# Matrix-vector multiplication/cross product problem

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.

Mathematics Asked by Kurt Muster on November 12, 2021

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: $$mathbf{w}timesmathbf{v} =[mathbf{w}]_times mathbf{v}=begin{bmatrix},0&!-w_{3}&,,w_{2}\,,w_{3}&0&!-w_{1}\-w_{2}&,,w_{1}&,0end{bmatrix}mathbf{v}.$$ 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 $$w_2neq 0$$ or $$w_3neq 0$$, you can transform this system and find a particular solution. However, this solution can be found only if $$langlemathbf{w},mathbf{Aw}rangle=0$$. In particular, this implies that $$mathbf{A}^top=-mathbf{A}$$.

Answered by orthxx on November 12, 2021

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