This chapter provides an overview of matrices. Basic matrix operations are introduced first, such as addition, multiplication, transposition, and so on. Determinants and matrix inverses are then defined. The rank and Kruskal rank of matrices are defined and explained. The connection between rank, determinant, and invertibility is elaborated. Eigenvalues and eigenvectors are then reviewed. Many equivalent meanings of singularity (non-invertibility) of matrices are summarized. Unitary matrices are reviewed. Finally, linear equations are discussed. The conditions under which a solution exists and the condition for the solution to be unique are also explained and demonstrated with examples.

Matrices are introduced and it is explained how matrix addition, scalar multiplication and multiplication of two matrices works. As an example of matrix multiplication, it is demonstrated how investment portfolios can be modelled, and their returns in various states quantified by multiplication with a returns matrix. The concept of an arbitrage portfolio is explained.

This chapter reviews vectors and matrices, and basic properties like shape, orthogonality, determinant, eigenvalues, and trace. It also reviews operations like multiplication and transpose. These operations are used throughout the book and are pervasive in the literature. In short, arranging data into vectors and matrices allows one to apply powerful data analysis techniques over a wide spectrum of applications. Throughout, this chapter (and book) illustrates how the ideas are implemented in practice in Julia.

This chapter investigates how Napoleon funded his war effort during the fifteen years that he was in power. Coming to power in the wake of the French monarchy’s financial insolvency and the profound turmoil of the French Revolution, Napoleon chose to become personally involved in financial affairs of the nation, opting for radical but effective solutions. This aspect of his accomplishments is usual forgotten. This is all the more unfair as his successes were real and stark. By rationalizing old principles, the winner of Austerlitz gave himself the means to embody the “God of war” but failed to win the battle for credit due in large part to the disastrous legacy he received from regimes that had preceded him.

This chapter looks at the tools and procedures that programmes use to assess claims. Assessing a redress claim involves deciding which injuries to redress and how much money to pay. Those judgements are difficult. Since people will reasonably disagree, good procedure is essential. Enabling survivors to choose how the programme will assess their claims can help protect survivors’ privacy and well-being and make programmes more effective and efficient.

Many scientific questions lead to hypotheses about random vectors. For instance, the question of whether global warming has occurred over a geographic region is a question about whether temperature has changed at each spatial location within the region. One approach to addressing such a question is to apply a univariate test to each location separately and then use the results collectively to make a decision. This approach is called multiple testing or multiple comparisons and is common in genomics for analyzing gene expressions. The disadvantage of this approach is that it does not fully account for correlation between variables. Multivariate techniques provide a framework for hypothesis testing that takes into account correlations between variables. Although multivariate tests are more comprehensive, they require estimating more parameters and therefore have low power when the number of variables is large. Multivariate statistical analysis draws heavily on linear algebra and includes a generalization of the normal distribution, called the multivariate normal distribution, whose population parameters are the mean vector and the covariance matrix.

We introduce the mathematical modeling process. We also set the stage for the rest of the book by discussing systems of linear equations and their solutions, matrices, Gauss-Jordan elimination, linear combinations of vectors, basis vectors of Euclidean space, and their connection to basic solutions of a linear system. We conclude with simple optimization problems with quadratic functions.

This is a brief chapter, establishing basic concepts of matrices and vectors, and particularly focusing on differentiation of matrices and vectors.

Chapter 6 covers global flat foldability.This includes determining how we can tell if a crease pattern with multiple vertices will fold flat without forcing the paper to self-intersect, as well as discovering properties that all such crease patterns have, beyond what was covered in the previous chapter.Justin’s Theorem, which is a generalization of Kawasaki’s and Maekawa’s Theorems and whose proof uses elements of basic knot theory, is covered, as are Justin’s non-crossing conditions that provide necessary and sufficient conditions for a general crease pattern to fold flat.The matrix model from Chapter 5 is developed further to create a formal folding map for flat origami.Finally, Bern and Hayes’ seminal proof that determining flat foldability of a given crease pattern is NP-Hard is presented and updated with more recent results on box pleating.

The field of flat-foldable origami is introduced, which involves a mix of geometry and combinantorics.This chapter focuses on local properties of flat origami, meaning the study of how and when a single vertex in an origami crease pattern will be able to fold flat.The classic theorems of Kawasaki and Maekawa are proved and generalizations are made to folding vertices on cone-shaped (i.e., non-developable) paper.The problem of counting valid mountain-valley assignments of flat-foldable vertices is solved, and the configuration space of flat-foldable vertices of a fixed degree is characterized.A matrix model for formalizing flat-vertex folds is introduced, and the chapter ends with historical notes on this topic.

This chapter takes the concept of rigid origami and puts it in motion, studying how a crease pattern flexes from the unfolded state to a continuum of rigid origami states.The treatment presented starts with the more general theory of flexible polyhedral surfaces, then moves to the special case of origami.The configuration space of the rigid foldings of a crease pattern is introduced, and the tools of reciprocal-parallel and reciprocal diagrams are used to establish conditions for infinitesimal and second-order rigid foldability.This is used to prove that a single-vertex origami crease pattern has a rigid folding from the unfolded state if and only if it has a nontrivial zero-area reciprocal diagram.These results are then used to establish equations for the folding angles of a degree-4 flat-foldable vertex that are linear when parameterized by the tangent of half the folding angles, also known as the Weierstrass transformation.An intrinsic condition for an origami vertex crease pattern to be rigidly foldable from the unfolded state is also given.

Rigid origami describes origami where each face of the crease pattern is flat, as if made from stiff metal.Modeling rigid origami with matrices allows one to describe materials that have been folded into a three-dimensional shape, as opposed to flat origami.This chapter describes this matrix model and proves its key features.In addition, a generalization of Maekawa’s Theorem for three-dimensional rigid origami is introduced, as is modeling rigid origami with the Gauss map from differential geometry.The latter turns out to be a useful tool for the remainder of the book.

We turn next to the case of additional masses.In one dimension, we can attach masses by springs to achieve collective motions that occur at a single frequency, the normal modes.Building general solutions, using superposition, from this ``basis" of solutions is physically relevant and requires a relatively formal treatment of linear algebra, the mathematical topic of the chapter.

In this chapter, an overview is provided of the types of fibre and matrix in common use and of how they are assembled into composites. Many types of reinforcement, mostly fibres, are available commercially. Their properties are related to atomic structure and the presence of defects, which must be controlled during manufacture. Matrices may be based on polymers, metals or ceramics. Choice of matrix is usually related to required properties, component geometry and method of manufacture. Certain composite properties may be sensitive to the nature of the reinforcement/matrix interface; this topic is covered in Chapter 7. Properties are also dependent on the arrangement and distribution of fibres, i.e. the fibre architecture, an expression that encompasses intrinsic features of the fibres, such as their diameter and length, as well as their volume fraction, alignment and spatial distribution. Fibre arrangements include laminae (sheets containing aligned long fibres) and laminates that are built up from these. Other continuous fibre systems, such as woven configurations, are also covered. Short fibre systems can be more complex and methods of characterising them are also briefly described.