# Factor Analysis and Related Methods

By Roderick P. McDonald | Go to book overview

Appendix
Some Matrix Algebra

A.1. MATRICES, VECTORS, SCALARS

From the point of view of the mathematician, who prefers on principle not to know what he is talking about, a matrix is an unidentified mathematical object whose definition consists solely in the rules (of addition, multiplication, and the like) that it obeys. From the point of view of practical persons, who usually like to believe that they know what they are talking about, a matrix is primarily a rectangular array of numbers. We can have the best of both worlds if we say that a matrix is an array of numbers that is subject to the fundamental operations, given in the following, of matrix algebra.

A rectangular array of mn numbers, written so as to have m rows and n columns, is said to be an (m × n) matrix or, in other words, a matrix of order (m × n). [We read (m × n) as "m by n," not as "m times n," and carefully distinguish this from mn, as when we say "there are mn numbers altogether, arranged to form the m × n matrix.") If in particular m = n, the matrix is said to be square of order n. We follow the conventions of lexical (reading) order and count rows of the matrix from the top down and columns of the matrix from left to right. The number in the jth row and the kth column is the (j, k)th element of the matrix. Collectively, one or more of the numbers in the matrix are called elements of the matrix. We shall follow the convention of denoting a matrix by a boldface uppercase letter.

We indicate a matrix by writing its elements inside square brackets. Where there is no ambiguity, we may denote an entire matrix by writing its general element inside square brackets. For example, the six numbers a11, a12, a13, a21, a22, a23 are elements of the (2 × 3) matrix:

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Factor Analysis and Related Methods

• Title Page iii
• Contents v
• Preface ix
• 1 - Introduction 1
• Glossary 40
• 2 - Exploratory Common Factor Analysis 50
• 3 - The Analysis of Covariance Structures: Confirmatory Factor Analysis and Pattern Hypotheses 96
• 4 - Models for Linear Structural Relations 113
• 5 - The Problem of Factor Scores 156
• 6 - Problems of Relationship Between Factor Analyses 171
• 7 - Item Response Theory 198
• 8 - Summary 223
• Appendix Some Matrix Algebra 232
• References 249
• Author Index 253
• Subject Index 255
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