Chapter 19: In this chapter, we introduce new examples of norms, with special attention to submultiplicative norms on matrices. These norms are well-adapted to applications involving power series of matrices and iterative numerical algorithms. We use them to prove a formula for the spectral radius that is the key to a fundamental theorem on positive matrices in the next chapter.

Chapter 17: In this chapter, we investigate applications and consequences of the singular value decomposition. For example, it provides a systematic way to approximate a matrix by a matrix of lower rank. It also permits us to define a generalized inverse for matrices that are not invertible (and need not even be square). The singular value decomposition has a pleasant special form for complex symmetric matrices. The largest singular value is especially important; it turns out to be a norm (the spectral norm) on matrices. We use the spectral norm to study how the solution of a linear system changes if the system is perturbed, and how the eigenvalues of a matrix can change if it is perturbed.

Chapter 5: Many abstract concepts that make linear algebra a powerful mathematical tool have their roots in plane geometry, so we begin the study of inner product spaces with a review of basic properties of lengths and angles in the real two-dimensional plane. Guided by these geometrical properties, we formulate axioms for inner products and norms, which provide generalized notions of length (norm) and perpendicularity (orthogonality) in abstract vector spaces.

Various CHA scholars have contributed to causal process tracing and helped establish it as principal alternative to VBA. A recent, Bayesian-informed version is particularly relevant to CHA because it shares the same historiographical sensibilities of seeing knowledge evolve through a close dialogue between new findings and the existing foreknowledge. It makes causal inferences conditional on a pre-testing articulation of testworthy hypotheses and juxtaposes new test results onto older ones at the post-testing stage. Process tracing defines the testworthiness of hypotheses according to the number and diversity of empirical implications a theory has before the testing starts. It also defines test strength in terms of the specificity of the hypotheses that are tested and of their uniqueness vis-à-vis the alternative explanations against whichthey are paired. The chapter introduces a new tool, the theory ledger, to help evaluate test strength and to update confidence in causal inferencing.

While the classical credibility theory addresses the important problem of combining claim experience and prior information to update the prediction for loss, it does not provide a very satisfactory solution. The method is based on arbitrary selection of the coverage probability and the accuracy parameter. Furthermore, for tractability some restrictive assumptions about the loss distribution have to be imposed.

Historical time is a plural entity, not a singular one. Methodologicals rest on difficult ontological assumptions, which translate into four potential notions of history: cyclical, bounded, serial, and eventful. Cyclical history freezes history the most, by assuming that the present merely repeats the past, and thus makes history de facto reversible. It in effect freezes history, and CHA regards this notion of history as ahistorical. Bounded history freezes certain time intervals during which it assumes that history stands still and the past in effect repeats itself, at least for a limited period of time. It is not interested in how the bounded period is qualitatively different from the periods preceeding or following it. Serial history partially unfreezes history and uses time series data to track secular trends through time. Eventful history is the most unfrozen treatment of the past and tries to identify continuities and discontinuities, or distinct periods. Historical tourism refers to notions of history so static and so frozen that they cease to be historical in any meaningful sense of the word. It identifies variants of historical tourism in history proper and in CHA.

Chapter 8: Many problems in applied mathematics involve finding a minimum-norm solution or a best approximation, subject to certain constraints. Orthogonal subspaces arise frequently in solving such problems. Among the topics we discuss in this chapter are the minimum-norm solution to a consistent linear system, a least-squares solution to an inconsistent linear system, and orthogonal projections.

Chapter 4: In this chapter, we collect some important facts about matrices: the rank-nullity theorem; the intersection and sum of column spaces; rank inequalities for sums and products of matrices; the LU factorization and solutions of linear systems; row equivalence, the pivot column decomposition, and the reduced row echelon form. In a final capstone section, we use linear dependence, the trace, block matrices, induction, and similarity to characterize matrices that are commutators. Throughout the chapter, we emphasize block-matrix methods.

Chapter 7: Unitary matrices play important roles in theory and computation. The adjoint of a unitary matrix is its inverse, so unitary matrices are easy to invert. They preserve lengths and angles, and have remarkable stability properties in many numerical algorithms. In this chapter, we explore the properties of unitary matrices and present several special cases. We derive an explicit formula for a unitary matrix whose first column is given. We give a constructive proof of the QR factorization and show that every square complex matrix is unitarily similar to an upper Hessenberg matrix.

The main focus of this chapter is the estimation of the distribution function and probability (density) function of duration and loss variables. The methods used depend on whether the data are for individual or grouped observations, and whether the observations are complete or incomplete.