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?

The Schur product theorem states that the Hadamard (i.e. entrywise) product of two positive definite matrices is positive definite.
It follows that the entrywise exponential of a positive definite matrix is positive definite. By continuity, the entrywise exponential of a positive semidefinite matrix is positive semidefinite.
