On the Differentials of Certain Matrix Functions

Abstract

In [5], Rinehart showed that if X is an n × n complex matrix with distinct eigenvalues, then a suitably defined diagonalizing matrix P and the diagonal matrix Λ of eigenvalues in P–lXP = Λ are both Hausdorff differentiate functions in an open set containing X. Furthermore, if the scalar function ƒ(z) is analytic at the eigenvalues of X, then the primary matrix function ƒ(X) is Hausdorff differentiable, and its differential may be represented in terms of the differentials of P and Λ [4]. Rinehart noted that the actual computation of differentials was difficult and ad hoc. This difficulty clearly arises because of the definition given for the diagonalizing matrix. Therefore, our aim in this note is to give a different definition of the diagonalizing matrix, one which simplifies the computations.