4 - Matrix Algebra and Multiple Linear Regression: Part 1

Abstract

This chapter demonstrates that matrix expressions can be readily verified through straightforward application of the basic matrix operations. A factor to be considered is the relationship between the quasi-algebraic expressions that matrix operations are normally written in and the computations that are used to implement those relationships. The complications and the amount of computation involved in actually doing a matrix inversion are enough to make even the most intrepid mathematician/statistician/chemometrician run for the nearest computer with a preprogrammed algorithm for the task. From the definition of the inverse of a matrix, a unit matrix is obtained by multiplying the inverse of a given matrix by the matrix itself. When there are fewer equations than unknowns, it is clear that there is not enough information available to determine the values of the unknown variables. Spectroscopists are concerned with the application of these mathematical techniques to the solution of spectroscopic problems, particularly, the use of spectroscopy to perform quantitative analysis, which is done by applying these concepts to a set of linear equations.