Linear algebra and matrix theory have long been fundamental tools in mathematical disciplines as well as fertile fields for research in their own right. In this book, and in the companion volume, Topics in Matrix Analysis, we present classical and recent results of matrix analysis that have proved to be important to applied mathematics. The book may be used as an undergraduate or graduate text and as a self-contained reference for a variety of audiences. We assume background equivalent to a one-semester elementary linear algebra course and knowledge of rudimentary analytical concepts. We begin with the notions of eigenvalues and eigenvectors; no prior knowledge of these concepts is assumed.

Facts about matrices, beyond those found in an elementary linear algebra course, are necessary to understand virtually any area of mathematical science, whether it be differential equations; probability and statistics; optimization; or applications in theoretical and applied economics, the engineering disciplines, or operations research, to name only a few. But until recently, much of the necessary material has occurred sporadically (or not at all) in the undergraduate and graduate curricula. As interest in applied mathematics has grown and more courses have been devoted to advanced matrix theory, the need for a text offering a broad selection of topics has become more apparent, as has the need for a modern reference on the subject.

There are several well-loved classics in matrix theory, but they are not well suited for general classroom use, nor for systematic individual study. A lack of problems, applications, and motivation; an inadequate index; and a dated approach are among the difficulties confronting readers of some traditional references.

One historical motivation for introducing the complex numbers C was that polynomials with real coefficients might not have real zeroes. For example, a calculation reveals that {1 + i, 1 − i} are zeroes of the polynomial p(t) = t2 – 2t + 2, which has no real zeroes. All zeroes of any polynomial with real coefficients are, however, contained in C. In fact, all zeroes of all polynomials with complex coefficients are in C. Thus, C is an algebraically closed field: There is no field F such that C is a subfield of F, and such that there is a polynomial with coefficients from C and with a zero in F that is not in C.

The fundamental theorem of algebra states that any polynomial p with complex coefficients and of degree at least 1 has at least one zero in C. Using synthetic division, if p(z) = 0, then t − z divides p(t); that is, p(t) = (t − z)q(t), in which q(t) is a polynomial with complex coefficients, whose degree is 1 smaller than that of p. The zeroes of p are z, together with the zeroes of q. The following theorem is a consequence of the fundamental theorem of algebra.

Theorem. A polynomial of degree n ≥ 1 with complex coefficients has, counting multiplicities, exactly n zeroes among the complex numbers.

The multiplicity of a zero z of a polynomial p is the largest integer k for which (t − z)k divides p(t). If a zero z has multiplicity k, then it is counted k times toward the number n of zeroes of p. It follows that a polynomial with complex coefficients may always be factored into a product of linear factors over the complex numbers.

All the crystal structures that we have considered so far can be described by means of the traditional Bravais lattice and space group formalism; they are all periodic in three dimensions. In this chapter we take a closer look at quasi-periodic structures, which were introduced briefly in Chapter 17. In essence, we will determine how one can obtain a structure that has no single unit cell, but at the same time has a diffraction pattern with sharp peaks. We will discuss 2-D and 3-D quasiperiodic tilings, and the concept of quasicrystals.

Introductory remarks

Icosahedral orientational order in a sharply peaked diffraction pattern was first observed for a rapidly solidified Al−14% Mn alloy (Shechtman et al., 1984; Shechtman and Blech, 1985). These materials were called quasicrystals, and the Al-14% Mn alloy phase was named Shechtmanite. Quasicrystals have long-range orientational order but no 3-D translational periodicity. The discovery of quasicrystals was somewhat unexpected and forced the crystallography community to reexamine some of the basic tenets of its field. Observations of icosahedral symmetry also spurred inquiries into its implications on electronic structure and magnetism (McHenry et al., 1986; McHenry, 1988).

In the previous chapter, we introduced a general method to compute distances between lattice points and angles between lattice directions in an arbitrary crystal system. In the present chapter, we take a closer look at lattice planes. We know that planes will be important, simply by looking at natural faceted crystals. We begin by introducing a notational system, known as Miller indices, that simplifies the identification of lattice planes. In the hexagonal crystal system, there is an ambiguity over which basis vectors to choose, which leads to the introduction of four-component indices. We conclude this chapter with a review of the external shapes of crystals, known as crystal forms.

Miller indices

In the previous chapters, we have seen how directions in a crystal lattice can be labeled, and how we can compute the distance between points, and the angle between lattice directions. What about planes? Figure 5.1 shows 2 × 2 × 2 unit cells of the cF Bravais lattice. In (a), the central horizontal plane of lattice sites is highlighted in gray. In (b), a different plane is highlighted. We can take any three non-collinear lattice points, and create a plane through those points. Such a plane is known as a lattice plane.

In the previous chapters we have learned how X-rays can be used to study the structure of materials, both in terms of the unit cell dimensions and the atom types and positions. In the present chapter we will describe how other types of radiation can be used to obtain the same and, sometimes, additional information. We will begin with neutron diffraction, which has the added benefit of being sensitive to the magnetic structure of a material. Then we cover electron diffraction, which is typically carried out inside a transmission electron microscope. We conclude with a description of the use of synchrotron X-ray sources.

Introductory remarks

Experimental techniques used to study the structure of materials nearly always involve the scattering of electromagnetic radiation or particle waves from atomic configurations. The Bragg equation, along with the concept of the structure factor, forms the basis of a welldeveloped theory that enables us to understand these scattering processes and the structural information that can be derived from them. X-rays are the most commonly used waves for diffraction experiments. Other important and widely used scattering techniques employ the wave-particle duality of electrons and neutrons.

X-ray diffraction experiments are typically the most economical means of determining crystal structures. X-ray diffractometers are commonly found in university, national, and industrial laboratories. Electron diffraction is typically performed using transmission electron microscopes, which are considerably more expensive than typical X-ray diffractometers, but still common in competitive laboratory facilities. Neutron diffraction, on the other hand, is typically performed at national or international reactor facilities. Highenergy, high-flux X-ray scattering experiments are also used to study materials, but they too require advanced and expensive facilities.