S. De Jong

Chapter 29 – Vectors Matrices Operations on Matrices

Abstract This chapter is an extension and generalization of the material presented in Chapter 9. Here we deal with the calculus of vectors and matrices from the point of view of the analysis of a two-way multivariate data table, as defined in Chapter 28 . Such data arise when several measurements are made simultaneously on each object in a set [1]. Usually these raw data are collected in tables in which the rows refer to the objects and the columns to the measurements. For example, one may obtain physicochemical properties such as lipophilicity, electronegativity, molecular volume, etc., on a number of chemical compounds. The resulting table is called a measurement table . Note that the assignment of objects to rows and of measurements to columns is only conventional. It arises from the fact that often there are more objects than measurements, and that printing of such a table is more convenient with the smallest number of columns. In a cross - tabulation each element of the table represents a count, a mean value or some other summary statistic for the various combinations of the categories of the two selected measurements. In the above example, one may cross the categories of lipophilicity with the categories of electronegativity (using appropriate intervals of the measurement scales). When each cell of such a cross-tabulation contains the number of objects that belong to the combined categories, this results in a contingency table or frequency table which is discussed extensively in Chapter 32 . In a more general cross-tabulation, each cell of the table may refer, for example, to the average molecular volume that has been observed for the combined categories of lipophilicity and electronegativity. One of the aims of multivariate analysis is to reveal patterns in the data, whether they are in the form of a measurement table or in that of a contingency table. In this chapter we will refer to both of them by the more algebraic term ‘matrix’. In what follows we describe the basic properties of matrices and of operations that can be applied to them. In many cases we will not provide proofs of the theorems that underlie these properties, as these proofs can be found in textbooks on matrix algebra (e.g. Gantmacher [2]). The algebraic part of this section is also treated more extensively in textbooks on multivariate analysis (e.g. Dillon and Goldstein [1], Giri [3], Cliff [4], Harris [5], Chatfield and Collins [6], Srivastana and Carter [7], Anderson [8]).