Linear Models

Shayle R. Searle

• Publisher: Wiley-Interscience
• Year: 2012
• Print ISBN: 9780471184997, 9780471769507
• eBook ISBN: 9781118491768
• Edition: 1
• Topic: Mathematics
• Subtopic: Probability & Statistics
• Index: Mathematics

This 1971 classic on linear models is once again available--as a Wiley Classics Library Edition. It features material that can be understood by any statistician who understands matrix algebra and basic statistical methods.

CHAPTER 1

GENERALIZED INVERSE MATRICES

1. INTRODUCTION

The application of generalized inverse matrices to linear statistical models is of relatively recent occurrence. As a mathematical tool such matrices aid in understanding certain aspects of the analysis procedures associated with linear models, especially the analysis of unbalanced data, a topic to which considerable attention is given in this book. An appropriate starting point is therefore a summary of the features of generalized inverse matrices that are important to linear models. Other ancillary results in matrix algebra are also discussed.

a. Definition and existence

A generalized inverse of a matrix A is defined, in this book, as any matrix G that satisfies the equation

(1)

The name “generalized inverse” for matrices G defined by (1) is unfortunately not universally accepted, although it is used quite widely. Names such as “conditional inverse”, “pseudo inverse” and “g-inverse” are also to be found in the literature, sometimes for matrices defined as is G of (1) and sometimes for matrices defined as variants of G. However, throughout this book the name “generalized inverse” of A is used exclusively for any matrix G satisfying (1).

Notice that (1) does not define G as “the” generalized inverse of A but as “a” generalized inverse. This is because G, for a given matrix A, is not unique. As shown below, there is an infinite number of matrices G that satisfy (1) and so we refer to the whole class of them as generalized inverses of A.

One way of illustrating the existence of G and its non-uniqueness starts with the equivalent diagonal form of A. If A has order p × q the reduction to this diagonal form can be written as

or, more simply, as

As usual, P and Q are products of elementary operators [see, for example, Searle (1966), Sec. 5.7], r is the rank of A and D_r is a diagonal matrix of order r. In general, if d₁ d₂, …, d_r, are the diagonal elements of any diagonal matrix D we will use the notation D{d_i} for D_r; i.e.,

(2)

Furthermore, as in Δ, null matrices will be represented by the symbol 0, with order being determined by context on each occasion.

Derivation of G comes easily from Δ. Analogous to Δ we define Δ⁻ (to be read as “Δ minus”) as

Then, as shown below,

(3)

satisfies (1). Hence G is a generalized inverse of A. Clearly G as given by (3) is not unique, for neither...

Table of contents

• Cover
• Half Title page
• Title page
• Copyright page
• Preface
• Chapter 1: Generalized Inverse Matrices
• Chapter 2: Distributions and Quadratic Forms
• Chapter 3: Regression, or the Full Rank Model
• Chapter 4: Introducing Linear Models: Regression on Dummy Variables
• Chapter 5: Models Not of Full Rank
• Chapter 6: Two Elementary Models
• Chapter 7: The 2-Way Crossed Classification
• Chapter 8: Some Other Analyses
• Chapter 9: Introduction to Variance Components
• Chapter 10: Methods of Estimating Variance Components from Unbalanced Data
• Chapter 11: Variance Component Estimation from Unbalanced Data: Formulae
• Literature Cited
• Statistical Tables
• Index