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Chapters I-V are (possibly first year) undergraduate material. For some of the better prepared A-level holders, much of Chapter I-IV in particular is more of a refresher, and provides a common framework of notation and formal theory with the rest of the book.
Indeed, the book begins, and again reinforces something which one would actually hope to be known to a reasonable numerate student: the resolution of a system of linear equations by the elimination method.
There are two main reasons for including, and even repeating this rather elementary material. There are students who actually need a brief on how to solve three unknown variables from a system of three linear equations in a systematic way. For students of average and above-average numeracy, a further educational reason is that familiarity with the particular version of the elimination method which is used in a large part of the book, is essential for the appreciation of some of its implications. Chapter V on rank and Chapter VI on definiteness both contain proofs, which refer to a particular tabulation-arrangement.
While Chapters I to IV are fairly elementary, V begins to introduce a somewhat higher level of abstraction, and proofs by recursive induction. Chapters VI -XI cover material, which is in terms of its substance generally classified as more advanced: definiteness, the characteristic equations of both symmetric and of non-symmetric matrices, similarity and diagonalization, triangularization, the inversion of complex matrices, the Moore-Penrose inverse, principal components, rotation-operators and the properties of non-negative square matrices.
Within these later chapters, there is still some degree of gradual increase in sophistication.
Chapter VI deals with definiteness without using latent roots.
The exposition of latent roots and characteristic vectors is split into two chapters, VII and VIII, one meant for undergraduates and the other not. The seventh chapter contains the basics and does not go much beyond the definition of characteristic vectors and the derivation of the characteristic equation. It does contain an introduction to complex numbers, the main purpose of which is to provide a background for showing that symmetric matrices have only real roots.
I The development of theory in the seventh chapter is largely
limited to what is prerequisite for Chapter IX, which deals with the symmetric Eigenvalue problem.
Chapter X (geometrical interpretations) falls itself into a fairly elementary part (starting with elementary coordinate geometry), and a somewhat more advanced part, rotation-operators and n-dimensional geometry. The n-dimensional concept of area is obviously relevant in econometrics, where it arises as the ratio between the density functions in the spaces of observed variables and structural residuals.
The eleventh chapter deals with non-negative square matrices. It briefly deals with the relationship with Input / Output analysis, but only insofar as this concerns the specific mathematical properties of this type of matrices. Non-economists, who are not concerned with that particular application, will obviously skip those few passages. As far as economists, who are indeed concerned with the I/O analysis application are concerned, it should be stressed that this chapter is not meant as an introduction to the basic accounting framework. There are other textbooks which cover that, including chapter III of mine own [22].
There are two sections of the book, which contain a possibly novel contribution to the subject.
In section 7.7 it is shown that, in the special cases of triangular or symmetric matrices, a repeated root is associated with the separate vanishment of each of the principal minors of [1Л-А], which are of the appropriate order, not just the sum of a group. For a triangular matrix that is obvious, but for a symmetric matrix that is not so immediately obvious. Yet I have not found that theorem anywhere else.
Section 11.5 identifies the structure of a matrices which are non-negative and in-decomposable, but not also semi-positive.
However, it should be stressed that the main purpose of the book is to provide briefing material for students and research workers.
One other issue which is useful to discuss here, is the not entirely standard approach to defining the inverse. There is, in the literature generally, a development away from starting with determinants, and it probably is now conventional to define the inverse asAB = BA=I. |