A complex matrix is a matrix whose entries are complex numbers. A complex vector space is one for which the scalars are complex numbers. We shall see that many of the results we have established for real matrices and real vector spaces carry over immediately to complex ones, but there are also some significant differences.

In this chapter, we explore these similarities and differences. We look at eigenvalues and eigenvectors of a complex matrix and investigate unitary diagonalisation, the complex analogue of orthogonal diagonalisation. Certain results for real matrices and vector spaces (such as the result that the eigenvalues of a symmetric matrix are real) are easily seen as special cases of their complex counterparts.

We begin with a careful review of complex numbers.

Complex numbers

Consider the two quadratic polynomials, p(x) = x2 – 3x +2 and q(x) = x2 + x + 1. If you sketch the graph of p(x), you will find that the graph intersects the x axis at the two real solutions (or roots) of the equation p(x) = 0, and that the polynomial factorises into two linear factors: p(x) = x2 – 3x + 2 = (x - 1)(x - 2). Sketching the graph of q(x), you will find that it does not intersect the x axis. The equation q(x) = 0 has no solution in the real numbers, and it cannot be factorised over the reals. Such a polynomial is said to be irreducible. In order to solve this equation, we need to use complex numbers.

Chapter Overview

Soil has been described as the excited skin of the Earth and indeed it is, for like our own skin it is constantly changing as it takes in and gives up heat, water, chemicals, and organic matter. And like our skin, soil is a transition medium between two large spheres, and it shares traits of both, including air and water from above and rock and minerals from below. In this chapter we examine soil as a complex of systems, geomorphic, ecological, hydrologic, and biochemical. These systems are responsible for giving soil its basic form and composition, transforming what often begins as a chaotic mix of organic matter, particles of sediment, water, and other substances into an ordered whole. And since these systems are driven by a larger body of geographic systems, such as climate and hydrology, soils tend to vary correspondingly with these systems. That is, prairie soils are different than rainforest soils, which are different than desert soils, and so on.

Introduction

It was a late afternoon call from Jack Goodnoe. “Bill, what do you know about soils on Staten Island?” “Next to nothing,” I replied. “Why do you ask?” “Well, we've got a project there planning a new cemetery. It calls for thousands of burials, as well mausoleums, roads, waterlines, and landscaping.

Chapter Overview

Our planet is laced with shorelines, more than a million miles of them in all. They are infinitely varied in form, composition, and geographic character, but all have one trait in common: a capacity for unending change. This change comes from a wide variety of sources including earthquakes, tides, volcanoes, glaciers, land use, and hurricanes, but one system stands above all these as the premiere coastal change agent: the geomorphic system of waves and currents. This system operates almost everywhere all the time and is responsible for doing the lion's share of the work in eroding coastal land and transporting and depositing sediment. Our mission here is to understand how that system works, what drives it, and how it is capable of shaping the landforms of this celebrated environment. We also want to know how all this relates to humanity, because humans are particularly fond of the sea coast. Each year more and more people crowd into coastal lands throughout the world. We begin with a brief examination of the various systems that move sediment along the coast and then go on to the master system of wind waves, currents, wave erosion, and coastal landforms.

Introduction

It was a glorious summer morning. We loaded our little boat for a trip along the Lake Superior shore. “What are we looking for, anyway?” Jim asked. “Shoreline features,” I said without thinking.

Chapter Overview

Plate tectonics is the window to understanding the geographic arrangement of so many of the things we once took for granted when looking at maps of the world. Why the sizes and shapes of the continents and ocean basins and what about those large islands, chains of islands, and the great belts of mountains hugging the edges of the continents? We begin with a brief review on the development of the theory itself and then go on to describe the gross features of the Earth's crust to provide a geographic framework for the ensuing discussion. This discussion looks into the nature of plate motion, the conditions and features on the plate borders, and the processes that produce earthquakes and volcanoes. We end the chapter briefly examining a few of the many geographic implications of plate tectonics, in particular the distribution of marsupial and placental mammals and the climate and drainage of monsoon Asia.

Introduction

This is the story of plate tectonics, a theory that ranks with evolution as one of the monumental advances of natural science in the past two centuries. For many reasons, some we have already touched on and many that lie in the pages ahead, this concept is foundational to physical geography. Although the theory of plate tectonics has been with us for only several decades, the roots of this revolutionary idea actually go back several centuries or more.

Chapter Overview

Our planet's surface is a work of natural sculpture created by many sculptors, rivers, glaciers, waves, currents, wind, working in teams under a broad range of conditions and influences. But unlike human sculptors who aim to chisel and mold pieces of art for us to enjoy, unchanged, through time, Earth's sculptures, landforms, are works in progress, constantly changing in response to changing systems and the forces that drive them. Every sculptor begins with raw stone. So it is fitting that we begin this chapter with a brief look at the origin of the rocks that are brought to the Earth's surface as part of the rock cycle, the cycle driven by tectonic forces from the Earth's interior. These rocks are heavy and hard and, in order for them to be sculpted by water and air, they must first be weakened, broken up like an old sidewalk. This is the first phase of all geomorphic systems, one that changes solid rock into loose particles. The second phase involves the movement of those particles downhill by rolling, sliding, and falling under the force of gravity. The chapter concludes with an examination of how mass movements and related processes shape mountainsides and hillslopes and feed rock particles to stream, glacier, wave, current, and wind systems.

Introduction

Landscape is the term commonly applied to the complex of forms and materials that make up the terrestrial environment around us. It is the habitat of humanity, where we and a vast number of other organisms play out our lives.