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Given a symmetric positive definite matrix $\bf \Sigma \in \mathbb{R}^{n \times n}$, I want to find a matrix ${\bf \Gamma} \in \mathbb{R}^{n \times n}$ and a vector ${\bf m} \in \mathbb{R}^n$ such that if ${\bf X} \sim N(\bf{m}, \bf{\Gamma})$, then the random vector $\bf{Y} = (\exp(X_1), ..., \exp(X_n))$ has covariance matrix $\bf \Sigma$. For $n=1$, this is possible, as shown e.g. in this blog entry

In higher dimensions $n > 1$, this is probably not possible for all matrices $\bf \Sigma$, so I would be interested in conditions for $\bf \Sigma$ under which such a matrix $\bf \Gamma$ exists, and how to find it. I have found this previous question, but maybe after 3 years somebody has a better answer, and also I have no restriction on the mean of $\bf Y$.

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    $\begingroup$ As matrix exponentiation is something completely different, ie $\exp(\mathbf{A})\neq\{\exp(A_{i,j}\}$ there is no suitable way rather than to use the elements of the matrix to represent the covariance of $\mathbf{Y}$. Thus it might be tricky to get sigma at the end $\endgroup$
    – Onyambu
    Commented Mar 10, 2023 at 16:21

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