I have a function where i determine the hessian with finite difference using the hessian function in the Calculus package. Then I test if the hessian is positive definite with isposdef this always return false because, ishermitian is called at: https://github.com/JuliaLang/julia/blob/5e2a6173eca65e78f787a3525bae18857d283229/base/linalg/dense.jl#L27-L28
before the LAPACK.potrf! is called. Due to numerical inaccuracy will ishermitian always return false because A[i,j] != ctranspose(A[j,i]) in https://github.com/JuliaLang/julia/blob/5e2a6173eca65e78f787a3525bae18857d283229/base/linalg/generic.jl#L282, will return true for some i,j due to numerical inaccuracy. If I change the line to !isapprox(A[i,j],ctranspose(Aj,i])) the ishermitian returns true and hence also call the LAPACK.potrf! to check if it is positive definite.

So from my point of view, should ishermitian test approximately equality.