I have the following vector z = ( x y ) . I also have the function f = ( − 5 β x y 5 β x y ) 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 β ( 0 − x y 0 ) zI 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?

Question

I have the following vector   z  =      (                            x                                      y                      )  . I also have the function  f  =      (                            −          5          β          x          y                                      5          β          x          y                      )  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  β      (                            0                          −          x                                      y                          0                      )    zI can&#039;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 · Accepted Answer

There are many ways to write this. Here&#039;s one. For  z  =      [                            x                                      y                      ]    ,define  f  (  z  )  =  (      z    T        σ    x    z  )      [                            −          5          β                      /                    2                                      5          β                      /                    2                      ]    ,where           T   denotes the transpose (so       z    T   is a row vector), and       σ    x   is a Pauli spin matrix, specifically      σ    x    =      [                            0                          1                                      1                          0                      ]    .Note that the quantity in parentheses is a number equal to xy.