Positive definite matrix from product of positve diagonal matrix with asymmetric matrix with positive eignevalues

Question

Let $B$ be a symmetric $n \times n$ matrix, and let $B=DA$, where $D$ is an $n \times n$ diagonal matrix with positive real values, and $A$ is an $n \times n$ matrix, not necessarily symmetric, but with all ($n$) real eigenvalues. Note that $BD^{-1}=A$ so $D^{1/2}[D^{-1/2}BD^{-1/2}]D^{-1/2}=A$, so if $B$ is positive definite, then $A$ will have $n$ positive real eigenvalues using matrix similarity of $[D^{-1/2}BD^{-1/2}]$ and $A$. My question is, if $A$ has all positive real eigenvalues, is that sufficient for $B$ to be positive definite? What I conjecture is: $B$ positive definite $\Longleftrightarrow$ $A$ has positive real eigenvalues. However, I am stuck on the $\Longleftarrow$ part, and beginning to doubt if it is true.