# Preparatory Reading on Matrix Algebra

Multivariate Data Analysis
Psychology 6140

The study of multivariate statistical methods relies heavily on the use of matrix algebra. Prior knowledge of matrix algebra is not a pre-requisite for the course; indeed, the first 4-5 weeks of the course will be devoted to teaching the basic matrix skills needed. Most likely we will use Green & Carroll (1976), Mathematical tools for applied multivariate analysis as the text for this segment of the course.

Nevertheless, students who are unfamiliar with matrices might well devote some extra time over the summer to familiarize themselves with the notation, terminology and basic operations of matrix algebra. The following notes and comments are provided for this purpose.

## Topics & Study Plans

The learning of matrices at an elementary level, for the purposes of Psychology 6140 consists of four things:
1. notation and terminology;
2. the algebra of matrices and vectors;
3. the interpretation of matrices;
4. applications of matrices to multivariate statistics.
Concentrate on areas (1) and (2) in summer reading. I suggest one of two plans of study, based on your previous background.

### A. The Short Course

For those with any previous exposure to matrix algebra, it is probably sufficient to read and work through the summary of matrix algebra provided in one of the following multivariate texts (see Annotated list of multivariate texts for further details). The books below are listed in order of increasing difficulty.
• Cliff, Chapter 1
• Green & Carroll, Chapter 2
• Bock, Chapter 2
• Timm, Chapter 1
Drill youself on some of the problems/exercises. For more extensive review, proceed to the Standard Course, selecting specific areas from one of the syllabuses below.

### B. The Standard Course

Suggested readings (R) and problems (P) are given below from three books on matrix algebra. For the problems, an unqualified section number, e.g. 2.6 means to select a reasonable number of problems from that section, aiming toward drill and consolidating your understanding of that section; qualified section numbers, e.g. 2.3 (1, 12) are meant to direct your attention to specific problems in that section.

It is worthwhile to go through a sufficient number of problems to familiarize yourself with the concepts and operations involved. We will be going over this material at the beginning of the course, so do not get discouraged if some aspects of matrix algebra seem horribly complicated or obtuse.

• Davis, P. J. (1965). The mathematics of matrices. Toronto: Ginn-Blaisdel.
• Chapter 1. What is a matrix?
• R: 1.1 - 1.8
• P: 1.1 - 1.5, 1.7
• Chapter 2. Arithmetic of matrices, I
• R: 2.1 - 2.6, 2.7(49-51), 2.8 - 2.10
• P: 2.1(1), 2.2(9,10), 2.3(1,12), 2.4-2.6, 2.7(1,3) 2.8-2.10(3,4)
• Chapter 3. Arithmetic of matrices, II
• R: 3.1 - 3.4, 3.6 - 3.9
• P: 3.1(1,3), 3.2(p82: 1,3), 3.3(p87: 1,4; p90: 1,4; p94: 1,2), 3.4, 3.6(8), 3.7 - 3.8
• Chapter 4. Linear transformations
• R: 4.1(skim), 4.2 - 4.9
• P: 4.2 - 4.9
• Chapter 5. Determinants
• R: 5.1 - 5.7
• P: 5.1 - 5.7
• Chapter 6. Vectors and inner products
• R: 6.1 - 6.3, 6.7
• P: 6.1 - 6.3, 6.7
• Chapter 8. Characteristic values (Eigenvalues; latent roots)
• R: 8.1
• P: 8.1
• Fuller, L. E. (1962). Basic matrix theory. Englewood Cliffs, N.J.: Prentice Hall.
• Chapter 1. Basic properties of matrices
• R: 1.1-1.5, 1.7-1.10(ignore Hermitian), 1.11, Appendix
• P: 1, 11, 16, 21, 22
• Chapter 2. Elementary matrix operations
• R: 2.1 - 2.5
• P: 1(ab), 4(ab), 6
• Chapter 3. Vector spaces and linear transformations
• R: 3.1 - 3.5
• P: 1, 3, 7
• Chapter 4. Determinants
• R: 4.1 - 4.3, 4.6
• P: 1(abde), 3
• Chapter 5. Characteristic roots and vectors
• R: 5.1 - 5.3, 5.6 - 5.9
• P: 1(c), 2
• Chapter 6. Inversion of matrices
• R: 6.1, 6.5, 6.6
• P: 4(a,b)
• Chapter 8. Homogeneous forms
• R: 8.1 - 8.3
• P: 1, 2
• Searle, S. R. (1966). Matrix algebra for the biological sciences. N.Y.: Wiley.
• Chapter 1. Introduction
• R: 1 - 9
• P: 1.10(1, 2, 4)
• Chapter 2. Elementary matrix operations
• R: 1-12
• P: 2.13(4-6)
• Chapter 3. Determinants
• R: 1, 3, 7
• P: 3.9(1a,b, 3, 4)
• Chapter 4. The inverse of a matrix
• R: 1-6, 8-9
• P: 4.10
• Chapter 5. Rank and linear independence
• R: 1-9
• P: 5.14(1a,b, c, g, 3, 4A, B)
• Chapter 6. Linear equations and generalized inverses
• R: 1-3, 4(a,b), 5(a,b)
• Chapter 7. Latent roots and vectors
• R: 1-3, 5-8
• P: 7.9
• Chapter 8. Miscellanea
• R: 1-3, 5, 7, 10
• P: 8.11(3-6, 9)
• Chapter 9. The matrix algebra of regression analysis
• R: 1-11, 14
• Chapter 10. Some matrix algebra of linear statistical models
• R: 1-10