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9781133110873_0801.qxp 3/10/12 8 8.1 8.2 8.3 8.4 8.5 6:52 AM Page 391 Complex Vector Spaces Complex Numbers Conjugates and Division of Complex Numbers Polar Form and DeMoivre’s Theorem Complex Vector Spaces and Inner Products Unitary and Hermitian Matrices Quantum Mechanics (p. 425) Signal Processing (p. 417) Elliptic Curve Cryptography (p. 406) Mandelbrot Set (p. 400) Electric Circuits (p. 395) 391 Clockwise from top left, James Thew/Shutterstock.com; Mikhail/Shutterstock.com; Sybille Yates/Shutterstock.com; Lisa F. Young/Shutterstock; Štepán Kápl/Shutterstock.com 9781133110873_0801.qxp 392 3/10/12 Chapter 8 6:52 AM Page 392 Complex Vector Spaces 8.1 Complex Numbers Use the imaginary unit i to write complex numbers. Graphically represent complex numbers in the complex plane as points and as vectors. Add and subtract two complex numbers, and multiply a complex number by a real scalar. Multiply two complex numbers, and use the Quadratic Formula to find all zeros of a quadratic polynomial. Perform operations with complex matrices, and find the determinant of a complex matrix. COMPLEX NUMBERS REMARK When working with products involving square roots of negative numbers, be sure to convert to a multiple of i before multiplying. For instance, consider the following operations. 冪1冪1 i i i2 1 Correct 冪1冪1 冪共1兲共1兲 冪1 1 Incorrect So far in the text, the scalar quantities used have been real numbers. In this chapter, you will expand the set of scalars to include complex numbers. In algebra it is often necessary to solve quadratic equations such as x 2 3x 2 0. The general quadratic equation is ax 2 bx c 0, and its solutions are given by the Quadratic Formula x b 冪b2 4ac 2a where the quantity under the radical, b2 4ac, is called the discriminant. If b2 4ac 0, then the solutions are ordinary real numbers. But what can you conclude about the solutions of a quadratic equation whose discriminant is negative? For instance, the equation x2 4 0 has a discriminant of b2 4ac 16, but there is no real number whose square is 16. To overcome this deficiency, mathematicians invented the imaginary unit i, defined as i 冪1 where i 2 1. In terms of this imaginary unit, 冪16 4冪1 4i. With this single addition of the imaginary unit i to the real number system, the system of complex numbers can be developed. Definition of a Complex Number If a and b are real numbers, then the number a bi is a complex number, where a is the real part and bi is the imaginary part of the number. The form a bi is the standard form of a complex number. Some examples of complex numbers written in standard form are 2 2 0i, 4 3i, and 6i 0 6i. The set of real numbers is a subset of the set of complex numbers. To see this, note that every real number a can be written as a complex number using b 0. That is, for every real number, a a 0i. A complex number is uniquely determined by its real and imaginary parts. So, two complex numbers are equal if and only if their real and imaginary parts are equal. That is, if a bi and c di are two complex numbers written in standard form, then a bi c di if and only if a c and b d. 9781133110873_0801.qxp 3/10/12 6:52 AM Page 393 8.1 Imaginary axis Complex Numbers 393 THE COMPLEX PLANE (a, b) or a + bi b a Real axis Because a complex number is uniquely determined by its real and imaginary parts, it is natural to associate the number a bi with the ordered pair 共a, b兲. With this association, complex numbers can be represented graphically as points in a coordinate plane called the complex plane. This plane is an adaptation of the rectangular coordinate plane. Specifically, the horizontal axis is the real axis and the vertical axis is the imaginary axis. The point that corresponds to the complex number a bi is 共a, b兲, as shown in Figure 8.1. The Complex Plane Plotting Numbers in the Complex Plane Figure 8.1 Plot each number in the complex plane. a. 4 3i b. 2 i c. 3i d. 5 SOLUTION Figure 8.2 shows the numbers plotted in the complex plane. a. Imaginary axis Imaginary axis b. 2 4 4 + 3i or (4, 3) 3 2 1 −3 −2 −1 1 −2 −1 1 2 3 4 2 3 Real axis −2 − i or (−2, −1) Real axis −3 −2 −4 Imaginary axis c. 1 d. Imaginary axis 2 4 1 −3 −2 −1 1 2 3 2 1 −2 −3 3 Real axis −3i or (0, −3) −4 −1 5 or (5, 0) 1 2 3 4 5 Real axis −2 Figure 8.2 Another way to represent the complex number a bi is as a vector whose horizontal component is a and whose vertical component is b. (See Figure 8.3.) (Note that the use of the letter i to represent the imaginary unit is unrelated to the use of i to represent a unit vector.) Imaginary axis 1 Horizontal component Real axis −1 −2 Vertical component 4 − 2i −3 Vector Representation of a Complex Number Figure 8.3 9781133110873_0801.qxp 394 3/10/12 Chapter 8 6:52 AM Page 394 Complex Vector Spaces ADDITION, SUBTRACTION, AND SCALAR MULTIPLICATION OF COMPLEX NUMBERS Because a complex number consists of a real part added to a multiple of i, the operations of addition and multiplication are defined in a manner consistent with the rules for operating with real numbers. For instance, to add (or subtract) two complex numbers, add (or subtract) the real and imaginary parts separately. Definition of Addition and Subtraction of Complex Numbers The sum and difference of a bi and c di are defined as follows. REMARK 共a bi兲 共c di兲 共a c兲 共b d 兲i Sum 共a bi兲 共c di兲 共a c兲 共b d 兲i Difference Note in part (a) of Example 2 that the sum of two complex numbers can be a real number. Adding and Subtracting Complex Numbers a. 共2 4i兲 共3 4i兲 共2 3兲 共4 4兲i 5 b. 共1 3i兲 共3 i兲 共1 3兲 共3 1兲i 2 4i Using the vector representation of complex numbers, you can add or subtract two complex numbers geometrically using the parallelogram rule for vector addition, as shown in Figure 8.4. Imaginary axis Imaginary axis z = 3 + 4i 4 2 3 1 −1 w=3+i 1 2 z+w=5 1 2 3 4 5 6 −2 Real axis −3 1 −3 −3 −4 w = 2 − 4i Addition of Complex Numbers 2 3 Real axis z = 1 − 3i z − w = −2 − 4i Subtraction of Complex Numbers Figure 8.4 Many of the properties of addition of real numbers are valid for complex numbers as well. For instance, addition of complex numbers is both associative and commutative. Moreover, to find the sum of three or more complex numbers, extend the definition of addition in the natural way. For example, 共2 i兲 共3 2i兲 共2 4i兲 共2 3 2兲 共1 2 4兲i 3 3i. 9781133110873_0801.qxp 3/10/12 6:52 AM Page 395 8.1 Complex Numbers 395 Another property of real numbers that is valid for complex numbers is the distributive property of scalar multiplication over addition. To multiply a complex number by a real scalar, use the definition below. Definition of Scalar Multiplication If c is a real number and a bi is a complex number, then the scalar multiple of c and a bi is defined as c共a bi兲 ca cbi. Scalar Multiplication with Complex Numbers a. 3共2 7i兲 4共8 i兲 6 21i 32 4i 38 17i b. 4共1 i兲 2共3 i兲 3共1 4i兲 4 4i 6 2i 3 12i 1 6i Geometrically, multiplication of a complex number by a real scalar corresponds to the multiplication of a vector by a scalar, as shown in Figure 8.5. Imaginary axis Imaginary axis 3 4 2 3 2z = 6 + 2i 2 1 −1 z=3+i 1 z=3+i 1 2 3 4 5 6 Real axis 1 −z = −3 − i 2 3 Real axis −2 −3 −2 Multiplication of a Complex Number by a Real Number Figure 8.5 With addition and scalar multiplication, the set of complex numbers forms a vector space of dimension 2 (where the scalars are the real numbers). You are asked to verify this in Exercise 55. LINEAR ALGEBRA APPLIED Complex numbers have some useful applications in electronics. The state of a circuit element is described by two quantities: the voltage V across it and the current I flowing through it. To simplify computations, the circuit element’s state can be described by a single complex number z V li, of which the voltage and current are simply the real and imaginary parts. A similar notation can be used to express the circuit element’s capacitance and inductance. When certain elements of a circuit are changing with time, electrical engineers often have to solve differential equations. These can often be simpler to solve using complex numbers because the equations are less complicated. Adrio Communications Ltd/shutterstock.com 9781133110873_0801.qxp 396 3/10/12 Chapter 8 6:52 AM Page 396 Complex Vector Spaces MULTIPLICATION OF COMPLEX NUMBERS The operations of addition, subtraction, and scalar multiplication of complex numbers have exact counterparts with the corresponding vector operations. By contrast, there is no direct vector counterpart for the multiplication of two complex numbers. Definition of Multiplication of Complex Numbers The product of the complex numbers a bi and c di is defined as 共a bi兲共c di兲 共ac bd 兲 共ad bc兲i. Rather than try to memorize this definition of the product of two complex numbers, simply apply the distributive property, as follows. TECHNOLOGY Many graphing utilities and software programs can calculate with complex numbers. For example, on some graphing utilities, you can express a complex number a bi as an ordered pair 共a, b兲. Try verifying the result of Example 4(b) by multiplying 共2, 1兲 and 共4, 3兲. You should obtain the ordered pair 共11, 2兲. 共a bi兲共c di兲 a共c di兲 bi共c di兲 Distributive property ac 共ad 兲i 共bc兲i 共bd 兲i Distributive property ac 共ad 兲i 共bc兲i 共bd 兲共1兲 Use i 2 1. ac bd 共ad 兲i 共bc兲i Commutative property 共ac bd 兲 共ad bc兲i Distributive property 2 Multiplying Complex Numbers a. 共2兲共1 3i兲 2 6i b. 