Matrix Differential Calculus with Applications in Statistics and Econometrics, 3rd Edition

Chapter 8Some important differentials

1 INTRODUCTION

Now that we know what differentials are and have adopted a convenient and simple notation for them our next step is to determine the differentials of some important functions. We shall discuss the differentials of some scalar functions of X (eigenvalue, determinant), a vector function of X (eigenvector), and some matrix functions of X (inverse, Moore‐Penrose (MP) inverse, adjoint matrix).

But first, we must list the basic rules.

2 FUNDAMENTAL RULES OF DIFFERENTIAL CALCULUS

The following rules are easily verified. If u and v are real‐valued differentiable functions and α is a real constant, then we have

(1)

(2)

(3)

and also

(4)

(5)

The differential of the power function is

(6)

where the domain of definition of the power function u^α depends on the arithmetical nature of α. If α is a positive integer, then u^α is defined for all real u; but ...