Let $\mathbf{v}$ be a vector:
$$ \mathbf{v}:=\pmatrix{x+ia\\y+ib\\z+ic} $$
I am looking for an explicit $M$ such that:
$$ f(\mathbf{v})= \mathbf{v}^*M\mathbf{v}=\det\pmatrix{\sigma_x&\sigma_y&\sigma_z\\x&y&z\\a&b&c} $$
Here $\sigma_x,\sigma_y,\sigma_z$ are the Pauli matrices.
Then, I would like to make the replacement $f(\mathbf{v})\to f(U\mathbf{v})$ and identify the invariance group of the determinant.
I have so far investigated writing $f(\mathbf{v})$ as follows:
$$ f(\mathbf{v})= \sigma_x \mathbf{v}^*M_x\mathbf{v} + \sigma_y \mathbf{v}^*M_y\mathbf{v}+ \sigma_z \mathbf{v}^*M_z\mathbf{v} $$
Then finding 3 different $M_x,M_y,M_z$.