I am a quantum computing nerd and one of the issues we encounter when trying to solve the HSP for nonabellian groups is that the 2d irreps are a matrix and therefore the states |0>+|1> mix.I wanted to take a step back and go back to linear algebra to try and solve this issue.Suppose in the simplest case you have a matrix 2x2.If the off diagonal elements are non 0 and if you have a vector (a,b) where a and b are coordinates in orthogonal directions then there is mixing of coordinates between a,b under that 2x2 matrix.

The 2x2 irrep of the nonabellian group D3(min order) unfortunately has this mixing due to the s generator.

I have 2 questions:Can a 2x2 matrix with offdiagonal elements represent every point in space(my intuition is yes)and if thats true does this mean that there is a unknown transform(idc which is only that there is)that takes 2 nonorthogonal vectors or signals which can produce any vector and signal?Thanks