The requirement that the off-diagonal elements of the jth column be non-negative follows from the geometry of the simplex, as per the following.

Observe that \\(\Gamma_j(0) = j\\) = the corner of the simplex which is 1 at j, and 0 elsewhere.

The derivative \\(\Gamma'_j(0)\\) is a tangent vector which must lead from this corner further into the simplex.

(Or else the tangent vector is zero.)

So it must be decreasing the jth component to start going down from 1, and increasing all the other components to start increasing from 0.

The latter implies that the off-diagonal elements are all non-negative.

That shows why H must be infinitesimal stochastic.

Putting the two conditions together, we see that the diagonal entry will be the negative of the sum of the off-diagonal entries.

Observe that \\(\Gamma_j(0) = j\\) = the corner of the simplex which is 1 at j, and 0 elsewhere.

The derivative \\(\Gamma'_j(0)\\) is a tangent vector which must lead from this corner further into the simplex.

(Or else the tangent vector is zero.)

So it must be decreasing the jth component to start going down from 1, and increasing all the other components to start increasing from 0.

The latter implies that the off-diagonal elements are all non-negative.

That shows why H must be infinitesimal stochastic.

Putting the two conditions together, we see that the diagonal entry will be the negative of the sum of the off-diagonal entries.