Chapter 16: This chapter is about the singular value decomposition, a matrix factorization that shares some features with the spectral decomposition of a normal matrix. However, every real or complex matrix, normal or not, square or not, has a singular value decomposition. It was discovered in the nineteenth century and was applied in the 1930s to problems in psychometrics. Widespread use of the singular value decomposition in applications began only in the 1960s, when accurate and efficient algorithms were developed for its numerical computation. Today, singular value subroutines are embedded in thousands of computer algorithms to solve problems in data analysis, image compression, chemical physics, least-squares problems, low-dimensional approximation of high-dimensional data, genomics, robotics, and many other fields.

Chapter 20: This chapter is about some remarkable properties of positive matrices, by which we mean square matrices with real positive entries. Positive matrices are found in economic models, genetics, biology, team rankings, network analysis, Google's PageRank, and city planning. The spectral radius of any matrix is the absolute value of an eigenvalue, but for a positive matrix the spectral radius itself is an eigenvalue, and it is positive and dominant. It is associated with a positive eigenvector, whose ordered entries have been used for ranking sports teams, priority setting, and resource allocation in multicriteria decision-making. Since the spectral radius is a dominant eigenvalue, an associated positive eigenvector can be computed by the power method. Some properties of positive matrices are shared by nonnegative matrices that satisfy certain auxiliary conditions. One condition that we investigate in this chapter is that some positive power has no zero entries.

A useful way to solve a complex problem – whether in physics, mathematics, or life in general – is to break it down into smaller pieces that can be handled more easily. This is especially true of the Ising model. In this chapter, we investigate various partial-summation techniques in which a subset of Ising spins is summed over to produce new, effective couplings among the remaining spins. These methods are useful in their own right and are even more important when used as a part of position-space renormalization-group techniques.

Chapter 9: In the next four chapters, we develop tools to show that each square complex matrix is similar to an essentially unique direct sum of special bidiagonal matrices (the Jordan canonical form). The first step is to show that each square complex matrix has a one-dimensional invariant subspace and explore some consequences of that fact.

Chapter 15: Many interesting mathematical ideas evolve from analogies. If we think of matrices as analogs of complex numbers, then the representation z = a + bi suggests the Cartesian decomposition A = H + iK of a square complex matrix, in which Hermitian matrices play the role of real numbers. Hermitian matrices with nonnegative eigenvalues are natural analogs of nonnegative real numbers. They arise in statistics (correlation matrices and the normal equations for least-squares problems), Lagrangian mechanics (the kinetic energy functional), and quantum mechanics (density matrices). They are the subject of this chapter.

Eventful analysis employs the most unfrozen and hence the most exploratory strand of CHA. It employs historical comparisons and explores transformation patterns, that is, patterns of qualitative change. It uses two key tools: historical description and conceptualization. The aim of historical description is to figure out what is going on, to gain a basic understand of a phenomenon before proceeding to explain it. Often this involves de-redescribing a phenomena that has qualitatively changed over time. Historical description, in turn, involves six concrete steps: fact gathering, chronicling, concatenation, periodizing, looking for intercurrence patterns, and rethinking research questions. Conceptualization serves to make historical description more comparativist and to explore broader patterns. The chapter discusses how to replace proper names with broader concepts by defining both the positive and the negative pole of concepts. It lists criteria for assessing the content and temporcal validity of concepts.

Some problems arising from loss modeling may be analytically intractable. Many of these problems, however, can be formulated in a stochastic framework, with a solution that can be estimated empirically. This approach is called Monte Carlo simulation. It involves drawing samples of observations randomly according to the distribution required, in a manner determined by the analytic problem.

In this chapter, we discuss some applications of Monte Carlo methods to the analysis of actuarial and financial data. We first revisit the tests of model misspecification introduced in Chapter 13.

In the chapters so far, we have studied a number of exact methods of calculation for Ising models. These studies culminated in the exact solution for an infinite one-dimensional Ising model, as well as the corresponding solution on a 2 × ∞ lattice. Neither of these systems shows a phase transition, however. In this chapter, we start with Onsager’s exact solution for the two-dimensional lattice, which quite famously does have a phase transition. Next, we explore exact series expansions from low and high temperature, and show how these results can be combined, via the concept of duality, to give the exact location of the phase transition in two dimensions.

Some models assume that the failure-time or loss variables follow a certain family of distributions, specified up to a number of unknown parameters. To compute quantities such as average loss or VaR, the parameters of the distributions have to be estimated. This chapter discusses various methods of estimating the parameters of a failure-time or loss distribution.

Chapter 18: If a Hermitian matrix is perturbed by adding a rank-1 Hermitian matrix, or by bordering to obtain a larger Hermitian matrix, the eigenvalues of the respective matrices are related by interlacing inequalities. We use subspace intersections to study eigenvalue interlacing and the related inequalities of Weyl. We discuss applications of eigenvalue interlacing, including Sylvester's principal-minor criterion for positive definiteness, singular value interlacing between a matrix and a submatrix, and majorization inequalities between the eigenvalues and diagonal entries of a Hermitian matrix. We also prove Sylvester's inertia theorem, and then use the polar decomposition to prove a generalization of the inertia theorem for normal matrices.