ii MATRIX EXCERPTS FROM MATRIX AND POWER SERIES METHODS Mathematics 306 All You Ever Wanted to Know About Matrix Algebra and Infinite Series But Were Afraid To Ask By John W. Lee Department of Mathematics Oregon State University January 2006 iii iv Contents I Background and Review 1 1ComplexNumbers 3 1.1Goals................................. 3 1.2Overview............................... 3 1.3TheComplexNumberSystem.................... 4 1.4PropertiesofAbsoluteValuesandComplexConjugates..... 8 1.5TheComplexPlane......................... 8 1.6CirclesintheComplexPlane.................... 12 1.7ComplexFunctions.......................... 14 1.8SuggestedProblems......................... 15 2 Vectors, Lines, and Planes 19 2.1Goals................................. 19 2.2Overview............................... 19 2.3Vectors................................ 19 2.4DotProducts............................. 24 2.5RowandColumnVectors...................... 27 2.6LinesandPlanes........................... 28 2.7SuggestedProblems......................... 30 II Matrix Methods 33 3 Linear Equations 35 3.1Goals................................. 35 3.2Overview............................... 35 3.3ThreeBasicPrinciples........................ 36 3.4SystematicEliminationofUnknowns................ 40 3.5Non-SquareSystems......................... 45 3.6 E!cientEvaluationofDeterminants................ 47 3.7ComplexLinearSystems....................... 48 3.8SuggestedProblems......................... 48 v vi CONTENTS 4 Matrices and Linear Systems 51 4.1Goals................................. 51 4.2Overview............................... 51 4.3BasicMatrixOperations....................... 51 4.4LinearSystemsinMatrixForm................... 55 4.5TheInverseofaSquareMatrix................... 57 4.6TransposeofaMatrix........................ 60 4.7SuggestedProblems......................... 60 5 Linear Dependence and Independence 65 5.1Goals................................. 65 5.2Overview............................... 65 5.3DependenceandIndependence................... 65 5.4SuggestedProblems......................... 71 6 Matrices and Linear Transformations 73 6.1Goals................................. 73 6.2Overview............................... 73 6.3MatricesasTransformations..................... 73 6.4MatrixofaLinearTransformation................. 76 6.5Projections.............................. 80 6.6SuggestedProblems......................... 83 7 Eigenvalue Problems 87 7.1Goals................................. 87 7.2Overview............................... 87 7.3BasicProperties........................... 88 7.4SymmetricMatrices......................... 94 7.5SuggestedProblems......................... 99 8 Catch Up and Review 103 III Appendix 107 9SelectedAnswers 109 Part I Background and Review 1 Lesson 1 Complex Numbers, Variables, and Functions 1.1 Goals • Be able to add, subtract, multiply, and divide complex numbers • Represent complex numbers as points or vectors in the complex plane and illustrate addition and subtraction with the parallelogram (or triangle) law • Learn the algebraic properties of absolute values and complex conjugates given in the lesson • Write equations or inequalities that define any circle, disk, or the interior or exterior of a disk in the complex plane • Learn the terms real part, imaginary part, and complex conjugate of a complex number and be able to illustrate them geometrically • Express complex numbers in polar form and use polar forms to calculate powers and roots of complex numbers 1.2 Overview Most of Lesson 1 is designed for self-study and/or review. Your instructor will only cover a few highlights. Complex numbers play an important role in engineering and scientific prob- lems. Mathematical models of real-world problems that involve oscillatory be- havior typically lead to calculations and graphical visualizations in which com- plex numbers and functions come up in very natural ways. In particular this is true for problems involving matrix methods and power series expansions, 3 4 LESSON 1. COMPLEX NUMBERS which are the principal areas of study in MTH 306. Thus, we begin our study with a review of some basic properties of complex numbers and with a brief introduction to complex functions. 1.3 The Complex Number System The number systems you use today evolved over several thousand years. New types of numbers and their related rules of arithmetic were developed in order to solve increasingly sophisticated problems that were encountered as human societies became increasingly complex. You should be familiar will most of the kinds of numbers listed below (in the order of their historical development). 1. natural numbers: 1, 2, 3, 4, 5, ... 2. integers: ..., 5> 4> 3> 2> 1> 0> 1> 2> 3> 4> 5> === . 