Matrix/vector multiplication

Question

I have the following vector $z=\begin{pmatrix} x \\ y \end{pmatrix}$. I also have the function \begin{equation} f=\begin{pmatrix} -5\beta xy \\ 5\beta xy \end{pmatrix} \end{equation} I need to rewrite this such that $x$ and $y$ will be simply denoted as the vector $z$. The closest I am getting to is \begin{equation} f=5\beta\begin{pmatrix} 0 & -x \\ y & 0 \end{pmatrix}z \end{equation} I can't seem to rearrange this matrix however such that it can be written in terms of $z$. I am starting to question whether what I am trying to do is even possible. Is it possible?

Answer 1

There are many ways to write this. Here's one. For $$ z=\begin{bmatrix}x\\y\end{bmatrix}, $$ define $$ f(z) = (z^T\sigma_x z)\begin{bmatrix} -5\beta/2\\5\beta/2\end{bmatrix}, $$ where $^T$ denotes the transpose (so $z^T$ is a row vector), and $\sigma_x$ is a Pauli spin matrix, specifically $$ \sigma_x = \begin{bmatrix} 0 & 1\\1 & 0 \end{bmatrix}. $$ Note that the quantity in parentheses is a number equal to $xy$.