In this note, elements of matrix calculus are extended to complex matrices. With this extension, the Cauchy-Riemann conditions in matrix notation are introduced. It is shown that for many cases where the complex matrix derivative does not exist, a complex gradient and Hessian matrix-which are defined by partial derivatives with respect to real and imaginary matrix parts-do exist. Further, these first and second-order derivative matrices are used to provide necessary and sufficient conditions for an extremum of a real scalar function of a complex matrix. Some illustrative examples are provided.