# Mathematical Fundamentals

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## Abstract

Matrix algebra is one of the essential tools of energy systems analysis. A grasp of even the most fundamental concepts allows much that would otherwise be extremely complex to be reduced to a few simple matrix equations; and the definition of most statistical tools, such as least squares regression, becomes extremely simple when the tedium of scalar algebra is replaced by the elegance of matrix expressions.

## Keywords

Extreme Point Matrix Algebra Feasible Space Principal Diagonal Residential Energy Consumption
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## References

- 1.This Chapter is designed solely as a brief refresher of the basic tools of matrix algebra, statistical analysis, and linear programming. We make no claim of comprehensiveness; indeed, the idea here is to present the minimum amount of information necessary for the comprehension of the techniques presented in subsequent chapters.Google Scholar
- 2.Transposition is sometimes denoted by use of a prime — thus A
^{+}is the transpose of A.Google Scholar - 3.The general definition of the determinant of an n-th order matrix A is given by as |A| = Σ ± a
_{1}α c_{2}β… a_{nu}the sum being taken over all permutations of the second subscripts, with even permutations having positive signs, add permutations a negative sign. A permutation is said to be odd when the number of inversions is odd: an inversion, in turn, is said to occur when of, say, two integer subscripts the larger precedes the former. For further discussion, see e.g. Johnston (1963).Google Scholar - 5.This, too, is an enourmous subject, to which many textbooks have been dedicated. Our purpose here is simply to review the basics, introduce those techniques most commonly encountered in the statistical analysis of energy economic phenomena, and to lay the groundwork for some of the more complex estimation problems (of energy demand elasticities, and the like), presented in Chapter 4. Any student familiar with this material can proceed to the next section since only fundamentals are discussed here (as they might be treated in any graduate level course on econometrics).Google Scholar
- 6.Minimizing the sum of the absolute values of the deviations leads to a linear programming problem (known as MAD estimation, for minimum absolute deviations). For an excellent introduction to this approach, see e.g. Rogers (1968). The MAD approach is sometimes useful in situations in which conventional least squares estimation is beset by serious problems, some of which we shall encounter later in these pages. The MAD technique has perhaps found greatest application in problems of mathematical demography.Google Scholar
- 8.We follow here the notational convention used in Johnston (1963)Google Scholar
- 9.The rules of matrix and vector differentiation are analogous to that of scalars: compare (2.36) and (2.37) with (2.25) and (2.27), and recall footnote 7, supra.Google Scholar
- 10.See e.g. Johnston (1963) p. 112.Google Scholar
- 11.This is the basic form of regression model in M. Chenery and M. SyrguinGoogle Scholar
- (1975), although we have omitted here the non linear terms and time dummy variables for illustrative purposes.Google Scholar
- 12.The classic illustration of the difference between a type I and type II error is in terms of the axiom of anglo-saxon law: the presumption being that an accused is innocent (the null hypothesis) unless proven guilty (the alternative). The requirement for unanimity in juries is an effort to minimize type I error; under no circumstances do we wish to reject the null hypothesis if it is in fact true (i.e. we wish to minimize convictions of persons actually innocent) even if it means committing a large type II error (accepting innocence even though the individual may be guilty).Google Scholar
- 13.Also known sometimes as the region of significance.Google Scholar
- 14.This test is most commonly used to test for sample means where the sample size is small.Google Scholar
- 15.The value of t is tabulated in most statistics textbooks.Google Scholar
- 16.For a more leisurely exposition, but also based on a single numerical abstraction (of “Malawi”) see e.g. Todaro (1971), p. 87ff.Google Scholar
- 17.This “country” will be used for illustrative purposes throughout this book.Google Scholar
- 18.A convex space is one for which it is impossible to find a line drawn between two points on the boundary of the space that intersects another part of the boundary. For example, the space is not convex, since the line AB intersects the boundaryGoogle Scholar
- 19.Subject of course to the exception of having an objective function that has the identical slope to one of the constraints: then the objective function will coincide with that constraint, and any one of the combinations of X
_{1}and X_{2}that lie along the feasible segment of such a constraint will be optimal. For example, if in our example, the objective function were Max S = 4x_{1}+ 8x_{2}, then the optimum would be anywhere along the line YY of Figure 2.6.Google Scholar - 20.For a full discussion of the numerical problems of computing inverse matrices, see Forsythe and Moier (1967).Google Scholar
- 21.But see exercise 4 for a discussion of significant figures.Google Scholar

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© Springer-Verlag Berlin Heidelberg 1984