共2 i兲共4 3i兲 8 6i 4i 3i 2 8 6i 4i 3共1兲 8 3 6i 4i 11 2i Complex Zeros of a Polynomial Use the Quadratic Formula to find the zeros of the polynomial p共x兲 x 2 6x 13 and verify that p共x兲 0 for each zero. SOLUTION Using the Quadratic Formula, x b 冪b2 4ac 6 冪16 6 4i 3 2i. 2a 2 2 Substitute each value of x into the polynomial p共x兲 to verify that p共x兲 0. REMARK A well-known result from algebra states that the complex zeros of a polynomial with real coefficients must occur in conjugate pairs. (See Review Exercise 81.) p共3 2i兲 共3 2i兲2 6共3 2i兲 13 共3 2i兲共3 2i兲 6共3 2i兲 13 9 6i 6i 4 18 12i 13 0 p共3 2i兲 共3 2i兲2 6共3 2i兲 13 共3 2i兲共3 2i兲 6共3 2i兲 13 9 6i 6i 4 18 12i 13 0 In Example 5, the two complex numbers 3 2i and 3 2i are complex conjugates of each other (together they form a conjugate pair). More will be said about complex conjugates in Section 8.2. 9781133110873_0801.qxp 3/10/12 6:52 AM Page 397 8.1 Complex Numbers 397 COMPLEX MATRICES Now that you are able to add, subtract, and multiply complex numbers, you can apply these operations to matrices whose entries are complex numbers. Such a matrix is called complex. Definition of a Complex Matrix A matrix whose entries are complex numbers is called a complex matrix. All of the ordinary operations with matrices also work with complex matrices, as demonstrated in the next two examples. Operations with Complex Matrices Let A and B be the complex matrices A 冤2 3ii 1i 4 冥 and B 冤2ii 冥 0 1 2i and determine each of the following. b. 共2 i兲B a. 3A c. A B d. BA SOLUTION 1i 3i 3 3i 4 6 9i 12 冤2 3ii 冥 冤 a. 3A 3 0 2 4i 0 1 2i 1 2i 4 3i 冤2ii 冥 冤 b. 共2 i兲B 共2 i兲 c. A B 冤2 3ii 冥 冥 1i 2i 0 3i 1 i 4 i 1 2i 2 2i 5 2i 冥 冤 冥 冤 冥 i 1i 冤 i 1 2i冥冤2 3i 4冥 2 0 2i 2 0 冤 1 2 3i 4i 6 i 1 4 8i冥 2 2 2i 冤 7i 3 9i冥 d. BA 2i 0 Finding the Determinant of a Complex Matrix Find the determinant of the matrix TECHNOLOGY Many graphing utilities and software programs can perform matrix operations on complex matrices. Try verifying the calculation of the determinant of the matrix in Example 7. You should obtain the same answer, 共8, 26兲. A 冤2 4i3 SOLUTION ⱍ 冥 2 . 5 3i ⱍ 2 4i 2 3 5 3i 共2 4i兲共5 3i兲 共2兲共3兲 10 20i 6i 12 6 8 26i det共A兲 9781133110873_0801.qxp 398 3/10/12 Chapter 8 6:52 AM Page 398 Complex Vector Spaces 8.1 Exercises In Exercises 1–6, determine the value of the expression. 1. 冪2冪3 2. 冪8冪8 3. 冪4冪4 3 4 4. i 5. i 6. 共i兲7 Simplifying an Expression In Exercises 7–10, determine x such that the complex numbers in each pair are equal. 7. x 3i, 6 3i 8. 共2x 8兲 共x 1兲i, 2 4i 9. 共x 2 6兲 共2x兲i, 15 6i 10. 共x 4兲 共x 1兲i, x 3i Equality of Complex Numbers In Exercises 11–16, plot the number in the complex plane. 11. z 6 2i 12. z 3i 13. z 5 5i 14. z 7 15. z 1 5i 16. z 1 5i Plotting Complex Numbers Adding and Subtracting Complex Numbers In Exercises 17–24, find the sum or difference of the complex numbers. Use vectors to illustrate your answer. 17. 共2 6i兲 共3 3i兲 18. 共1 冪2i兲 共2 冪2i兲 19. 共5 i兲 共5 i兲 20. i 共3 i兲 21. 6 共2i兲 22. 共12 7i兲 共3 4i兲 23. 共2 i兲 共2 i兲 24. 共2 i兲 共2 i兲 Scalar Multiplication In Exercises 25 and 26, use vectors to illustrate the operations geometrically. Be sure to graph the original vector. 25. u and 2u, where u 3 i 26. 3u and 32u, where u 2 i Multiplying Complex Numbers find the product. 27. 共5 5i兲共1 3i兲 29. 共冪7 i兲共冪7 i兲 31. 共a bi兲2 33. 共1 i兲3 28. 30. 32. 34. In Exercises 27–34, 共3 i兲共 23 i兲 共4 冪2i兲共4 冪2 i兲 共a bi兲共a bi兲 共2 i兲共2 2i兲共4 i兲 In Exercises 35–40, determine all the zeros of the polynomial function. 35. p共x兲 2x 2 2x 5 36. p共x兲 x 2 x 1 37. p共x兲 x 2 5x 6 38. p共x兲 x 2 4x 5 39. p共x兲 x 4 16 40. p共x兲 x 4 10x 2 9 Finding Zeros Finding Zeros In Exercises 41–44, use the given zero to find all zeros of the polynomial function. 41. p共x兲 x3 3x2 4x 2 Zero: x 1 42. p共x兲 x3 2x2 11x 52 43. p共x兲 2x3 3x2 50x 75 44. p共x兲 x3 x2 9x 9 Zero: x 4 Zero: x 5i Zero: x 3i Operations with Complex Matrices In Exercises 45–54, perform the indicated matrix operation using the complex matrices A and B. Aⴝ 45. 47. 49. 51. 53. 冤21ⴚⴙ2ii 冥 1 ⴚ3i AB 2A 2iA det共A B兲 5AB and Bⴝ 46. 48. 50. 52. 54. 冤1 ⴚⴚ3i 冥 3i ⴚi BA 1 2B 1 4 iB det共B兲 BA 55. Proof Prove that the set of complex numbers, with the operations of addition and scalar multiplication (with real scalars), is a vector space of dimension 2. 56. Consider the functions p共x兲 x 2 6x 10 and q共x兲 x 2 6x 10. (a) Without graphing either function, determine whether the graphs of p and q have x-intercepts. Explain your reasoning. (b) For which of the given functions is x 3 i a zero? Without using the Quadratic Formula, find the other zero of this function and verify your answer. 57. (a) Evaluate i n for n 1, 2, 3, 4, and 5. (b) Calculate i 2010. (c) Find a general formula for i n for any positive integer n. 58. Let A 冤0i 0i 冥. (a) Calculate An for n 1, 2, 3, 4, and 5. (b) Calculate A2010. (c) Find a general formula for An for any positive integer n. True or False? In Exercises 59 and 60, determine whether each statement is true or false. If a statement is true, give a reason or cite an appropriate statement from the text. If a statement is false, provide an example that shows the statement is not true in all cases or cite an appropriate statement from the text. 59. 冪2冪2 冪4 2 60. 共冪10兲2 冪100 10 61. Proof Prove that if the product of two complex numbers is zero, then at least one of the numbers must be zero. 9781133110873_0802.qxp 3/14/12 8:23 AM Page 399 8.2 Conjugates and Division of Complex Numbers 399 8.2 Conjugates and Division of Complex Numbers Find the conjugate of a complex number. Find the modulus of a complex number. Divide complex numbers, and find the inverse of a complex matrix. COMPLEX CONJUGATES In Section 8.1, it was mentioned that the complex zeros of a polynomial with real coefficients occur in conjugate pairs. For instance, in Example 5 you saw that the zeros of p共x兲 ⫽ x 2 ⫺ 6x ⫹ 13 are 3 ⫹ 2i and 3 ⫺ 2i. In this section, you will examine some additional properties of complex conjugates. You will begin with the definition of the conjugate of a complex number. Definition of the Conjugate of a Complex Number The conjugate of the complex number z ⫽ a ⫹ bi is denoted by z and is given by z ⫽ a ⫺ bi. REMARK In part (d) of Example 1, note that 5 is its own complex conjugate. In general, it can be shown that a number is its own complex conjugate if and only if the number is real. (See Exercise 39.) Imaginary axis z = − 2 + 3i 3 −1 1 Real axis 2 −3 Imaginary axis −3 −2 THEOREM 8.1 Properties of Complex Conjugates zz ⫽ 共a ⫹ bi兲共a ⫺ bi兲 ⫽ a 2 ⫹ abi ⫺ abi ⫺ b2i 2 ⫽ a 2 ⫹ b2. 2 3 −2 −3 −4 −5 z ⫽ ⫺2 ⫺ 3i z ⫽ 4 ⫹ 5i z ⫽ 2i z⫽5 PROOF To prove the first property, let z ⫽ a ⫹ bi. Then z ⫽ a ⫺ bi and z = 4 + 5i 5 4 3 2 1 Conjugate z ⫽ ⫺2 ⫹ 3i z ⫽ 4 ⫺ 5i z ⫽ ⫺2i z⫽5 For a complex number z ⫽ a ⫹ bi, the following properties are true. 1. zz ⫽ a2 ⫹ b2 2. zz ⱖ 0 3. zz ⫽ 0 if and only if z ⫽ 0. 4. 共z兲 ⫽ z −2 z = − 2 − 3i Complex Number a. b. c. d. Geometrically, two points in the complex plane are conjugates if and only if they are reflections in the real (horizontal) axis, as shown in Figure 8.6. Complex conjugates have many useful properties. Some of these are shown in Theorem 8.1. 2 −4 −3 Finding the Conjugate of a Complex Number 5 6 7 Real axis The second and third properties follow directly from the first. Finally, the fourth property follows from the definition of the complex conjugate. That is, 共 z 兲 ⫽ 共 a ⫹ bi 兲 ⫽ a ⫺ bi ⫽ a ⫹ bi ⫽ z. z = 4 − 5i Finding the Product of Complex Conjugates Conjugate of a Complex Number Figure 8.6 When z ⫽ 1 ⫹ 2i, you have zz ⫽ 共1 ⫺ 2i兲共1 ⫹ 2i兲 ⫽ 12 ⫹ 22 ⫽ 1 ⫹ 4 ⫽ 5. 9781133110873_0802.qxp 400 3/10/12 Chapter 8 6:57 AM Page 400 Complex Vector Spaces THE MODULUS OF A COMPLEX NUMBER REMARK The modulus of a complex number is also called the absolute value of the number. In fact, when z is a real number, ⱍzⱍ 冪a2 02 ⱍaⱍ. Because a complex number can be represented by a vector in the complex plane, it makes sense to talk about the length of a complex number. This length is called the modulus of the complex number. Definition of the Modulus of a Complex Number ⱍⱍ The modulus of the complex number z a bi is denoted by z and is given by ⱍzⱍ 冪a2 b2. Finding the Modulus of a Complex Number For z 2 3i and w 6 i, determine the value of each modulus. ⱍⱍ a. z ⱍ ⱍ b. w ⱍ ⱍ c. zw SOLUTION a. z 冪22 32 冪13 b. w 冪6 2 共1兲2 冪37 c. Because zw 共2 3i兲共6 i兲 15 16i, you have zw 冪15 2 16 2 冪481. ⱍⱍ ⱍ ⱍ ⱍ ⱍ ⱍ ⱍ ⱍ ⱍⱍ ⱍ Note that in Example 3, zw z w . In Exercise 41, you are asked to prove that this multiplicative property of the modulus always holds. Theorem 8.2 states that the modulus of a complex number is related to its conjugate. THEOREM 8.2 The Modulus of a Complex Number ⱍⱍ For a complex number z, z 2 zz. PROOF Let z a bi, then z a bi and zz 共a bi兲共a bi兲 a2 b2 z 2. ⱍⱍ LINEAR ALGEBRA APPLIED Fractals appear in almost every part of the universe. They have been used to study a wide variety of applications such as bacteria cultures, the human lungs, the economy, and galaxies. The most famous fractal is called the Mandelbrot Set, named after the Polish-born mathematician Benoit Mandelbrot (1924 – 2010). The Mandelbrot Set is based on the following sequence of complex numbers. zn 共zn1兲2 c, z1 c The behavior of this sequence depends on the value of the complex number c. For some values of c, the modulus of each term zn in the sequence is less than some fixed number N, and the sequence is bounded. This means that c is in the Mandelbrot Set, and its point is colored black. For other values of c, the moduli of the terms of the sequence become infinitely large, and the sequence is unbounded. This means that c is not in the Mandelbrot Set, and its point is assigned a color based on “how quickly” the sequence diverges. Andrew Park/Shutterstock.com 