I have encountered a problem of determining the condition for matrix $A+A^T$ to be positive definite, where
$$A= \operatorname{diag} \{a_1, a_2, \ldots, a_n\} + \mathbb{1}\cdot [b_1, b_2, \ldots, b_n]$$ where $\mathbb{1}$ is a column vector with element $1$, and $b_i>0$.
Is it true that $A$ is positive definite if and only if $\operatorname{diag} \{a_1, a_2, \ldots, a_n\}$ is positive definite, i.e., $\min(a_i)>0$? Or it has something to do with the $b_i$
Thanks in advance!