Let's look at the master equation again:

\\[\Gamma_{\sigma}'(t) = H(\Gamma_{\sigma}(t))\\]

The linear operator \\(H\\) is just matrix multiplication by \\(\[H_{i,j}\]\\).

Instantiating to t=0, we get:

\\[\Gamma_{\sigma}'(0) = H(\Gamma_{\sigma}(0)) = H(\sigma)\\]

For a definite state \\(j\\), this gives us:

\\[\Gamma_{j}'(0) = H(\Gamma_{j}(0)) = H(j)\\]

which equals the jth column of \\(\[H_{i,j}\]\\).

\\[\Gamma_{\sigma}'(t) = H(\Gamma_{\sigma}(t))\\]

The linear operator \\(H\\) is just matrix multiplication by \\(\[H_{i,j}\]\\).

Instantiating to t=0, we get:

\\[\Gamma_{\sigma}'(0) = H(\Gamma_{\sigma}(0)) = H(\sigma)\\]

For a definite state \\(j\\), this gives us:

\\[\Gamma_{j}'(0) = H(\Gamma_{j}(0)) = H(j)\\]

which equals the jth column of \\(\[H_{i,j}\]\\).