9781133110873_0802.qxp 3/10/12 6:57 AM Page 401 8.2 Conjugates and Division of Complex Numbers 401 DIVISION OF COMPLEX NUMBERS One of the most important uses of the conjugate of a complex number is in performing division in the complex number system. To define division of complex numbers, consider z a bi and w c di and assume that c and d are not both 0. For the quotient z x yi w to make sense, it has to be true that z w共x yi兲 共c di兲共x yi兲 共cx dy兲 共dx cy兲i. But, because z a bi, you can form the linear system below. cx dy a dx cy b Solving this system of linear equations for x and y yields x ac bd ww and y bc ad . ww Now, because zw 共a bi兲共c di兲 共ac bd兲 共bc ad兲i, the following definition is obtained. Definition of Division of Complex Numbers The quotient of the complex numbers z a bi and w c di is defined as z a bi w c di ac bd bc ad 2 2 i c d2 c d2 1 共zw 兲 w2 REMARK If c2 d 2 0, then c d 0, and w 0. In other words, as is the case with real numbers, division of complex numbers by zero is not defined. ⱍ ⱍ provided c2 d 2 0. In practice, the quotient of two complex numbers can be found by multiplying the numerator and the denominator by the conjugate of the denominator, as follows. a bi a bi c di c di c di c di 共a bi兲共c di兲 共c di兲共c di兲 共ac bd兲 共bc ad兲i c2 d 2 ac bd bc ad 2 2 i c d2 c d2 冢 冣 Division of Complex Numbers 冢 1 1 1 1i 1i 1 2i 2i b. 3 4i 3 4i a. 1i 1i 1 1 i 2 i i 1 i2 2 2 2 2 11i 3 4i 2 11 i 3 4i 9 16 25 25 冢 冣 冣 9781133110873_0802.qxp 402 3/10/12 Chapter 8 6:57 AM Page 402 Complex Vector Spaces Now that you can divide complex numbers, you can find the (multiplicative) inverse of a complex matrix, as demonstrated in Example 5. Finding the Inverse of a Complex Matrix Find the inverse of the matrix A 2i 5 2i 6 2i 冤3 i 冥 and verify your solution by showing that AA 1 I2. SOLUTION Using the formula for the inverse of a 2 A1 6 2i 冤 ⱍ ⱍ 3 i 1 A 2 matrix from Section 2.3, 5 2i . 2i 冥 Furthermore, because ⱍAⱍ 共2 i兲共6 2i兲 共5 2i兲共3 i兲 共12 6i 4i 2兲 共15 6i 5i 2兲 3i it follows that TECHNOLOGY If your graphing utility or software program can perform operations with complex matrices, then you can verify the result of Example 5. If you have matrix A stored on a graphing utility, evaluate A1. 1 6 2i 5 2i 3 i 3 i 2 i 1 1 共6 2i兲共3 i兲 共5 2i兲共3 i兲 3i 3i 共3 i兲共3 i兲 共2 i兲共3 i兲 1 20 17 i . 10 10 7i 冤 冢 A1 冤 冥 冣冤 冥 冥 To verify your solution, multiply A and A1 as follows. AA1 冤23 ii 5 2i 1 20 6 2i 10 10 冥 冤 1 10 17 i 10 0 7i 冥 冤 冥 冤 0 1 10 0 冥 0 1 The last theorem in this section summarizes some useful properties of complex conjugates. THEOREM 8.3 Properties of Complex Conjugates For the complex numbers z and w, the following properties are true. 1. z w z w 2. z w z w 3. zw z w 4. z兾w z兾w PROOF To prove the first property, let z a bi and w c di. Then z w 共a c兲 共b d兲i 共a c兲 共b d 兲i 共a bi兲 共c di兲 z w. The proof of the second property is similar. The proofs of the other two properties are left to you. 9781133110873_0802.qxp 3/10/12 6:57 AM Page 403 8.2 403 Exercises 8.2 Exercises Finding the Conjugate In Exercises 1–6, find the complex conjugate z and geometrically represent both z and z. 1. z 6 3i 2. z 2 5i 3. z 8i 4. z 2i 5. z 4 6. z 3 In Exercises 7–12, find the indicated modulus, where z ⴝ 2 ⴙ i, w ⴝ ⴚ3 ⴙ 2i, and v ⴝ ⴚ5i. 7. z 8. z2 9. zw 10. wz 11. v 12. zv 2 Finding the Modulus ⱍⱍ ⱍ ⱍ ⱍⱍ 13. ⱍ ⱍ ⱍ ⱍ ⱍ ⱍ Verify that ⱍwzⱍ ⱍwⱍⱍzⱍ ⱍzwⱍ, where z 1 i and w 1 2i. 14. Verify that zv 2 z v 2 z v 2, where z 1 2i and v 2 3i. ⱍ ⱍ ⱍ ⱍⱍ ⱍ ⱍ ⱍⱍ ⱍ In Exercises 15 – 20, Dividing Complex Numbers Finding the Inverse of a Complex Matrix In Exercises 31–36, determine whether the complex matrix A has an inverse. If A is invertible, find its inverse and verify that AAⴚ1 ⴝ I. 6 3i 2i 2 i 31. A 32. A 2i i 3 3i 冤 冥 1i 2 33. A 冤 1 1 i冥 冤 i 35. A 0 0 0 i 0 0 0 i 冥 冤 1i 34. A 冤 0 冤 1 36. A 0 0 冥 冥 2 1i 0 0 1i 0 0 1i 冥 In Exercises 37 and 38, determine all values of the complex number z for which A is singular. (Hint: Set det冇A冈 ⴝ 0 and solve for z.) Singular Matrices 37. A 冤 39. Proof 冥 5 z 3i 2 i 冤 2 2i 1 i 38. A 1 i 1 i z 1 0 0 冥 Prove that z z if and only if z is real. perform the indicated operations. 15. 2i i 16. 1 6 3i 17. 3 冪2i 3 冪2i 18. 5i 4i 19. 共2 i兲共3 i兲 4 2i 20. 3i 共2 i兲共5 2i兲 Operations with Complex Rational Expressions In Exercises 21–24, perform the operation and write the result in standard form. 21. 2 3 1i 1i 22. 2i 5 2i 2i 23. i 2i 3i 3i 24. 1i 3 i 4i In Exercises 25–28, use the given zero to find all zeros of the polynomial function. 25. p共x兲 3x3 4x2 8x 8 Zero: 1 冪3i 3 2 26. p共x兲 4x 23x 34x 10 Zero: 3 i 4 3 2 27. p共x兲 x 3x 5x 21x 22 Zero: 3 冪2i 28. p共x兲 x3 4x2 14x 20 Zero: 1 3i Finding Zeros Powers of Complex Numbers In Exercises 29 and 30, find each power of the complex number z. (a) z 2 (b) z 3 (c) zⴚ1 (d) zⴚ2 29. z 2 i 30. z 1 i 1i . 6 2i (a) Without performing any calculations, describe how to find the quotient. (b) Explain why the process described in part (a) results in a complex number of the form a bi. (c) Find the quotient. 40. Consider the quotient 41. Proof Prove that for any two complex numbers z and w, each of the statements below is true. (a) zw z w (b) If w 0, then z兾w z 兾 w . 42. Graphical Interpretation Describe the set of points in the complex plane that satisfies each of the statements below. (a) z 3 (b) z 1 i 5 (c) z i 2 (d) 2 z 5 n 43. (a) Evaluate 共1兾i兲 for n 1, 2, 3, 4, and 5. (b) Calculate 共1兾i兲2000 and 共1兾i兲2010. (c) Find a general formula for 共1兾i兲n for any positive integer n. 1i 2 44. (a) Verify that i. 冪2 (b) Find the two square roots of i. (c) Find all zeros of the polynomial x 4 1. ⱍ ⱍ ⱍ ⱍⱍ ⱍ ⱍ ⱍ ⱍⱍⱍ ⱍ ⱍⱍ ⱍ ⱍ ⱍ 冢 冣 ⱍⱍ ⱍ 9781133110873_0803.qxp 404 3/10/12 Chapter 8 6:54 AM Page 404 Complex Vector Spaces 8.3 Polar Form and DeMoivre’s Theorem Determine the polar form of a complex number, convert between the polar form and standard form of a complex number, and multiply and divide complex numbers in polar form. Use DeMoivre’s Theorem to find powers and roots of complex numbers in polar form. POLAR FORM OF A COMPLEX NUMBER Imaginary axis At this point you can add, subtract, multiply, and divide complex numbers. However, there is still one basic procedure that is missing from the algebra of complex numbers. To see this, consider the problem of finding the square root of a complex number such as i. When you use the four basic operations (addition, subtraction, multiplication, and division), there seems to be no reason to guess that (a, b) r b 冪i θ a 0 Real axis That is, 1i Standard Form: a + bi Polar Form: r(cos θ + i sin θ ) Figure 8.7 1i . 冪2 冢 冪2 冣 i. 2 To work effectively with powers and roots of complex numbers, it is helpful to use a polar representation for complex numbers, as shown in Figure 8.7. Specifically, if a bi is a nonzero complex number, then let be the angle from the positive real axis to the radial line passing through the point 共a, b兲 and let r be the modulus of a bi. This leads to the following. a r cos b r sin r 冪a 2 b2 REMARK The polar form of z 0 is expressed as z 0共cos i sin 兲, where is any angle. So, a bi 共r cos 兲 共r sin 兲i, from which the polar form of a complex number is obtained. Definition of the Polar Form of a Complex Number The polar form of the nonzero complex number z a bi is given by z r共cos i sin 兲 where a r cos , b r sin , r 冪a 2 b2, and tan b兾a. The number r is the modulus of z and is the argument of z. Because there are infinitely many choices for the argument, the polar form of a complex number is not unique. Normally, the values of that lie between and are used, although on occasion it is convenient to use other values. The value of that satisfies the inequality < Principal argument is called the principal argument and is denoted by Arg(z). Two nonzero complex numbers in polar form are equal if and only if they have the same modulus and the same principal argument. 9781133110873_0803.qxp 3/10/12 6:54 AM Page 405 8.3 Polar Form and DeMoivre’s Theorem 405 Finding the Polar Form of a Complex Number Find the polar form of each of the complex numbers. (Use the principal argument.) a. z 1 i b. z 2 3i c. z i SOLUTION a. Because a 1 and b 1, then r 2 1 2 共1兲2 2, which implies that r 冪2. From a r cos and b r sin , cos 冪2 a 1 冪2 r 2 and sin 冪2 b 1 . 冪2 r 2 So, 兾4 and 冤 冢 4 冣 i sin冢 4 冣冥. z 冪2 cos b. Because a 2 and b 3, then r 2 2 2 32 13, which implies that r 冪13. So, cos a 2 冪13 r and sin b 3 冪13 r and it follows that ⬇ 0.98. So, the polar form is 冤 冥 z ⬇ 冪13 cos 共0.98兲 i sin共0.98兲 . c. Because a 0 and b 1, it follows that r 1 and 兾2, so 冢 z 1 cos i sin . 2 2 冣 The polar forms derived in parts (a), (b), and (c) are depicted graphically in Figure 8.8. Imaginary axis Imaginary axis −2 −1 2 4 1 3 θ −1 2 Real axis 2 1 z=1−i −2 冤 冢 4 冣 i sin 冢 4 冣冥 Imaginary axis z=i θ Real axis 1 冢 c. z 1 cos Figure 8.8 θ 1 a. z 冪2 cos 1 z = 2 + 3i i sin 2 2 冣 2 Real axis b. z ⬇ 冪13 关cos共0.98兲 i sin共0.98兲兴 9781133110873_0803.qxp 406 3/10/12 Chapter 8 6:54 AM Page 406 Complex Vector Spaces Converting from Polar to Standard Form Express the complex number in standard form. 