 Entropy increase under doubly stochastic mixing

Let $A$ be a doubly stochastic matrix, let $q$ be a probability vector, and put $w=Aq$. In row $i$ with $w_i>0$, define $r_i(j)=a_{ij}{q}_j/w_i$ on the support where $a_{ij}>0$. The positive row entries ${\alpha }_i=(a_{ij})$ form a probability vector. Expanding relative entropy and using the column sums givesZero-output rows contribute zero because all their $a_{ij}{q}_j$ vanish. The row sums supply the normalized comparison laws, while the column sums preserve the input entropy contribution. Equality holds exactly when $q_j=w_i$ for every positive matrix entry $a_{ij}$, including the automatic zero values in zero-output rows.