In this chapter, we establish the celebrated Jordan decomposition theorem which allows us to reduce a linear mapping over the complex numbers into a canonical form in terms of its eigenspectrum. As a preparation we first recall some facts regarding factorization of polynomials. Then we show how to reduce a linear mapping over a set of its invariant subspaces determined by a prime factorization of the characteristic polynomial of the mapping. Next we reduce a linear mapping over its generalized eigenspaces. Finally, we prove the Jordan decomposition theorem by understanding how a mapping behaves itself over each of its generalized eigenspaces.

The theory of kernels offers a rich mathematical framework for the archetypical tasks of classification and regression. Its core insight consists of the representer theorem that asserts that an unknown target function underlying a dataset can be represented by a finite sum of evaluations of a singular function, the so-called kernel function. Together with the infamous kernel trick that provides a practical way of incorporating such a kernel function into a machine learning method, a plethora of algorithms can be made more versatile. This chapter first introduces the mathematical foundations required for understanding the distinguished role of the kernel function and its consequence in terms of the representer theorem. Afterwards, we show how selected popular algorithms, including Gaussian processes, can be promoted to their kernel variant. In addition, several ideas on how to construct suitable kernel functions are provided, before demonstrating the power of kernel methods in the context of quantum (chemistry) problems.

In this chapter, we change our viewpoint and focus on how physics can influence machine learning research. In the first part, we review how tools of statistical physics can help to understand key concepts in machine learning such as capacity, generalization, and the dynamics of the learning process. In the second part, we explore yet another direction and try to understand how quantum mechanics and quantum technologies could be used to solve data-driven task. We provide an overview of the field going from quantum machine learning algorithms that can be run on ideal quantum computers to kernel-based and variational approaches that can be run on current noisy intermediate-scale quantum devices.

In this chapter, we introduce one of the most important computational tools in linear algebra – the determinants. First, we discuss some motivational examples. Next we present the definition and basic properties of determinants. Then we study some applications of determinants, including the determinant characterization of an invertible matrix or mapping, Cramer’s rule for solving a system of nonhomogeneous equations, and a proof of the Cayley–Hamilton theorem.

Three models of a partially ionised fluid are considered by examining together three sets of (M)HD equations for the neutral, ionised, and electron components of a fluid. The first assumes low ionisation and isothermality leading to the one-fluid, isothermal model where all three non-ideal terms–resistance, the Hall effect, ambipolar diffusion–appear in the induction equation. New quantities introduced include: the ambipolar force density; coupling, rate, and ambipolar coefficients; and resistivity, all helping to determine the relative role of each non-ideal term. For resistive MHD, the Sweet–Parker model for magnetic reconnection, and dynamo theory are discussed. For the Hall effect, a two-fluid, isothermal model is introduced that refines the Sweet–Parker model to give a reconnection time scale in better keeping with observations of solar flares. Finally, the section on ambipolar diffusion derives the full two-fluid, non-isothermal model applicable for a fluid with arbitrary ionisation. Here, exchange terms are introduced to account for mass, momentum, and energy transfers when neutrals ionise or ions recombine.

This chapter looks at four important fluid instabilities – Kelvin–Helmholtz, Rayleigh–Taylor, magneto-rotational, and Parker–where normal mode analysis of the lin-earised equations is taught using each instability as an exemplar. All are examined from the linear regime in which conditions for instability and rates of growth of the fastest mode are developed from first principles. For the KHI, RTI, and MRI, numerical simulations are presented which recover the results of linear analysis from the early stages of a non-linear calculation. For the KHI and RTI, numerical simulations well into the non-linear regime are presented where the onset of fluid turbulence is noted. For the MRI, a section describing how it solved the angular momentum transport problem for accretion discs is included. For the Parker instability, an account is given how this purely astrophysical phenomenon explains the clumpy structure of the interstellar medium.

After some historical perspective on the subject, the introduction attempts to define, distinguish, and link in the broadest terms the various areas of physics related to fluid dynamics. These include fluid mechanics, hydrodynamics, gas dynamics, magnetohydrodynamics, and plasma physics. In particular, the link between ordinary hydrodynamics and magnetohydro-dynamics is made, and the approach this text takes in teaching both, namely wave mechanics, is revealed.