冤 冢 3 冣 i sin冢 3 冣冥 z 8 cos SOLUTION Because cos共 兾3兲 1兾2 and sin共兾3兲 冪3兾2, obtain the standard form 冤 冢 3 冣 i sin冢 3 冣冥 8冤12 i z 8 cos 冪3 2 冥 4 4冪3i. The polar form adapts nicely to multiplication and division of complex numbers. Suppose you have two complex numbers in polar form z1 r1共cos 1 i sin 1兲 and z2 r2共cos 2 i sin 2 兲. Then the product of z1 and z2 is expressed as z1 z 2 r1 r2共cos 1 i sin 1兲共cos 2 i sin 2兲 r1r2 关共cos 1 cos 2 sin 1 sin 2兲 i 共cos 1 sin 2 sin 1 cos 2兲兴. Using the trigonometric identities cos共1 2兲 cos 1 cos 2 sin 1 sin 2 and sin共1 2兲 sin 1 cos 2 cos 1 sin 2 , you have z1z 2 r1r2 关cos共1 2兲 i sin共1 2兲兴. This establishes the first part of the next theorem. The proof of the second part is left to you. (See Exercise 75.) THEOREM 8.4 Product and Quotient of Two Complex Numbers Given two complex numbers in polar form z1 r1共cos 1 i sin 1兲 and z2 r2共cos 2 i sin 2 兲 the product and quotient of the numbers are as follows. z1z2 r1r2 关cos共1 2兲 i sin共1 2兲兴 Product z1 r1 z 2 r2 关cos共1 2兲 i sin共1 2兲兴, z2 0 Quotient LINEAR ALGEBRA APPLIED Elliptic curves are the foundation for elliptic curve cryptography (ECC), a type of public key cryptography for secure communications over the Internet. ECC has gained popularity due to its computational and bandwidth advantages over traditional public key algorithms. One specific variety of elliptic curve is formed using Eisenstein integers. Eisenstein integers are complex 1 冪3 numbers of the form z a b, where i, 2 2 and a and b are integers. These numbers can be graphed as intersection points of a triangular lattice in the complex plane. Dividing the complex plane by the lattice of all Eisenstein integers results in an elliptic curve. thumb/Shutterstock.com 9781133110873_0803.qxp 3/10/12 6:54 AM Page 407 8.3 407 Polar Form and DeMoivre’s Theorem Theorem 8.4 says that to multiply two complex numbers in polar form, multiply moduli and add arguments. To divide two complex numbers, divide moduli and subtract arguments. (See Figure 8.9.) Imaginary axis Imaginary axis z1 z2 z1z2 θ1 + θ 2 r 2 r1r2 z2 r2 z1 θ 2 r1 θ1 r1 θ1 θ2 z1 z2 r1 r2 θ1 − θ 2 Real axis Real axis To multiply z1 and z2 : Multiply moduli and add arguments. To divide z1 and z2 : Divide moduli and subtract arguments. Figure 8.9 Multiplying and Dividing in Polar Form Find z1z2 and z1兾z2 for the complex numbers 冢 z1 5 cos i sin 4 4 冣 and z2 1 cos i sin . 3 6 6 冢 冣 SOLUTION Because z1 and z2 are in polar form, apply Theorem 8.4, as follows. multiply z1z2 共5兲 5 5 冢3冣 冤cos冢 4 6 冣 i sin冢 4 6 冣冥 3 冢cos 12 i sin 12 冣 1 add 5 add divide z1 5 cos i sin z2 1兾3 4 6 4 6 冤 冢 冣 冢 subtract 冣冥 15冢cos 12 i sin 12冣 subtract Use the standard forms of z1 and z2 to check the multiplication in Example 3. For instance, z1z2 冢5 2 2 5 2 2 i冣冢 63 16 i冣 5126 5122 i 5126 i 5122 i 冪 冪 冪 冪 冪 冪 冪 2 5冪6 5冪2 5冪6 5冪2 i 12 12 5 冪6 冪2 冪6 冪2 i . 3 4 4 冢 REMARK Try verifying the division in Example 3 using the standard forms of z1 and z2. 冣 To verify that this answer is equivalent to the result in Example 3, use the formulas for cos 共u v兲 and sin 共u v兲 to obtain cos 冢512冣 cos冢6 4 冣 冪6 冪2 4 and sin 冢512冣 sin冢6 4 冣 冪6 冪2 4 . 9781133110873_0803.qxp 408 3/10/12 Chapter 8 6:54 AM Page 408 Complex Vector Spaces DEMOIVRE’S THEOREM The final topic in this section involves procedures for finding powers and roots of complex numbers. Repeated use of multiplication in the polar form yields z r 共cos i sin 兲 z 2 r 共cos i sin 兲r 共cos i sin 兲 r 2共cos 2 i sin 2兲 and z3 r 共cos i sin 兲r 2 共cos 2 i sin 2兲 r 3共cos 3 i sin 3兲. Similarly, z4 r 4共cos 4 i sin 4兲 and z 5 r 5共cos 5 i sin 5兲. This pattern leads to the next important theorem, named after the French mathematician Abraham DeMoivre (1667–1754). You are asked to prove this theorem in Review Exercise 85. THEOREM 8.5 DeMoivre’s Theorem If z r 共cos i sin 兲 and n is any positive integer, then zn r n共cos n i sin n兲. Raising a Complex Number to an Integer Power Find 共1 冪3 i兲 and write the result in standard form. 12 SOLUTION First convert to polar form. For 1 冪3i, r 冪共1兲2 共冪3 兲2 2 and tan 冪3 1 冪3 which implies that 2兾3. So, 冢 1 冪3i 2 cos 2 2 . i sin 3 3 冣 By DeMoivre’s Theorem, 2 2 12 i sin 3 3 12共2兲 12共2兲 212 cos i sin 3 3 4096共cos 8 i sin 8兲 4096 关1 i 共0兲兴 4096. 共1 冪3i兲12 冤 2冢cos 冤 冣冥 冥 9781133110873_0803.qxp 3/10/12 6:54 AM Page 409 8.3 Polar Form and DeMoivre’s Theorem 409 Recall that a consequence of the Fundamental Theorem of Algebra is that a polynomial of degree n has n zeros in the complex number system. So, a polynomial such as p共x兲 x6 1 has six zeros, and in this case you can find the six zeros by factoring and using the Quadratic Formula. x6 1 共x3 1兲共x3 1兲 共x 1兲共x 2 x 1兲共x 1兲共x 2 x 1兲 Consequently, the zeros are x ± 1, x 1 ± 冪3i , 2 and x 1 ± 冪3i . 2 Each of these numbers is called a sixth root of 1. In general, the nth root of a complex number is defined as follows. Definition of the nth Root of a Complex Number The complex number w a bi is an nth root of the complex number z when z w n 共a bi兲n. DeMoivre’s Theorem is useful in determining roots of complex numbers. To see how this is done, let w be an nth root of z, where w s共cos i sin 兲 and z r 共cos i sin 兲. Then, by DeMoivre’s Theorem w n s n共cos n i sin n 兲 and because w n z, it follows that s n 共cos n i sin n 兲 r 共cos i sin 兲. Now, because the right and left sides of this equation represent equal complex numbers, n r, and equate principal equate moduli to obtain s n r, which implies that s 冪 arguments to conclude that and n must differ by a multiple of 2. Note that r is a n r is also a positive real number. Consequently, for positive real number and so s 冪 some integer k, n 2k, which implies that REMARK Note that when k exceeds n 1, the roots begin to repeat. For instance, when k n, the angle is 2n 2 n n which yields the same values for the sine and cosine as k 0. 2 k . n Finally, substituting this value of into the polar form of w produces the result stated in the next theorem. THEOREM 8.6 The nth Roots of a Complex Number For any positive integer n, the complex number z r 共cos i sin 兲 has exactly n distinct roots. These n roots are given by 2 k 2k i sin n n 冤 冢 n r cos 冪 冣 where k 0, 1, 2, . . . , n 1. 冢 冣冥 9781133110873_0803.qxp 410 3/10/12 Chapter 8 6:54 AM Complex Vector Spaces Imaginary axis n 2π n r 2π n Page 410 Real axis The nth Roots of a Complex Number Figure 8.10 The formula for the nth roots of a complex number has a nice geometric interpretation, as shown in Figure 8.10. Because the nth roots all n r, they lie on a have the same modulus (length) 冪 n circle of radius 冪r with center at the origin. Furthermore, the n roots are equally spaced around the circle, because successive nth roots have arguments that differ by 2兾n. You have already found the sixth roots of 1 by factoring and the Quadratic Formula. Try solving the same problem using Theorem 8.6 to get the roots shown in Figure 8.11. When Theorem 8.6 is applied to the real number 1, the nth roots have a special name—the nth roots of unity. Imaginary axis −1 + 3i 2 2 −1 1+ 3 i 2 2 1 Real axis 1− 3 −1 − 3i i 2 2 2 2 The Sixth Roots of Unity Figure 8.11 Finding the nth Roots of a Complex Number Determine the fourth roots of i. SOLUTION In polar form, 冢 i 1 cos i sin 2 2 冣 so r 1 and 兾2. Then, by applying Theorem 8.6, 冤 冢4兾2 2k4冣 i sin冢4兾2 2k4冣冥 k k cos冢 冣 i sin冢 冣. 8 2 8 2 4 1 cos i1兾4 冪 Setting k 0, 1, 2, and 3, i sin 8 8 5 5 z 2 cos i sin 8 8 9 9 z3 cos i sin 8 8 13 13 z4 cos i sin 8 8 z1 cos REMARK In Figure 8.12, note that when each of the four angles 兾8, 5兾8, 9兾8, and 13兾8 is multiplied by 4, the result is of the form 共兾2兲 2k. as shown in Figure 8.12. cos 5π + i sin 5π 8 8 Imaginary axis cos π + i sin π 8 8 Real axis cos 9π + i sin 9π 8 8 cos 13π + i sin 13π 8 8 Figure 8.12 9781133110873_0803.qxp 3/10/12 6:54 AM Page 411 8.3 411 Exercises 8.3 Exercises In Exercises 1–4, express the complex number in polar form. Imaginary 1. Imaginary 2. Converting to Polar Form axis axis 1 Real axis 2 1 + 3i 3 2 −1 1 −2 2 − 2i 3. Imaginary axis −1 4. −6 1 Real axis 2 3 3i 2 Real axis − 6 −5 −4 −3 − 2 −2 −3 1 −1 Real axis 1 Graphing and Converting to Polar Form In Exercises 5–16, represent the complex number graphically, and give the polar form of the number. (Use the principal argument.) 5. 2 2i 6. 2 2i 7. 2共1 冪3i 兲 8. 52 共冪3 i 兲 9. 6i 10. 2i 11. 7 12. 4 13. 3 冪3i 14. 2冪2 i 15. 1 2i 16. 5 2i Graphing and Converting to Standard Form In Exercises 17–26, represent the complex number graphically, and give the standard form of the number. 17. 2 cos i sin 2 2 冢 冣 3 3 18. 5 cos i sin 4 4 冢 冣 3 5 5 19. cos i sin 2 3 3 冢 20. 3 7 7 cos i sin 4 4 4 21. 15 cos i sin 4 4 4 冢 冢 冢 冢 i sin 6 6 冢 5 5 i sin 6 6 冣 22. 8 cos 3 3 i sin 2 2 24. 6 cos 25. 7共cos 0 i sin 0兲 冣 冣 26. 9共cos i sin 兲 冣 28. 冤 4 冢cos 2 i sin 2 冣冥冤6 冢cos 4 i sin 4 冣冥 3 29. 关0.5共cos i sin 兲兴关 0.5共cos 关兴 i sin关兴兲兴 30. 冤3冢cos 3 i sin 3 冣冥冤 3 冢cos 31. 2关cos共2兾3兲 i sin共2兾3兲兴 4[cos共5兾6兲 i sin共5兾6兲] 32. cos共5兾3兲 i sin共5兾3兲 cos i sin 1 2 2 i sin 3 3 冣冥 12关cos共兾3兲 i sin共兾3兲兴 3关cos共兾6兲 i sin共兾6兲兴 9关cos共3兾4兲 i sin共3兾4兲兴 34. 5关cos共 兾4兲 i sin共 兾4兲兴 33. In Exercises 35–44, use DeMoivre’s Theorem to find the indicated powers of the complex number. Express the result in standard form. 35. 共1 i兲 4 36. 共2 2i兲 6 37. 共1 i兲 10 38. 共冪3 i 兲 7 3 39. 共1 冪3i 兲 Finding Powers of Complex Numbers 40. 冤5冢cos 9 i sin 9 冣冥 41. 冤3冢cos 42. 冢cos 43. 冤2冢cos 2 i sin 2 冣冥 5 5 i sin 6 6 5 5 i sin 4 4 44. 5 cos 冣 冤3冢cos 3 i sin 3 冣冥冤4冢cos 6 i sin 6 冣冥 冤冢 冣 27. Imaginary axis 3 2 1 23. 4 cos In Exercises 27–34, perform the indicated operation and leave the result in polar form. Multiplying and Dividing in Polar Form 冣 3 冣冥 4 10 3 3 i sin 2 2 8 冣冥 4 Finding Square Roots of a Complex Number In Exercises 45–52, find the square roots of the complex number. 45. 2i 46. 5i 47. 3i 48. 6i 49. 2 2i 50. 2 2i 冪 51. 1 3i 52. 1 冪3i 9781133110873_0803.qxp 412 3/10/12 Chapter 8 6:54 AM Page 412 Complex Vector Spaces Finding and Graphing nth Roots In Exercises 53–64, (a) use Theorem 8.6 to find the indicated roots, (b) represent each of the roots graphically, and (c) express each of the roots in standard form. 