Matrix-vector multiplication/cross product problem

Question

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

Answer 1

OK, let's put it other way as $\mathbf{w}\times \mathbf{v}=-\mathbf{A}\mathbf{w}$. We can write the the cross product as vector-matrix multiplication: $$\mathbf{w}\times\mathbf{v} =[\mathbf{w}]_\times \mathbf{v}=\begin{bmatrix}\,0&\!-w_{3}&\,\,w_{2}\\\,\,w_{3}&0&\!-w_{1}\\-w_{2}&\,\,w_{1}&\,0\end{bmatrix}\mathbf{v}.$$ So you can write your equation as a system of linear equations $$[\mathbf{w}]_\times \mathbf{v}=-\mathbf{A}\mathbf{w}.$$ Matrix $[\mathbf{w}]_\times$ has rank $2$ and its nullspace is spanned by $[w_1,\,w_2,\,w_3]^\top$.

Now depending on whether you assume $w_2\neq 0$ or $w_3\neq 0$, you can transform this system and find a particular solution. However, this solution can be found only if $\langle\mathbf{w},\mathbf{Aw}\rangle=0$. In particular, this implies that $\mathbf{A}^\top=-\mathbf{A}$.