Matrix Calculus Operations and Taylor Expansions

In problems of large dimensional complexities, matrix methods are frequently the favored mathematical tools. In this paper some extensions of matrix methods to calculus operations are introduced. Consistent array structural definitions are given for derivatives of matrix-valued functions with respect to matrices, for matrix differentials, and for matrix integrals, and some operational properties arising therefrom are detailed. Novel structures are developed for Taylor expansions of a matrix-valued function, which have some attractive features both for manipulative and for computational purposes.