冢 53. Square roots: 16 cos 冢 54. Square roots: 9 cos 冢 冢 57. 58. 59. 60. 61. 62. 63. 64. 冣 2 2 i sin 3 3 55. Fourth roots: 16 cos 56. Fifth roots: 32 cos i sin 3 3 5 5 i sin 6 6 76. Proof Show that the complex conjugate of z r共cos i sin 兲 is z r 关cos共 兲 i sin共 兲兴. 77. Use the polar forms of z and z in Exercise 76 to find each of the following. (a) zz (b) z兾z, z 0 78. Proof Show that the negative of z r共cos i sin 兲 is 冣 4 4 i sin 3 3 冣 冣 Square roots: 25i Fourth roots: 625i Cube roots: 125 2 共1 冪3i兲 Cube roots: 4冪2 共1 i兲 Cube roots: 8 Fourth roots: 81i Fourth roots: 1 Cube roots: 1000 z r 关cos共 兲 i sin共 兲兴. 79. Writing In Exercises 65–72, find all the solutions of the equation and represent your solutions graphically. 65. x 4 256i 0 66. x 4 16i 0 67. x 3 1 0 68. x3 27 0 69. x 5 243 0 70. x 4 81 0 71. x 3 64i 0 72. x 4 i 0 73. Electrical Engineering In an electric circuit, the formula V I Z relates voltage drop V, current I, and impedance Z, where complex numbers can represent each of these quantities. Find the impedance when the voltage drop is 5 5i and the current is 2 4i. Use the graph of the roots of a complex number. (a) Write each of the roots in trigonometric form. (b) Identify the complex number whose roots are given. Use a graphing utility to verify your results. Imaginary Imaginary (i) (ii) axis 30° 2 axis 2 3 30° 2 1 −1 Real axis i sin . 6 6 Sketch z, iz, and z兾i in the complex plane. (b) What is the geometric effect of multiplying a complex number z by i? What is the geometric effect of dividing z by i? 80. Calculus Recall that the Maclaurin series for e x, sin x, and cos x are x 2 x3 x 4 . . . ex 1 x 2! 3! 4! (a) Let Finding and Graphing Solutions 74. 75. Proof When provided with two complex numbers z 1 r1共cos 1 i sin 1兲 and z 2 r2共cos 2 i sin 2 兲, with z 2 0, prove that r1 z1 z 2 r 2 关cos共 1 2兲 i sin共 1 2兲兴. 3 45° 45° 45° 45° 3 3 Real axis 冢 z r共cos i sin 兲 2 cos sin x x 冣 x 3 x5 x 7 . . . 3! 5! 7! x 2 x 4 x6 . . . . 2! 4! 6! Substitute x i in the series for ex and show that ei cos i sin . Show that any complex number z a bi can be expressed in polar form as z rei. Prove that if z rei, then z rei. Prove the formula ei 1. cos x 1 (a) (b) (c) (d) True or False? In Exercises 81 and 82, determine whether each statement is true or false. If a statement is true, give a reason or cite an appropriate statement from the text. If a statement is false, provide an example that shows the statement is not true in all cases or cite an appropriate statement from the text. 81. Although the square of the complex number bi is given by 共bi兲2 b2, the absolute value of the complex number z a bi is defined as a bi 冪a2 b2. 82. Geometrically, the nth roots of any complex number z are all equally spaced around the unit circle centered at the origin. ⱍ ⱍ 9781133110873_0804.qxp 3/10/12 6:56 AM Page 413 8.4 Complex Vector Spaces and Inner Products 413 8.4 Complex Vector Spaces and Inner Products Recognize and perform vector operations in complex vector spaces C n, represent a vector in C n by a basis, and find the Euclidean inner product, Euclidean norm, and Euclidean distance in C n. Recognize complex inner product spaces. COMPLEX VECTOR SPACES All the vector spaces studied so far in the text have been real vector spaces because the scalars have been real numbers. A complex vector space is one in which the scalars are complex numbers. So, if v1, v2, . . . , vm are vectors in a complex vector space, then a linear combination is of the form c1v1 c 2v2 cmvm where the scalars c1, c2, . . . , cm are complex numbers. The complex version of Rn is the complex vector space C n consisting of ordered n-tuples of complex numbers. So, a vector in C n has the form v 共a1 b1i, a 2 b 2i, . . . , an bni 兲. It is also convenient to represent vectors in C n by column matrices of the form 冤 冥 a1 b1i a bi v 2 .. 2 . . an bni As with R n, the operations of addition and scalar multiplication in C n are performed component by component. Vector Operations in C n Let v 共1 2i, 3 i兲 and u 共2 i, 4兲 be vectors in the complex vector space C 2. Determine each vector. a. v u b. 共2 i兲v c. 3v 共5 i兲u SOLUTION a. In column matrix form, the sum v u is 1 2i 2 i 1 3i vu . 3i 4 7i 冤 冥 冤 冥 冤 冥 b. Because 共2 i兲共1 2i兲 5i and 共2 i兲共3 i兲 7 i, 共2 i 兲v 共2 i 兲共1 2i, 3 i 兲 共5i, 7 i 兲. c. 3v 共5 i兲u 3共1 2i, 3 i兲 共5 i兲共2 i, 4兲 共3 6i, 9 3i兲 共9 7i, 20 4i兲 共12 i, 11 i兲 9781133110873_0804.qxp 414 3/10/12 Chapter 8 6:56 AM Page 414 Complex Vector Spaces Many of the properties of R n are shared by C n. For instance, the scalar multiplicative identity is the scalar 1 and the additive identity in C n is 0 共0, 0, 0, . . . , 0兲. The standard basis for C n is simply e1 共1, 0, 0, . . . , 0兲 e 2 共0, 1, 0, . . . , 0兲 . . . en 共0, 0, 0, . . . , 1兲 which is the standard basis for Rn. Because this basis contains n vectors, it follows that the dimension of C n is n. Other bases exist; in fact, any linearly independent set of n vectors in C n can be used, as demonstrated in Example 2. Verifying a Basis Show that S 再v1, v2, v3冎 再共i, 0, 0兲, 共i, i, 0兲, 共0, 0, i兲冎 is a basis for C 3. SOLUTION Because the dimension of C 3 is 3, the set 再v1, v2, v3冎 will be a basis if it is linearly independent. To check for linear independence, set a linear combination of the vectors in S equal to 0, as follows. c1v1 c2v2 c3v3 共0, 0, 0兲 共c1i, 0, 0兲 共c2i, c2i, 0兲 共0, 0, c3i兲 共0, 0, 0兲 共共c1 c2 兲i, c2i, c3i兲 共0, 0, 0兲 This implies that 共c1 c 2 兲i 0 c2i 0 c3i 0. So, c1 c2 c3 0, and 再v1, v2, v3冎 is linearly independent. Representing a Vector in C n by a Basis Use the basis S in Example 2 to represent the vector v 共2, i, 2 i兲. SOLUTION By writing v c1v1 c2v2 c3v3 共共c1 c 2 兲 i, c2i, c3i兲 共2, i, 2 i兲 you can obtain 共c1 c 2 兲 i 2 c2i i c3i 2 i REMARK Try verifying that this linear combination yields 共2, i, 2 i 兲. which implies that c2 1, c1 2i 1 2i, i and So, v 共1 2i兲v1 v2 共1 2i兲v3. c3 2i 1 2i. i 9781133110873_0804.qxp 3/10/12 6:56 AM Page 415 8.4 Complex Vector Spaces and Inner Products 415 Other than C n, there are several additional examples of complex vector spaces. For instance, the set of m n complex matrices with matrix addition and scalar multiplication forms a complex vector space. Example 4 describes a complex vector space in which the vectors are functions. The Space of Complex-Valued Functions Consider the set S of complex-valued functions of the form f共x兲 f1共x兲 if 2共x兲 where f1 and f2 are real-valued functions of a real variable. The set of complex numbers forms the scalars for S, and vector addition is defined by f共x兲 g共x兲 关f1共x兲 if 2共x兲兴 关g1(x兲 i g2共x兲兴 关f1共x兲 g1共x兲兴 i 关f 2共x兲 g2共x兲兴. It can be shown that S, scalar multiplication, and vector addition form a complex vector space. For instance, to show that S is closed under scalar multiplication, let c a bi be a complex number. Then cf共x兲 共a bi兲关f1共x兲 if 2共x兲兴 关af1共x兲 bf 2共x兲兴 i 关bf1共x兲 af 2共x兲兴 is in S. REMARK Note that if u and v happen to be “real,” then this definition agrees with the standard inner (or dot) product in Rn. The definition of the Euclidean inner product in C n is similar to the standard dot product in R n, except that here the second factor in each term is a complex conjugate. Definition of the Euclidean Inner Product in C n Let u and v be vectors in C n. The Euclidean inner product of u and v is given by u v u1v1 u2v2 unvn. Finding the Euclidean Inner Product in C 3 Determine the Euclidean inner product of the vectors u 共2 i, 0, 4 5i兲 and v 共1 i, 2 i, 0兲. SOLUTION u v u1v1 u2v2 u 3v3 共2 i兲共1 i兲 0共2 i兲 共4 5i兲共0兲 3i Several properties of the Euclidean inner product C n are stated in the following theorem. THEOREM 8.7 Properties of the Euclidean Inner Product Let u, v, and w be vectors in C n and let k be a complex number. Then the following properties are true. 1. u v v u 2. 共u v兲 w u w v w 3. 共ku兲 v k共u v兲 4. u 共kv兲 k共u v兲 5. u u 0 6. u u 0 if and only if u 0. 9781133110873_0804.qxp 416 3/10/12 Chapter 8 6:56 AM Page 416 Complex Vector Spaces PROOF The proof of the first property is shown below, and the proofs of the remaining properties have been left to you (see Exercises 59–63). Let u 共u1, u 2, . . . , un 兲 and v 共v1, v2, . . . , vn 兲. Then v u v1u1 v2u2 . . . vn un v1u1 v2u2 . . . vnun v1u1 v2u 2 . . . vnun u1v1 u2v2 unvn u v. The Euclidean inner product in C n is used to define the Euclidean norm (or length) of a vector in C n and the Euclidean distance between two vectors in C n. Definitions of the Euclidean Norm and Distance in C n The Euclidean norm (or length) of u in C n is denoted by 储u储 and is 储u储 共u u兲1兾2. The Euclidean distance between u and v is d共u, v兲 储u v储 . The Euclidean norm and distance may be expressed in terms of components, as follows (see Exercise 51). ⱍ ⱍ ⱍ ⱍ ⱍ ⱍ ⱍ ⱍ ⱍ ⱍ 储u储 共 u1 2 u2 2 . . . u n 2兲1兾2 d共u, v兲 共 u1 v1 2 u 2 v2 2 . . . un vn 2兲1兾2 ⱍ ⱍ Finding the Euclidean Norm and Distance in C n Let u 共2 i, 0, 4 5i兲 and v 共1 i, 2 i, 0兲. a. Find the norms of u and v. b. Find the distance between u and v. SOLUTION a. 储u储 共 u1 2 u 2 2 u 3 2兲1兾2 关共22 12兲 共02 02兲 共42 共5兲 2兲兴1兾2 共5 0 41兲1兾2 冪46 ⱍ ⱍ ⱍ ⱍ ⱍ ⱍ ⱍ ⱍ ⱍ ⱍ ⱍ ⱍ 储v储 共 v1 2 v2 2 v3 2兲1兾2 关共12 12兲 共22 12兲 共02 02兲兴1兾2 共2 5 0兲1兾2 冪7 b. d共u, v兲 储u v储 储共1, 2 i, 4 5i兲储 关共12 02兲 共共2兲2 共1兲2兲 共42 共5兲2兲兴1兾2 共1 5 41兲1兾2 冪47 9781133110873_0804.qxp 3/10/12 6:56 AM Page 417 8.4 Complex Vector Spaces and Inner Products 417 COMPLEX INNER PRODUCT SPACES The Euclidean inner product is the most commonly used inner product in C n. On occasion, however, it is useful to consider other inner products. To generalize the notion of an inner product, use the properties listed in Theorem 8.7. Definition of a Complex Inner Product Let u and v be vectors in a complex vector space. A function that associates u and v with the complex number 具u, v典 is called a complex inner product when it satisfies the following properties. 