Let \(M_n\), \(M^\geq_n\), and \(M^>_n\) stand respectively for the sets of all complex, Hermitian nonnegative definite, and positive definite \(n \times n\) matrices. Assume \(S\) positive. Then the set \(M^\geq_n (S) = \{A \in M_n : SA \in M^\geq_n\}\) is a convex cone. On this cone the partial order defined by \(A \leq_{SL} B\) iff \((B - A) \in M^\geq_n (S)\) is considered; for \(S = I\) this is just the Loewner ordering \(\leq_L\) restricted to the cone of nonnegative definite matrices. The \(\leq_{SL}\)-ordering is characterized in several different ways using the spectral radius \(\rho\), various types of generalized inverses, and inclusion relations between range spaces; it is put in perspective with various other partial orderings considered e.g. by \textit{C. Stepniak} [Linear Algebra Appl. 94, 263-272 (1987; Zbl 0626.15008)] and \textit{J. K. Baksalary} and \textit{S. K. Mitra} [ibid. 149, 73-89 (1991; Zbl 0717.15004)] to name just a few. As sample results we mention \(A \leq_{SL} B\) iff \(\rho (B^- A) \leq 1\) for every generalized inverse \(B^-\) of \(B\) iff there exists an \(N \in M_n^>\) such that \(A N^{-1} A^* \leq_L BN^{-1} B^*\). The article concludes with some results for rectangular matrices.