# Chapter 18Matrix calculus: the essentials

## 1 INTRODUCTION

This chapter differs from the other chapters in this book. It attempts to summarize the theory and the practical applications of matrix calculus in a few pages, leaving out all the subtleties that the typical user will not need. It also serves as an introduction for (advanced) undergraduates or Master's and PhD students in economics, statistics, mathematics, and engineering, who want to know how to apply matrix calculus without going into all the theoretical details. The chapter can be read independently of the rest of the book.

We begin by introducing the concept of a differential, which lies at the heart of matrix calculus. The key advantage of the differential over the more common derivative is the following. Consider the linear vector function f(x) = Ax where A is an m × n matrix of constants. Then, f(x) is an m × 1 vector function of an n × 1 vector x, and the derivative Df(x) is an m × n matrix (in this case, the matrix A). But the differential df remains an m × 1 vector. In general, the differential df of a vector function f = f(x) has the same dimension as f, irrespective of the dimension of the vector x, in contrast to the derivative Df(x).

The advantage is even larger for matrices. The differential dF of a matrix function F(X) has the same dimension as F, irrespective of the dimension of the matrix X. The practical importance of working with differentials is huge and will be demonstrated through many examples. ...

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