1. 具u, v典 具v, u典 2. 具u v, w典 具u, w典 具v, w典 3. 具ku, v典 k具u, v典 4. 具u, u典 0 and 具u, u典 0 if and only if u 0. A complex vector space with a complex inner product is called a complex inner product space or unitary space. A Complex Inner Product Space Let u 共u1, u2兲 and v 共v1, v2兲 be vectors in the complex space C 2. Show that the function defined by 具u, v典 u1v1 2u2v2 is a complex inner product. SOLUTION Verify the four properties of a complex inner product, as follows. 1. 具v, u典 v1u1 2v2u2 u1v1 2u2v2 具u, v典 2. 具u v, w典 共u1 v1兲 w1 2共u2 v2兲w2 共u1w1 2u 2w2 兲 共v1w1 2v2w2 兲 具u, w典 具v, w典 3. 具ku, v典 共ku1兲v1 2共ku 2 兲v2 k 共u1v1 2u2v2 兲 k 具u, v典 4. 具u, u典 u1u1 2u2 u2 u1 2 2 u2 2 0 Moreover, 具u, u典 0 if and only if u1 u2 0. ⱍ ⱍ ⱍ ⱍ Because all the properties hold, 具u, v典 is a complex inner product. LINEAR ALGEBRA APPLIED Complex vector spaces and inner products have an important application called the Fourier transform, which decomposes a function into a sum of orthogonal basis functions. The given function is projected onto the standard basis functions for varying frequencies to get the Fourier amplitudes for each frequency. Like Fourier coefficients and the Fourier approximation, this transform is named after the French mathematician Jean-Baptiste Joseph Fourier (1768 –1830). The Fourier transform is integral to the study of signal processing. To understand the basic premise of this transform, imagine striking two piano keys simultaneously. Your ear receives only one signal, the mixed sound of the two notes, and yet your brain is able to separate the notes. The Fourier transform gives a mathematical way to take a signal and separate out its frequency components. Eliks/Shutterstock.com 9781133110873_0804.qxp 418 3/10/12 Chapter 8 6:56 AM Page 418 Complex Vector Spaces 8.4 Exercises Vector Operations In Exercises 1– 8, perform the indicated operation using u ⴝ 冇i, 3 ⴚ i冈, v ⴝ 冇2 ⴙ i, 3 ⴙ i兲, and w ⴝ 冇4i, 6冈. 1. 3u 2. 4iw 3. 共1 2i兲w 4. iv 3w 5. u 共2 i兲v 6. 共6 3i兲v 共2 2i兲w 7. u iv 2iw 8. 2iv 共3 i兲w u Finding the Euclidean Norm Linear Dependence or Independence In Exercises 9–12, determine whether the set of vectors is linearly independent or linearly dependent. 9. 再共1, i兲, 共i, 1兲冎 10. 再共1 i, 1 i, 1兲, 共i, 0, 1兲, 共2, 1 i, 0兲冎 11. 再共1, i, 1 i兲, 共0, i, i兲, 共0, 0, 1兲冎 12. 再共1 i, 1 i, 0兲, 共1 i, 0, 0兲, 共0, 1, 1兲冎 Finding the Euclidean Distance In Exercises 35–40, determine the Euclidean distance between u and v. 35. u 共1, 0兲, v 共i, i兲 36. u 共2 i, 4, i兲, v 共2 i, 4, i兲 37. u 共i, 2i, 3i兲, v 共0, 1, 0兲 38. u 共冪2, 2i, i兲, v 共i, i, i兲 39. u 共1, 0兲, v 共0, 1兲 40. u 共1, 2, 1, 2i兲, v 共i, 2i, i, 2兲 In Exercises 13 –16, determine whether S is a basis for C n. 13. S 再共1, i兲, 共i, 1兲冎 14. S 再共1, i兲, 共i, 1兲冎 15. S 再共i, 0, 0兲, 共0, i, i兲, 共0, 0, 1兲冎 16. S 再共1 i, 0, 1兲, 共2, i, 1 i兲, 共1 i, 1, 1兲冎 Verifying a Basis In Exercises 17–20, express v as a linear combination of each of the following basis vectors. (a) {冇i, 0, 0冈, 冇i, i, 0冈, 冇i, i, i冈} (b) {冇1, 0, 0冈, 冇1, 1, 0冈, 冇0, 0, 1 ⴙ i冈冎 17. v 共1, 2, 0兲 18. v 共1 i, 1 i, 3兲 19. v 共i, 2 i, 1兲 20. v 共i, i, i兲 Representing a Vector by a Basis Finding Euclidean Inner Products In Exercises 21 and 22, determine the Euclidean inner product u v. 21. u 共i, 2i, 1 i兲 22. u 共4 i, i, 0兲 v 共3i, 0, 1 2i兲 v 共1 3i, 2, 1 i兲 In Exercises 23–26, let u ⴝ 冇1 ⴚ i, 3i兲, v ⴝ 冇2i, 2 ⴙ i冈, w ⴝ 冇1 ⴙ i, 0冈, and k ⴝ ⴚi. Evaluate the expressions in parts (a) and (b) to verify that they are equal. 23. (a) u v 24. (a) 共u v兲 w (b) v u (b) u w v w 25. (a) 共ku兲 v 26. (a) u 共kv兲 (b) k共u v兲 (b) k 共u v兲 Properties of Euclidean Inner Products In Exercises 27–34, determine the Euclidean norm of v. 27. v 共i, i兲 28. v 共1, 0兲 29. v 3共6 i, 2 i兲 30. v 共2 3i, 2 3i兲 31. v 共1, 2 i, i兲 32. v 共0, 0, 0兲 33. v 共1 2i, i, 3i, 1 i兲 34. v 共2, 1 i, 2 i, 4i兲 In Exercises 41–44, determine whether the function is a complex inner product, where u ⴝ 冇u1, u2冈 and v ⴝ 冇v1, v2冈. 41. 具u, v典 u1 u2v2 42. 具u, v典 共u1 v1兲 2共u2 v2兲 43. 具u, v典 4u1v1 6u2v2 44. 具u, v典 u1v1 u2v2 Complex Inner Products In Exercises 45–48, use the inner product 具u, v典 ⴝ u1v1 ⴙ 2u2v2 to find 具u, v典. 45. u 共2i, i兲 and v 共i, 4i兲 46. u 共3 i, i兲 and v 共2 i, 2i兲 47. u 共2 i, 2 i兲 and v 共3 i, 3 2i兲 48. u 共4 2i, 3兲 and v 共2 3i, 2兲 Finding Complex Inner Products Finding Complex Inner Products In Exercises 49 and 50, use the inner product 具u, v典 ⴝ u11v11 u12v12 u21v21 u22v22 where u u12 v v12 and v ⴝ 11 u ⴝ 11 u21 u22 v21 v22 to find 具u, v典. [ ] [ ] 冤01 2ii冥 v 冤10 1 2ii冥 1 2i i 2i 50. u 冤 v冤 冥 1i 0 3i 1冥 49. u 9781133110873_0804.qxp 3/10/12 6:56 AM Page 419 8.4 51. Let u 共a1 b1i, a2 b2i, . . . , an bn i兲. (a) Use the definitions of Euclidean norm and Euclidean inner product to show that 储u储 共 u1 2 u2 2 . . . un 2兲1兾2. (b) Use the results of part (a) to show that d共u, v兲 共 u1 v1 2 u2 v2 2 . . . un vn 2兲1兾2. ⱍ ⱍ ⱍ ⱍ ⱍ ⱍ ⱍ ⱍ ⱍ ⱍ ⱍ ⱍ 52. The complex Euclidean inner product of u and v is sometimes called the complex dot product. Compare the properties of the complex dot product in C n and those of the dot product in R n. (a) Which properties are the same? Which properties are different? (b) Explain the reasons for the differences. 53. Let v1 共i, 0, 0兲 and v2 共i, i, 0兲. If v3 共z1, z2, z3兲 and the set 再v1, v2, v3冎 is not a basis for C 3, what does this imply about z1, z2, and z3? 54. Let v1 共i, i, i兲 and v2 共1, 0, 1兲. Determine a vector v3 such that 再v1, v2, v3冎 is a basis for C 3. In Exercises 55–58, verify the statement using the properties of a complex inner product. 55. 具u, kv w典 k具u, v典 具u, w典 56. 具u, 0典 0 57. 具u, v典 具u, v典 具v, 2u典 Properties of Complex Inner Products 58. 具u, kv典 k具u, v典 In Exercises 59–63, prove the property, where u, v, and w are vectors in C n and k is a complex number. 59. 共u v兲 w u w v w 60. 共ku兲 v k共u v兲 61. u 共kv兲 k 共u v兲 62. u u 0 63. u u 0 if and only if u 0. Proof 64. Writing Let 具u, v典 be a complex inner product and let k be a complex number. How are 具u, v典 and 具u, kv典 related? Finding a Linear Transformation In Exercises 65 and 66, determine the linear transformation T : C m → C n that has the given characteristics. 65. T共1, 0兲 共2 i, 1兲 T共0, 1兲 共0, i兲 66. T共i, 0兲 共2 i, 1兲 T共0, i兲 共0, i兲 Exercises 419 In Exercises 67–70, the linear transformation T : C m → C n is shown by T冇v冈 ⴝ Av. Find the image of v and the preimage of w. 1 0 1i 0 67. A , v , w i i 1i 0 Finding an Image and a Preimage 冤 68. A 冤 冥 0 i i 0 冤 冥 1 , v 0 冥 冤冥 冤 冥 i 1 0 , w 1 1i 冤冥 冤 冥 冤 冥 冤冥 冤 冥 冤冥 冤 冥 1 69. A i i 0 70. A i 0 0 2 2i , w 2i 0 , v 3 2i i 3i 1 i i 1 2 1i 1 , v 5 , w 1 i 0 0 i 71. Find the kernel of the linear transformation in Exercise 68. 72. Find the kernel of the linear transformation in Exercise 69. In Exercises 73 and 74, find the image of v ⴝ 冇i, i冈 for the indicated composition, where T1 and T2 are the matrices below. 0 i ⴚi i and T2 ⴝ T1 ⴝ i 0 i ⴚi Finding an Image [ ] 73. T2 T1 [ ] 74. T1 T2 75. Determine which of the sets below are subspaces of the vector space of 2 2 complex matrices. (a) The set of 2 2 symmetric matrices. (b) The set of 2 2 matrices A satisfying 共 A 兲T A. (c) The set of 2 2 matrices in which all entries are real. (d) The set of 2 2 diagonal matrices. 76. Determine which of the sets below are subspaces of the vector space of complex-valued functions (see Example 4). (a) The set of all functions f satisfying f 共i兲 0. (b) The set of all functions f satisfying f 共0兲 1. (c) The set of all functions f satisfying f 共i兲 f 共i兲. True or False? In Exercises 77 and 78, determine whether each statement is true or false. If a statement is true, give a reason or cite an appropriate statement from the text. If a statement is false, provide an example that shows the statement is not true in all cases or cite an appropriate statement from the text. 77. Using the Euclidean inner product of u and v in C n, u v u1v1 u2v2 . . . unvn. 78. The Euclidean norm of u in C n denoted by 储u储 is 共u u兲2. 9781133110873_0805.qxp 420 3/10/12 Chapter 8 6:55 AM Page 420 Complex Vector Spaces 8.5 Unitary and Hermitian Matrices Find the conjugate transpose A* of a complex matrix A. Determine if a matrix A is unitary. Find the eigenvalues and eigenvectors of a Hermitian matrix, and diagonalize a Hermitian matrix. CONJUGATE TRANSPOSE OF A MATRIX Problems involving diagonalization of complex matrices and the associated eigenvalue problems require the concepts of unitary and Hermitian matrices. These matrices roughly correspond to orthogonal and symmetric real matrices. In order to define unitary and Hermitian matrices, the concept of the conjugate transpose of a complex matrix must first be introduced. Definition of the Conjugate Transpose of a Complex Matrix The conjugate transpose of a complex matrix A, denoted by A*, is given by A* A T where the entries of A are the complex conjugates of the corresponding entries of A. Note that if A is a matrix with real entries, then A* AT. To find the conjugate transpose of a matrix, first calculate the complex conjugate of each entry and then take the transpose of the matrix, as shown in the following example. Finding the Conjugate Transpose of a Complex Matrix Determine A* for the matrix A 冤 3 7i 2i 冥 0 . 4i SOLUTION A 冤 3 7i 2i A* AT 冥 0 3 7i 4i 2i 冤3 7i0 冤 冥 0 4i 2i 4i 冥 Several properties of the conjugate transpose of a matrix are listed in the following theorem. The proofs of these properties are straightforward and are left for you to supply in Exercises 47–50. THEOREM 8.8 Properties of the Conjugate Transpose If A and B are complex matrices and k is a complex number, then the following properties are true. 1. 