3. rational numbers (fractions): numbers of the form s@t where s and t are integers and t 6=0= For example, 2/3, and 5@2 are rational numbers. 4. real numbers: numbers which can be represented in decimal form with a possibly infinite number of decimal digits following the decimal point. 5. complex numbers: numbers of the form d + le where d and e are real numbers and l, called the imaginary unit, satisfies l2 = 1= Evidently each natural number is an integer. Each integer is a rational num- ber. For example, the integer 5 can also be expressed as 5/1. Rational numbers are real numbers because each rational number has a finite or periodically re- peating decimal expansion that can be found by long division. For instance, 3@8=0=375 and 13@7=1= |857142{z } |857142{z } ··· . Finally, all real numbers are complex numbers. For example, 5=5+l0. The primary reason for extending the natural number system to include the other numbers above was the need to solve increasingly complex equations that came up in agriculture, astronomy, economics, science, and engineering. Let’s look at a few simple examples to motivate the need to introduce new number systems. The linear equation { +2=5 has solution { =3in the natural numbers. On the other hand, { +5=2 has no solution in the system of natural numbers. The same is true of the equation { +2=2= 1.3. THE COMPLEX NUMBER SYSTEM 5 The need to solve simple equations such as these led to the integers and their familiar arithmetic properties. More linear equations can be solved in the system of integers than in the system of natural numbers, but we still cannot solve all linear equations. For instance, 2{ 4=6 has solution { =5but we cannot solve 2{ 4=5 in the system of integers. You can see how the need to solve this last equation led naturally to the system of rational numbers, in which the equation has solution { =9@2= Indeed, any linear equation with integer coe!cients can be solved in the rational number system. The equation d{ + e = f> with d 6=0> has solution { =(f e) @d= If d> e> and f are integers, the solution is a rational number. Quadratic equations are more challenging. Not all quadratic equations can be solved in the rational number system. In fact, most quadratic equations do not have rational number solutions. This shortcoming of the rational numbers was known to the Greeks. In particular, the Pythagorean school knew that the quadratic equation {2 =2> in which { gives the length of a diagonal of a square with side 1> has no solution in the system of rational numbers. We simply cannot determine the diagonal of such a unit square within the rational number system. The real numbers system is needed in order to solve this equation and the solution is { = 2=1=414213562 ··· > an unending nonperiodic decimal. The introduction of the real number system enables us to solve more quadratic equations, but not all such equations. You probably encountered this short com- ing in connection with the quadratic formula which expresses the solutions to the quadratic equation 2 d{ + e{ + f =0>d6=0> as s e ± e2 4df { = = 2d The resulting solution formula give real number solutions only if e2 4df 0= What are we to make of the formula when e2 4df ? 0? For example, the quadratic formula applied to {2 +2{ +3=0yields s s 2 ± 4 12 2 ± 8 = = { 2 2 = Here is the dilemma: In the real numbers system, the product of a number by itself is always positive or zero. So if 8 is real, then its product with itself must bes positive or zero, but, on the other hand, its product with itself must be 8 if 8 stands for a number whose square is 8= You can appreciate that mathematicians of the fifteenth century were puzzled. Indeed it was not until 6 LESSON 1. COMPLEX NUMBERS the nineteenth century that this dilemma was fully resolved and the complex number system was placed on a firm foundation. There is no real number whose 8 squares is but there is a complex number with that property; one such number is l 8,wherel is the imaginary unit. Once complex numbers are available, all quadratic equations can be solved. In fact, the quadratic formula gives the solution not only for all quadratic equa- tions with real coe!cients, the same formula gives the solution to all quadratic equations with complex (number) coe!cients. The complex number system is even more powerful when it comes to solving equations. It is a fact, called the Fundamental Theorem of Algebra, that every polynomial (regardless of its de- gree) with complex coe!cients can be solved in the complex number system. That is, all the roots of any such polynomial equation are complex numbers. The historical confusion that preceded a clear understanding of the complex number system has led to some unfortunate terminology that persists to this day. For example, given the complex number