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.

Energy, Power, Force, and Pressure

Energy units and their equivalents

joule (abbreviation J); 1 joule = 1 unit of force (a newton) applied over a distance of 1 meter = 0.239 calorie

calorie (abbreviation cal); 1 calorie = heat needed to raise the temperature of 1 gram of water from 14.5 °C to 15.5 °C = 4.186 joules

British Thermal Unit (abbreviation BTU); 1 BTU = heat needed to raise the temperature of 1 pound of water 1 °Fahrenheit from 39.4 to 40.4 °F = 252 calories = 1055 joules

Power

watt (abbreviation W); 1 watt = 1 joule per second

horsepower (abbreviation hp); 1 hp = 746 watts

Force and Pressure

newton (abbreviation N); 1 newton = force needed to accelerate a 1-kilogram mass over a distance of 1 meter in 1 second squared bar (abbreviated b); 1 bar = pressure equivalent to 100,000 newtons on an area of 1 square meter

millibar (abbreviation mb); 1 millibar = one-thousandth of a bar pascal (abbreviation Pa); 1 pascal = force exerted by 1 newton on an area of 1 square meter

atmosphere (abbreviation Atmos.); 1 atmosphere = 14.7 pounds of pressure per square inch = 1013.2 millibars

Length, Area, and Volume

Length 1 micrometer (μm) = 0.000001 meter = 0.0001 centimeter

1 millimeter (mm) = 0.03937 inch = 0.1 centimeter

1 centimeter (cm) = 0.39 inch = 0.01 meter

Chapter Overview

The atmosphere is a complex system, sometimes described as chaotic in nature. In this chapter we examine one of the principal components of that system, the precipitation system, and find that it is indeed complex, but within that complexity, there is a good deal of order to be found in its processes and patterns. We want to learn how precipitation is produced and how and where it is delivered. We trace our way through the atmospheric moisture system, beginning with a brief examination of water vapor, humidity, and condensation and then go on to the processes and causes of precipitation and their geographic circumstances, including the nature of storm systems such as thunderstorms, tornadoes, and hurricanes. The chapter ends with some insights into the nature of violent storms and some of their consequences.

Introduction

The town was nothing more than a general store and saloon on a side road a few hundred feet off US-21 in northern Nebraska. A half mile or so to the north you could see a few houses and beyond them endless farm fields and plains stretching into the horizon. Thunderstorms with great billowing heads dominated the hot afternoon sky creating shadowy islands in an otherwise sunny landscape. Inside the store we milled around looking at antiques and chatting with the clerk. Suddenly, a siren sounded. Almost immediately the clerk said, “It's a severe storm warning, probably a thunderstorm, we get them all the time.”

Chapter Overview

This chapter takes us into the solid Earth and the system of forces that shapes the Earth's crust, the platform of rock upon which the landscape and the oceans rest. This is important to physical geography for crustal processes and features exert a powerful influence on surface systems including oceanic and atmospheric circulation, climate, biogeography, and even the location and operation of watersheds. The story begins in the Earth's interior, the source of energy that powers the crustal processes like volcanoes and earthquakes, and how that energy makes its way toward the planet's surface. To answer this question, it is helpful to take a brief look at Earth's early formation as a planet. This will explain Earth's gross composition and structure and how the internal energy system is constructed. We then go on to examine how the great internal energy system works, how it brings heat to the surface and causes the crust to break apart and move in great sections called tectonic plates.

Introduction

All the great systems that we have discussed so far in this book, including the hydrologic cycle, atmospheric pressure and winds, ocean circulation, and ecosystems, are powered by solar radiation. It is now time to look inside the Earth and examine the system driven by Earth's internal energy system. The term traditionally given to this source of energy is endogenous, that is, having origin within the planet.

Chapter Overview

Mark Twain warned that we risk losing touch with the romance and beauty of a river by knowing about and attaching meaning to its parts. It appears that is a risk we will have to face in this chapter as we go in search of answers to some big questions about streams, beginning with why are they so effective in lowering the landmasses and shaping landscapes? Part of the answer has to do with their geographic coverage. They are one of Earth's most ubiquitous terrestrial features, operating in humid and dry and mountains and flat terrains alike. Part also has to do with the fact that they are organized into great branching systems, large enough to link the interiors of the landmasses to coastal lands and the sea. Accordingly, we begin this chapter with a brief discussion on streams as systems, the nature of streamflow, and the erosional processes of water running in open channels. This is followed by a look at the forms and dynamics of stream channels, how streams cut and shape their valleys, and develop landforms over entire watersheds. Next is a description of some of the grand ideas that have been advanced by scientists over the years to explain how stream systems shape the surfaces of continents. The chapter closes with some remarks about human impacts on stream systems.