I could calculate a modular operator $Δ$ for a state $τ_α(x_{ij})= αx_{11}+(1-α)x_{22}(0<α<1)$. But, I cannot understand an automorphism $σ_t(x)＝Δ^{it}(x)Δ^{-it}$ of $M_2(\mathbb{C})$.
For example, suppose that $x=(x_{ij}) \in M_2(\mathbb{C})$. Then how can $σ_t(x)$ be written as matrix form?