共A*兲* A 2. 共A B兲* A* B* 3. 共kA兲* kA* 4. 共AB兲* B*A* 9781133110873_0805.qxp 3/10/12 6:55 AM Page 421 8.5 Unitary and Hermitian Matrices 421 UNITARY MATRICES Recall that a real matrix A is orthogonal if and only if A1 AT. In the complex system, matrices having the property that A1 A* are more useful, and such matrices are called unitary. Definition of Unitary Matrix A complex matrix A is unitary when A1 A*. A Unitary Matrix Show that the matrix A is unitary. A 1 1i 2 1i 冤 1i 1i 冥 SOLUTION Begin by finding the product AA*. 冤11 ii 冤40 04冥 1 0 冤 0 1冥 1 2 1 4 AA* 1i 1 1i 1i 2 1i 冥冤 1i 1i 冥 Because AA* 冤10 01冥 I 2 it follows that A* A1. So, A is a unitary matrix. Recall from Section 7.3 that a real matrix is orthogonal if and only if its row (or column) vectors form an orthonormal set. For complex matrices, this property characterizes matrices that are unitary. Note that a set of vectors 再v1, v2, . . . , vm冎 in C n (a complex Euclidean space) is called orthonormal when the statements below are true. 1. 储vi储 1, i 1, 2, . . . , m 2. vi vj 0, i j The proof of the next theorem is similar to the proof of Theorem 7.8 presented in Section 7.3. THEOREM 8.9 Unitary Matrices An n n complex matrix A is unitary if and only if its row (or column) vectors form an orthonormal set in C n. 9781133110873_0805.qxp 422 3/10/12 Chapter 8 6:55 AM Page 422 Complex Vector Spaces The Row Vectors of a Unitary Matrix Show that the complex matrix A is unitary by showing that its set of row vectors forms an orthonormal set in C 3. 冤 1i 2 i 冪3 3i 2冪15 1 2 A i 冪3 5i 2冪15 1 2 1 冪3 4 3i 2冪15 冥 SOLUTION Let r1, r2, and r3 be defined as follows. 1 1i 1 i i 1 , , r2 , , , 冪 冪 冪 2 2 3 3 3 r1 冢 2, 冣 r3 冢2冪15, 2冪15, 冢 3 i 4 3i 2冪15 5i 冣 冣 Begin by showing that r1, r2 , and r3 are unit vectors. 储r1储 冤 冢12冣冢12冣 冢1 2 i冣冢1 2 i冣 冢 12冣冢 21冣冥 1兾2 冤冢 冪3冣冢 冪3冣 冢冪3冣冢冪3冣 冢 冣冢 冣冥 1 1 1 冤 冥 3 3 3 储r2 储 i i i i 1 冪3 1 冪3 冤 14 42 14冥 1兾2 1兾2 1兾2 1 冤冢2冪5i15冣冢2冪5i15冣 冢23冪15i 冣冢23冪15i 冣 冢42冪153i冣冢42冪153i冣冥 25 10 25 冤 60 60 60冥 储r3储 1兾2 1 Then show that all pairs of distinct vectors are orthogonal. REMARK Try showing that the column vectors of A also form an orthonormal set in C 3. 1i i 1 2 冪3 2 i 1 1 i 2冪3 2冪3 2冪3 2冪3 0 r1 r2 冢 2 冣冢 冪3 冣 冢 r1 r3 冢 2 冣冢 r2 r3 冢 冪 1 i 冣冢 冣 冢 冣冢冪13冣 5i 1i 3i 1 2 2 2冪15 2冪15 5i 4 2i 4 3i 4冪15 4冪15 4冪15 0 冣 冢 1 i 冣冢 冣 冢 冣冢42冪153i冣 5i i 3i 1 4 3i 冢 冢 冣冢 冣 冣冢 冣 冣冢 冪3 2冪15 冪3 2冪15 冣 3 2冪15 5 1 3i 4 3i 6冪5 6冪5 6冪5 0 So, 再r1, r2, r3冎 is an orthonormal set. 1兾2 1 9781133110873_0805.qxp 3/10/12 6:55 AM Page 423 8.5 Unitary and Hermitian Matrices 423 HERMITIAN MATRICES A real matrix is symmetric when it is equal to its own transpose. In the complex system, the more useful type of matrix is one that is equal to its own conjugate transpose. Such a matrix is called Hermitian after the French mathematician Charles Hermite (1822–1901). Definition of a Hermitian Matrix A square matrix A is Hermitian when A A*. As with symmetric matrices, you can recognize Hermitian matrices by inspection. To see this, consider the 2 2 matrix A. A a2i b1 b2i d1 d2i 1 c2i 冤ac 冥 1 The conjugate transpose of A has the form A* A T 冤 a1 a2i 冥 c1 c2i b1 b2i d1 d2i a a2i c1 c2i 1 . b1 b2i d1 d2i 冤 冥 If A is Hermitian, then A A*. So, A must be of the form A 冤b 1 a1 b1 b2i . b2i d1 冥 Similar results can be obtained for Hermitian matrices of order n n. In other words, a square matrix A is Hermitian if and only if the following two conditions are met. 1. The entries on the main diagonal of A are real. 2. The entry aij in the ith row and the jth column is the complex conjugate of the entry aji in the jth row and the ith column. Hermitian Matrices Which matrices are Hermitian? 1 3i a. 3i i 冤 冤 冥 3 2i 3i c. 2 i 0 1i 3i 1 i 0 冥 b. 冤3 2i0 d. 冤 1 2 3 2 0 1 3 2i 4 冥 3 1 4 冥 SOLUTION a. This matrix is not Hermitian because it has an imaginary entry on its main diagonal. b. This matrix is symmetric but not Hermitian because the entry in the first row and second column is not the complex conjugate of the entry in the second row and first column. c. This matrix is Hermitian. d. This matrix is Hermitian because all real symmetric matrices are Hermitian. 9781133110873_0805.qxp 424 3/10/12 Chapter 8 6:55 AM Page 424 Complex Vector Spaces REMARK Note that this theorem implies that the eigenvalues of a real symmetric matrix are real, as stated in Theorem 7.7. One of the most important characteristics of Hermitian matrices is that their eigenvalues are real. This is formally stated in the next theorem. THEOREM 8.10 The Eigenvalues of a Hermitian Matrix If A is a Hermitian matrix, then its eigenvalues are real numbers. PROOF Let be an eigenvalue of A and let a1 b1i a bi v 2 . 2 .. an bni 冤 冥 be its corresponding eigenvector. If both sides of the equation Av v are multiplied by the row vector v*, then v*Av v*共v兲 共v*v兲 共a12 b12 a22 b22 . . . an2 bn2兲. Furthermore, because 共v*Av兲* v*A*共v*兲* v*Av it follows that v*Av is a Hermitian 1 number, so is real. 1 matrix. This implies that v*Av is a real To find the eigenvalues of complex matrices, follow the same procedure as for real matrices. Finding the Eigenvalues of a Hermitian Matrix Find the eigenvalues of the matrix A. 冤 3 2i 3i A 2i 0 1i 3i 1 i 0 冥 SOLUTION The characteristic polynomial of A is ⱍ 3 2 i 3i 2 i 1 i I A ⱍ ⱍ 3i 1 i 共 3兲共 ⱍ 2兲 共2 i兲关共2 i兲 共3i 3兲兴 3i 关共1 3i兲 3i兴 3 2 共 3 2 6兲 共5 9 3i兲 共3i 9 9兲 3 32 16 12 共 1兲共 6兲共 2兲. 2 So, the characteristic equation is 共 1兲共 6兲共 2兲 0, and the eigenvalues of A are 1, 6, and 2. 9781133110873_0805.qxp 3/10/12 6:55 AM Page 425 8.5 Unitary and Hermitian Matrices 425 To find the eigenvectors of a complex matrix, use a procedure similar to that used for a real matrix. For instance, in Example 5, to find the eigenvector corresponding to the eigenvalue 1, substitute the value for into the equation 冤 冥冤 冥 冤 冥 3 2 i 3i v1 0 2 i 1 i v2 0 3i 1 i v3 0 to obtain 冤 TECHNOLOGY Some graphing utilities and software programs have built-in programs for finding the eigenvalues and corresponding eigenvectors of complex matrices. 冥冤 冥 冤 冥 4 2 i 3i v1 0 2 i 1 1 i v2 0 . 3i 1 i 1 v3 0 Solve this equation using Gauss-Jordan elimination, or a graphing utility or software program, to obtain the eigenvector corresponding to 1 1, which is shown below. 1 v1 1 2i 1 冤 冥 Eigenvectors for 2 6 and 3 2 can be found in a similar manner. They are 冤 1 21i 6 9i 13 冥 LINEAR ALGEBRA APPLIED and 冤 1 3i 2 i , respectively. 5 冥 Quantum mechanics had its start in the early 20th century as scientists began to study subatomic particles and light. Collecting data on energy levels of atoms, and the rates of transition between levels, they found that atoms could be induced to more excited states by the absorption of light. German physicist Werner Heisenburg (1901–1976) laid a mathematical foundation for quantum mechanics using matrices. Studying the dispersion of light, he used vectors to represent energy levels of states and Hermitian matrices to represent “observables” such as momentum, position, and acceleration. He noticed that a measurement yields precisely one real value and leaves the system in precisely one of a set of mutually exclusive (orthogonal) states. So, the eigenvalues are the possible values that can result from a measurement of an observable, and the eigenvectors are the corresponding states of the system following the measurement. Let matrix A be a diagonal Hermitian matrix that represents an observable. Then consider a physical system whose state is represented by the column vector u. To measure the value of the observable A in the system of state u, you can find the product u*Au 关u1 u2 u3兴 冤 a11 0 0 0 a22 0 0 0 a33 冥冤 冥 u1 u2 u3 共u1 u1兲a11 共u2 u2兲a22 共u3u3兲a33. Because A is Hermitian and its values along the diagonal are real, u*Au is a real number. It represents the average of the values given by measuring the observable A on a system in the state u a large number of times. Jezper/Shutterstock.com 9781133110873_0805.qxp 426 3/10/12 Chapter 8 6:55 AM Page 426 Complex Vector Spaces Just as real symmetric matrices are orthogonally diagonalizable, Hermitian matrices are unitarily diagonalizable. A square matrix A is unitarily diagonalizable when there exists a unitary matrix P such that P1AP is a diagonal matrix. Because P is unitary, P1 P*, so an equivalent statement is that A is unitarily diagonalizable when there exists a unitary matrix P such that P*AP is a diagonal matrix. The next theorem states that Hermitian matrices are unitarily diagonalizable. THEOREM 8.11 Hermitian Matrices and Diagonalization If A is an n n Hermitian matrix, then 1. eigenvectors corresponding to distinct eigenvalues are orthogonal. 2. A is unitarily diagonalizable. PROOF To prove part 1, let v1 and v2 be two eigenvectors corresponding to the distinct (and real) eigenvalues 1 and 2. Because Av1 1v1 and Av2 2v2, you have the equations shown below for the matrix product 共Av1兲*v2. 