Obviously, there similarly exists an upper triangular matrix $U$ such that $A = U^H U$ since we can choose $U^H = L \text{.}$
The lower triangular matrix $L$ is known as the Cholesky factor and $L L^H$ is known as the Cholesky factorization of $A \text{.}$ It is unique if the diagonal elements of $L$ are restricted to be positive. Typically, only the lower (or upper) triangular part of $A$ is stored, and it is that part that is then overwritten with the result. In our discussions, we will assume that the lower triangular part of $A$ is stored and overwritten.