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792 Chapter 18 The Theory of Multiple Regression

c. Show that Bn = (X′MWX)-1X′MWY .
d. The Frisch–Waugh theorem (Appendix 6.2) says that Bn =

(X′X)-1X′Y. Use the result in (c) to prove the Frisch–Waugh
theorem.

A p p e n d i x

18.1 Summary of Matrix Algebra

This appendix summarizes vectors, matrices, and the elements of matrix algebra used in
Chapter 1. The purpose of this appendix is to review some concepts and definitions from
a course in linear algebra, not to replace such a course.

Definitions of Vectors and Matrices

A vector is a collection of n numbers or elements, collected either in a column (a column
vector) or in a row (a row vector). The n-dimensional column vector b and the n-dimensional
row vector c are

b1

b= ≥ b2 ¥ and c = 3c1 c2 g cn4,
f

bn

where b1 is the first element of b and in general bi is the ith element of b.
Throughout, a boldface symbol denotes a vector or matrix.

A matrix is a collection, or an array, of numbers or elements in which the elements are

laid out in columns and rows. The dimension of a matrix is n * m, where n is the number
of rows and m is the number of columns. The n * m matrix A is

a11 a12 g a1m
a21 a22 a2m
A = ≥ f f g f ¥ ,
f

an1 an2 g anm

where aij is the (i, j) element of A, that is, aij is the element that appears in the ith row and jth
column. An n * m matrix consists of n row vectors or, alternatively, of m column vectors.

To distinguish one-dimensional numbers from vectors and matrices, a one-dimensional

number is called a scalar.