On the Formal Differentiation of Traces and Determinants

A compact notation for obtaining and handling matrices of partial derivatives is suggested in an attempt to generalize "symbolic vector differentiation" to matrices of independent variables. The proposed technique differs from methods advocated by Dwyer and MacPhail (1948) and Wrobleski (1963) in several respects, notably in a deliberate limitation on the classes of scalar functions considered: traces and determinants. To narrow interest to these two classes of scalar matrix functions allows one to invoke certain algebraic identities which simplifies the problem, because (a) the treatment of traces of products of matrices can be reduced to that of a few representatives of large equivalence classes of such products, all having the same formal derivative, and because (b) the more involved task of differentiating determinants of matrix products can be translated into the more amenable problem of differentiating the traces of such products. A number of illustrative examples are included in an attempt to show that the above limitation is not as serious as might at first appear, because traces and determinants apply to a wide range of psychometric and statistical problems.