Basic matrix algebra

Abstract

It is important to understand that matrices cannot be manipulated in the same way as scalars. This is illustrated in Figure 1. The individual row or column of a matrix is denoted by a low case bold (and optionally as above) italic symbol. Unfortunately, there is no agreed convention as to whether this should refer to a row or column vector. Matrix transposition is used frequently in chemometrics. Some operations like vector multiplication or determining the magnitude of a vector are often expressed using transposes, although there are other ways of formulating these operations. Vector operations can be considered as matrix operations where one dimension is equal to 1. Diagonal matrices occur frequently in chemometrics, for example in principal components analysis. Some of these properties are useful to remember when deriving matrix expressions. Sometimes determinants are very small, but not zero. By small we mean that the magnitude is just a few orders of magnitude below or even comparable to the smallest number that can be stored by a computer. Many traditional texts spend considerable time on describing methods for calculating matrix inverses or determinants, but for the chemometrics expert (or indeed applied statistician), a detailed knowledge is rarely necessary unless one intends to program these procedures from scratch, something very few people will do in their careers. There is no agreed distinction in notation between whether the pseudoinverse is right or left. However, if the number of rows in X equals I and columns equals J, then The equation for the left pseudoinverse can be derived by simple algebra. Because (X′X)−1(X′X) = I and X+X = I, then (X′X)−1 X′ = X+ This article provides the main elements of matrix algebra necessary for a chemometrics expert. There are numerous more detailed web pages and articles and matrix algebra, of which 2 are cited,5, 6 but some involve unnecessary details and others omit operations that are important to chemometrics.