I would like to define a covariance matrix in RStan.
Similarly to how you can provide constraints to scalar and vector values, e.g. real a, I would like to provide constraints that the leading diagonal of the covariance matrix must be positive, but the off-diagonal components could take any real value.
Is there a way to enforce that the matrix must also be positive semi-definite? Otherwise, some of the samples produced will not be valid covariance matrices.
Yes, defining
cov_matrix[K] Sigma;
ensures that Sigma is symmetric and positive definite K x K matrix. It can reduce to semidefinite due to floating point, but we'll catch that and raise exceptions to ensure it stays strictly positive definite.
Under the hood, Stan uses the Cholesky factor transform---the unconstrained representation is a lower triangular matrix with positive diagonal. We just use that as the real parameters, then transform and apply the Jacobian implicitly under the hood as described the reference manual chapter on constrained variables to create a covariance matrix with an implicit (improper) uniform prior.