Faith Welch

2022-07-19

How can I generally solve equations of the form $\mathbf{A}\mathbf{w}=\left(\begin{array}{c}x\\ y\\ z\end{array}\right)×\mathbf{w}$ 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 $\left(\begin{array}{c}1\\ 2\\ 1\end{array}\right)$, but doing so still isn't helpful. Also, x, y, and z are entirely independent variables.

minotaurafe

Expert

OK, let's put it other way as $\mathbf{w}×\mathbf{v}=-\mathbf{A}\mathbf{w}$. We can write the the cross product as vector-matrix multiplication:
$\mathbf{w}×\mathbf{v}=\left[\mathbf{w}{\right]}_{×}\mathbf{v}=\left[\begin{array}{ccc}\phantom{\rule{thinmathspace}{0ex}}0& \phantom{\rule{negativethinmathspace}{0ex}}-{w}_{3}& \phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}{w}_{2}\\ \phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}{w}_{3}& 0& \phantom{\rule{negativethinmathspace}{0ex}}-{w}_{1}\\ -{w}_{2}& \phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}{w}_{1}& \phantom{\rule{thinmathspace}{0ex}}0\end{array}\right]\mathbf{v}.$
So you can write your equation as a system of linear equations
$\left[\mathbf{w}{\right]}_{×}\mathbf{v}=-\mathbf{A}\mathbf{w}.$
Matrix $\left[\mathbf{w}{\right]}_{×}$ has rank 2 and its nullspace is spanned by $\left[{w}_{1},\phantom{\rule{thinmathspace}{0ex}}{w}_{2},\phantom{\rule{thinmathspace}{0ex}}{w}_{3}{\right]}^{\mathrm{\top }}$
Now depending on whether you assume ${w}_{2}\ne 0$ or ${w}_{3}\ne 0$, you can transform this system and find a particular solution. However, this solution can be found only if $⟨\mathbf{w},\mathbf{A}\mathbf{w}⟩=0$. In particular, this implies that ${\mathbf{A}}^{\mathrm{\top }}=-\mathbf{A}$

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