How can I generally solve equations of the form A w = ( x y z ) × w for the matrix A, where w can be any vector? I recognize that you could just set w to a vector with simple values, such as ( 1 2 1 ) , but doing so still isn't helpful. Also, x, y, and z are entirely independent variables.

Question

How can I generally solve equations of the form       A        w    =      (                            x                                      y                                      z                      )    ×      w   for the matrix A, where w can be any vector? I recognize that you could just set w to a vector with simple values, such as       (                            1                                      2                                      1                      )  , but doing so still isn&#039;t helpful. Also, x, y, and z are entirely independent variables.

minotaurafe · Accepted Answer

OK, let&#039;s put it other way as       w    ×      v    =  −      A        w  . We can write the the cross product as vector-matrix multiplication:      w    ×      v    =  [      w        ]    ×        v    =      [                                      0                                    −                      w                          3                                                                                w                          2                                                                                            w                          3                                                0                                    −                      w                          1                                                            −                      w                          2                                                                                w                          1                                                          0                      ]        v    .So you can write your equation as a system of linear equations  [      w        ]    ×        v    =  −      A        w    .Matrix   [      w        ]    ×   has rank 2 and its nullspace is spanned by   [      w    1    ,        w    2    ,        w    3        ]    ⊤  Now depending on whether you assume       w    2    ≠  0 or       w    3    ≠  0, you can transform this system and find a particular solution. However, this solution can be found only if   ⟨      w    ,      A    w    ⟩  =  0. In particular, this implies that             A        ⊤    =  −      A