Is Kullback-Leibler Divergennce not equal to Relative Entropy?

Question

In many books, Kullback-Leibler Divergence is equal to Relative Entropy. $$ D_{kl}(u,v) = \sum_{i=1}^n(u_ilog(u_i/v_i). $$ However, I find in the book, Convex Optimization (Stephen Boyd) page 90, the KL Divergence is defined as, $$ D_{kl}(u,v) = \sum_{i=1}^n(u_ilog(u_i/v_i)-u_i+v_i). $$ Why KL Divergence has these two different definition? Which one is correct?

Answer 1

They're equivalent because $\sum_i u_i=\sum_i v_i=1$.