Nelson Jennings

2022-07-14

I have the following vector $z=\left(\begin{array}{c}x\\ y\end{array}\right)$. I also have the function
$f=\left(\begin{array}{c}-5\beta xy\\ 5\beta xy\end{array}\right)$
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
$f=5\beta \left(\begin{array}{cc}0& -x\\ y& 0\end{array}\right)z$
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?

Kitamiliseakekw

Expert

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

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