vec        ⁡        (        A        B        C        )        =        (                  C                      T                          ⊗        A        )        vec        ⁡        (        B        )        }                                                              =                  ∑                      i                                    ∑                      j                          vec        ⁡        (                  W          j                          )          H                          (          (                                                    h                ¯                                      j                                h            i            T                    )          ⊗          (                      g            j                                g            i            H                    )          )                vec        ⁡        (                  W          i                )                                          =                  w          H                Q        w        ,            Step 2where Q is a block matrix whose (j,i)-th (note: not (i,j)-th) subblock is   (                    h        ¯              j        h    i    T    )  ⊗  (      g    j        g    i    H    ).

Consider the set of N times N matrices {W_i}_{i=1}^{i=L}, set of N times 1 vectors {g_i}_{i=1}^{i=L} and {h_i}_{i=1}^{i=L}.

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New Questions in Algebra I