For $a_i,b_i\in\mathbb{R}^+$ for all $i=1,\cdots,n$, I want to find if and only if condition on $b_i$'s when \begin{equation} \sum_{i=1}^na_ib_i\geq\sum_{i=1}^na_i. \end{equation}
To transform this problem to linear algebra problem let $A=\mathrm{diag}\{a_1,\cdots,a_n\}$ and $B=\mathrm{diag}\{b_1,\cdots,b_n\}$. Then I have to find a condition on $B$, such that \begin{equation} \mathrm{tr}(AB)-\mathrm{tr}(A)\geq0. \end{equation} Is there any linear algebra theorem related to the last inequality? Is this problem formulated correctly or additional information is needed?