The matrix exponential $e^X$ of a square symmetric matrix $X$ is always positive-definite, where $$ e^X = \sum_{k=0}^\infty \frac{1}{k!} X^k $$
Does the exponentiation of each element of the symmetric matrix $X$ (not to be confused with the matrix exponential above) also result in a positive-definite matrix?
$$\begin{pmatrix} e^{x_1} & e^{x_{1,2}} & e^{x_{1,3}}\\ e^{x_{2,1}} & e^{x_2} & e^{x_{2,3}}\\ e^{x_{3,1}} & e^{x_{3,2}} & e^{x_3} \end{pmatrix}\neq e^X$$
After experimentally running this new question (different than the original question that was closed) with toy data, I am seeing that the element-wise exponential matrix is not always positive-definite like how the matrix exponential always is. but I could be wrong.
How can we prove that the element-wise exponential matrix does not guarantee positive-definiteness? (please general mathematical proofs only, rather than a counter-example)