Is it possible to define the operator $P$ to get the following for an input matrix with all elements $a,b,c,d,e,f$ positive:
$$\begin{bmatrix} a & b & c\\ d & e & f \\ \end{bmatrix} Q P =\begin{bmatrix} a & b & c\\ d & e & f \\ \end{bmatrix}\begin{bmatrix} 1 & 0 \\ -1 & 1 \\ 0&-1 \end{bmatrix} P=\begin{bmatrix} a-b & b- c\\ d -e & e- f \\ \end{bmatrix}P = \left[ \begin{bmatrix} 1 & 1 \\ 1 & 1 \\ \end{bmatrix}- \begin{bmatrix} \frac{b}{a} & \frac{c}{d}\\ \frac{e}{d} & \frac{f}{e} \\ \end{bmatrix}\right]P$$ Actually this is the relative change with matrix operators.
Thank you in advance.