Positive semi-definite matrix and exponent of its entries Let $(a_{ij})_{i,j=1,...,n}$ be a symmetric positive semi-definite matrix with real entries. Is the matrix $(\exp(a_{ij}))_{i,j=1,...,n}$ a positive semi-definite?

In fact, it is at least true that if a $2 \times 2$ matrix with non-negative entries is symmetric and positive definite, then so is its entry-wise exponential.  This follows from the AM-GM inequality.
I don't believe that this holds Apparently this holds in general. I don't have a proof either way, but I thought the partial result was worth posting.
