In this chapter, we extend our study on real quadratic forms and self-adjoint mappings to the complex situation. We begin by a discussion on the complex version of bilinear forms and the Hermitian structures. We will relate the Hermitian structure of a bilinear form with representing it by a unique self-adjoint mapping. Then we establish the main spectrum theorem for self-adjoint mappings. Next we focus again on the positive definiteness of self-adjoint mappings. We explore the commutativity of self-adjoint mappings and apply it to obtain the main spectrum theorem for normal mappings. We also show how to use self-adjoint mappings to study a mapping between two spaces.

In this chapter, we present two important and related problems in data analysis: the low-rank approximation and principal component analysis (PCA), both based on singular value decomposition. First, we consider the low-rank approximation problem for mappings between two vector spaces. Next we specialize on the low-rank approximation problem for matrices in both induced norm and the Frobenius norm, which are of independent interest for applications. Then we consider PCA. These results are also useful in machine learning. Furthermore, as an extension of the ideas and methods, we present a study of some related matrix nearness problems.

This chapter gives an overview of the different topics covered in the book. As an outlook, we also share our view on potential exciting directions.

Distinguishing between different phases of matter and detecting phase transitions are some of the most central tasks in many-body physics. Traditionally, these tasks are accomplished by searching for a small set of low-dimensional quantities capturing the macroscopic properties of each phase of the system, so-called order parameters. Because of the large state space underlying many-body systems, success generally requires a great deal of human intuition and understanding. In particular, it can be challenging to define an appropriate order parameter if the symmetry breaking pattern is unknown or the phase is of topological nature and thus exhibits nonlocal order. In this chapter, we explore the use of machine learning to automate the task of classifying phases of matter and detecting phase transitions. We discuss the application of various machine learning techniques, ranging from clustering to supervised learning and anomaly detection, to different physical systems, including the prototypical Ising model that features a symmetry-breaking phase transition and the Ising gauge theory which hosts a topological phase of matter.

In this chapter, we study vector spaces and their basic properties and structures. We start by stating the definition and discussing examples of vector spaces. Next we introduce the notions of subspaces, linear dependence, bases, coordinates, and dimensionality. And then we consider dual spaces, direct sums, and quotient spaces. Finally, we cover normed vector spaces.

In this chapter, we consider linear mappings over vector spaces. We begin by stating the definition and discussing the structural properties of linear mappings. Then we introduce the notion of adjoint mappings and illustrate some of their applications. Next we focus on linear mappings from a vector space into itself and study a series of important concepts such as invariance and reducibility, eigenvalues and eigenvectors, projections, nilpotent mappings, and polynomials of linear mappings. Finally, we discuss the use of norms of linear mappings and present a few analytic applications.

In this chapter, everything is brought together to solve the MHD Riemann problem, the most general 1-D MHD problem one can solve semi-analytically. Non-linear waves are introduced in which the 1-D primitive equations are neither steady-state nor linearised. The fast and slow eigenkets are evaluated and their normalisation to account for the not-strictly hyperbolic nature of MHD is emphasised. A method to determine profiles of the primitive variables across slow and fast rarefaction fans is described, including Euler, switch-on, and switch-off fans. A strategy for solving the MHD Riemann problem follows, including use of a multi-variate secant root finder, sixth- order Runge–Kutta, and inverting a 5 × 5 Jacobian matrix with emphasis on characteristic degeneracy and matrix singularity. The chapter concludes with an explicit algorithm for an MHD Riemann solver including numerous examples using a solver developed by the author.