共Av1兲*v2 v1*A*v2 v1*Av2 v1*2v2 2v1*v2 共Av1兲*v2 共1v1兲*v2 v1*1v2 1v1*v2 So, 2v1*v2 1v1*v2 0 共 2 1兲v1*v2 0 v1*v2 0 because 1 2 and this shows that v1 and v2 are orthogonal. Part 2 of Theorem 8.11 is often called the Spectral Theorem, and its proof is left to you. The Eigenvectors of a Hermitian Matrix The eigenvectors of the Hermitian matrix shown in Example 5 are mutually orthogonal because the eigenvalues are distinct. Verify this by calculating the Euclidean inner products v1 v2, v1 v3, and v2 v3. For example, v1 v2 共1兲共1 21i兲 共1 2i兲共6 9i兲 共1兲共13兲 共1兲共1 21i兲 共1 2i兲共6 9i兲 13 1 21i 6 9i 12i 18 13 0. The other two inner products v1 similar manner. v3 and v2 v3 can be shown to equal zero in a The three eigenvectors in Example 6 are mutually orthogonal because they correspond to distinct eigenvalues of the Hermitian matrix A. Two or more eigenvectors corresponding to the same eigenvalue may not be orthogonal. Once any set of linearly independent eigenvectors is obtained for an eigenvalue, however, the Gram-Schmidt orthonormalization process can be used to find an orthogonal set. 9781133110873_0805.qxp 3/10/12 6:55 AM Page 427 8.5 Unitary and Hermitian Matrices 427 Diagonalization of a Hermitian Matrix Find a unitary matrix P such that P*AP is a diagonal matrix where 冤 3 A 2i 3i 冥 2i 0 1i 3i 1i . 0 SOLUTION The eigenvectors of A are shown after Example 5. Form the matrix P by normalizing these three eigenvectors and using the results to create the columns of P. 储v1储 储共1, 1 2i, 1兲储 冪1 5 1 冪7 储v2储 储共1 21i, 6 9i, 13兲储 冪442 117 169 冪728 储v3储 储共1 3i, 2 i, 5兲储 冪10 5 25 冪40 So, P 冤 1 冪7 1 2i 冪7 1 冪7 冥 1 21i 冪728 6 9i 冪728 13 冪728 1 3i 冪40 2 i . 冪40 5 冪40 Try computing the product P*AP for the matrices A and P in Example 7 to see that P*AP 冤 1 0 0 0 6 0 0 0 2 冥 where 1, 6, and 2 are the eigenvalues of A. You have seen that Hermitian matrices are unitarily diagonalizable. It turns out that there is a larger class of matrices, called normal matrices, that are also unitarily diagonalizable. A square complex matrix A is normal when it commutes with its conjugate transpose: AA* A*A. The main theorem of normal matrices states that a complex matrix A is normal if and only if it is unitarily diagonalizable. You are asked to explore normal matrices further in Exercise 56. The properties of complex matrices described in this section are comparable to the properties of real matrices discussed in Chapter 7. The summary below indicates the correspondence between unitary and Hermitian complex matrices when compared with orthogonal and symmetric real matrices. Comparison of Symmetric and Hermitian Matrices A is a symmetric matrix (real) 1. Eigenvalues of A are real. 2. Eigenvectors corresponding to distinct eigenvalues are orthogonal. 3. There exists an orthogonal matrix P such that PTAP is diagonal. A is a Hermitian matrix (complex) 1. Eigenvalues of A are real. 2. Eigenvectors corresponding to distinct eigenvalues are orthogonal. 3. There exists a unitary matrix P such that P*AP is diagonal. 9781133110873_0805.qxp 428 3/10/12 Chapter 8 6:55 AM Page 428 Complex Vector Spaces 8.5 Exercises In Exercises 1–4, determine the conjugate transpose of the matrix. Finding the Conjugate Transpose 1. 冤2 i i 3i 冥 2. 冤 0 5 i 冪2i 3. 5i 6 4 冪 2i 4 3 冥 冤1 2i1 冤 2 4. 5 0 2i 1 冥 i 3i 6i 16. A 冥 In Exercises 5 and 6, use a software program or graphing utility to find the conjugate transpose of the matrix. Finding the Conjugate Transpose 冤 冤 1i 2i 5. 1i i 0 1 i 2i 2i 0 6. i 1 2i 1 0 2 1 1 2i i 0 1 2i 2i 4 i 2i 4i 0 冥 2i 1i 1 2i 冤0 i 0 0 冥 8. A 20. A 冤 1i 0 9. A 冪2 0 1 冤 0 i 冪3 冪2 1 2 i 1 冥 13. A 15. A 冤11 ii 冤i0 1i 1i 冥 冥 冥 0 i 冤 冥 4 5 3 i 5 3 5 4 i 5 12. A 冤11 ii 14. A 冤 1i 1i 冥 冥 i i 冪2 冪2 i 冪2 i i 冪2 冪3 冪6 i 0 冪3 i 冪6 冥 1 冤 冥 冪3 i 冥 1 冪3i 1 冪3i 2冪2 冪3 i 冤 冤 1 i 1 i 2 2 18. A 1 1 冪2 冪2 3 i 5 4 i 5 0 1 i 冪6 2 冪6 1 0 1i 冪3 1 冪3 0 0 i 冪2 冤i0 0i 冥 22. 0 23. 2 i 1 冤 2i i 0 1 0 1 冤2 1i 2i 2 3i 3i 25. Identifying Unitary Matrices In Exercises 11–16, determine whether A is unitary by calculating AA*. 11. A i 冥 Identifying Hermitian Matrices In Exercises 21–26, determine whether the matrix is Hermitian. 冥 1 21. 1 1i 2 2 1 i 冪3 冪3 1 1i 2 2 1 2 10. A i 冤i i 冪6 冤 冥 the matrix is not unitary. 7. A i 冪3 4 5 17. A 3 5 In Exercises 7–10, explain why Non-Unitary Matrices i 冪2 Row Vectors of a Unitary Matrix In Exercises 17–20, (a) verify that A is unitary by showing that its rows are orthonormal, and (b) determine the inverse of A. 19. A 冥 冤 冤 冥 冤 冥 0 i 0 24. 2 i 0 i i 1 1 0 0 冥 冥 冪2 i 1 26. 冪2 i 5 冤0i 5 3i 6 2 3i 冥 Finding Eigenvalues of a Hermitian Matrix In Exercises 27–32, determine the eigenvalues of the matrix A. 27. A 冤i0 0i冥 29. A 冤1 3i 冤 1 31. A 0 0 4 i 0 1i 2 冥 1i 3i 2i 冥 28. A 冤i3 3i冥 30. A 冤2 0i 2i 4 冥 9781133110873_0805.qxp 3/10/12 6:55 AM Page 429 8.5 32. A 冤 2 i 冪2 冪2 2 0 0 2 i 冪2 冥 i i 冪2 Unitary Matrices In Exercises 45 and 46, use the result of Exercise 44 to determine a, b, and c such that A is unitary. 1 1 45. A b 冪2 冤 33. Exercise 27 34. Exercise 30 35. Exercise 31 36. Exercise 28 In Exercises 37–41, find a unitary matrix P that diagonalizes the matrix A. Diagonalization of a Hermitian Matrix 冤i0 0i冥 39. A 冤 2 38. A i 冪2 冪2 2 0 0 2 i 冪2 i 冪2 冤2 0i 2i 4 冥 冥 i 4 2 2i 40. A 2 2i 6 冤 41. A 冤 冥 1 0 0 0 0 1 1 i 1 i 0 42. 冥 Consider the following matrix. 冤 2 3 i 4 i A 3i 1 1i 4i 1i 3 Is A unitary? Explain. Is A Hermitian? Explain. Are the row vectors of A orthonormal? Explain. The eigenvalues of A are distinct. Is it possible to determine the inner products of the pairs of eigenvectors by inspection? If so, state the value(s). If not, explain why not. (e) Is A unitarily diagonalizable? Explain. 43. Show that A In is unitary by computing AA*. 44. Let z be a complex number with modulus 1. Show that the matrix A is unitary. 冤 冥 冥 51. Proof Let A be a matrix such that A* A O. Prove that iA is Hermitian. 52. Show that det共A兲 det共A兲, where A is a 2 2 matrix. Determinants In Exercises 53 and 54, assume that the result of Exercise 52 is true for matrices of any size. 53. Show that det共A*兲 det共A兲. 54. Prove that if A is unitary, then det共A兲 1. ⱍ ⱍ 55. (a) Prove that every Hermitian matrix A can be written as the sum A B iC, where B is a real symmetric matrix and C is real and skew-symmetric. (b) Use part (a) to write the matrix A 冤1 2i 1i 3 冥 as the sum A B iC, where B is a real symmetric matrix and C is real and skew-symmetric. (c) Prove that every n n complex matrix A can be written as A B iC, where B and C are Hermitian. (d) Use part (c) to write the complex matrix A 56. (a) (b) (c) (d) (e) (f) 1 z z 冪2 iz iz 冥 6 3i 冪45 c In Exercises 47–50, prove the formula, where A and B are n ⴛ n complex matrices. 47. 共A*兲* A 48. 共A B兲* A* B* 49. 共kA兲* kA* 50. 共AB兲* B*A* 冥 (a) (b) (c) (d) A 冤 1 a 46. A 冪2 b a c Proof In Exercises 33 – 36, determine the eigenvectors of the matrix in the indicated exercise. Finding Eigenvectors of a Hermitian Matrix 37. A 429 Exercises (g) 冤2 ii 冥 2 1 2i as the sum A B iC, where B and C are Hermitian. Prove that every Hermitian matrix is normal. Prove that every unitary matrix is normal. Find a 2 2 matrix that is Hermitian, but not unitary. Find a 2 2 matrix that is unitary, but not Hermitian. Find a 2 2 matrix that is normal, but neither Hermitian nor unitary. Find the eigenvalues and corresponding eigenvectors of your matrix in part (e). Show that the complex matrix 冤0i 1i冥 is not diagonalizable. Is this matrix normal? 9781133110873_08REV.qxp 430 3/10/12 Chapter 8 8 6:53 AM Page 430 Complex Vector Spaces Review Exercises Operations with Complex Numbers In Exercises 1–6, perform the operation. 1. Find u z: u 2 4i, z 4i 2. Find u z: u 4, z 8i 3. Find uz: u 4 2i, z 4 2i 4. Find uz: u 2i, z 1 2i u 5. Find : u 6 2i, z 3 3i z u 6. Find : u 7 i, z i z Converting to Standard Form In Exercises 37–42, find the standard form of the complex number. In Exercises 11–18, perform the operation using 4ⴚi 2 1ⴙi i Aⴝ and B ⴝ . 3 3ⴙi 2i 2 ⴙ i 11. A B 12. A B 13. 2iB 14. iA 15. det共A B兲 16. det共A B兲 17. 3BA 18. 2AB 冤 冥 冤 ⱍ ⱍ ⱍ ⱍ ⱍ ⱍ In Exercises 25 –28, perform the indicated operation. 2i 2i 27. 共1 2i兲共1 2i兲 3 3i 1i 1 2i 28. 5 2i 共2 2i兲共2 3i兲 29 and 30, find Aⴚ1 (if it exists). 3i 1 2i 29. A 23 11 5 5i 2 3i 冤 30. A 冥 冤0 5 1i i 冥 5 5 i sin 4 4 冢 3 3 i sin 2 2 41. 7 cos 冢 2 2 i sin 3 3 冣 40. 6 cos 冣 42. 4关cos i sin 兴 冣 In Exercises 43–46, perform the indicated operation. Leave the result in polar form. Multiplying and Dividing in Polar Form i sin 2 2 冣冥冤3冢cos 6 i sin 6 冣冥 44. 冤 2 冢cos 2 i sin 2 冣冥冤2 冢cos 冢 2 冣 i sin 冢 2 冣冣冥 45. 9 关cos共兾2兲 i sin 共兾2兲兴 6 关cos共2兾3兲 i sin 共2兾3兲兴 46. 4 关cos 共兾4兲 i sin 共兾4兲兴 7 关cos 共兾3兲 i sin 共兾3兲兴 1 In Exercises 47–50, find the indicated power of the number and express the result in polar form. 47. 共1 i兲4 48. 共2i兲3 Finding Powers of Complex Numbers 冤 冢 49. 冪2 cos 26. Finding the Inverse of a Complex Matrix 冢 39. 4 cos 43. 4 cos In Exercises 19 – 24, perform the operation using w ⴝ 2 ⴚ 2i, v ⴝ 3 ⴙ i, and z ⴝ ⴚ1 ⴙ 2i. 19. z 20. v 21. w 22. vz 23. wv 24. zw 25. 冤 冢 3 冣 i sin冢 3 冣冥 38. 2 cos 冤冢 冥 Operations with Conjugates and Moduli Dividing Complex Numbers 冤 冢 6 冣 i sin冢 6 冣冥 37. 5 cos Finding Zeros In Exercises 7–10, use the given zero to find all zeros of the polynomial function. 7. p共x兲 x 3 2x 2 2x 4 Zero: 冪2i 8. p共x兲 x3 2x 4 Zero: 1 i 4 3 2 9. p共x兲 x x 3x 5x 10 Zero: 冪5i 4 3 2 10. p共x兲 x x x 3x 6 Zero: 冪3i Operations with Complex Matrices In Exercises 31– 36, determine the polar form of the complex number. 31. 4 4i 32. 2 2i 33. 冪3 i 34. 1 冪3i 35. 7 4i 36. 3 2i Converting to Polar Form i sin 6 6 冣冥 7 冤冢 50. 5 cos i sin 3 3 Finding Roots of Complex Numbers 51–54, express the roots in standard form. 冢 51. Square roots: 25 cos In Exercises 冢 52. Cube roots: 27 cos 2 2 i sin 3 3 i sin 6 6 冣 53. Cube roots: i 冢 54. Fourth roots: 16 cos i sin 4 4 冣 冣 冣冥 4 In Exercises 9781133110873_08REV.qxp 3/10/12 6:53 AM Page 431 Review Exercises Vector Operations in C n In Exercises 55–58, find the indicated vector using u ⴝ 冇4i, 2 ⴙ i冈, v ⴝ 冇3, ⴚi 冈, and w ⴝ 冇3 ⴚ i, 4 ⴙ i冈. 55. 7u v 56. 3iw 共4 i兲v 57. iu iv iw 58. 共3 2i兲u 共2i兲w Finding the Euclidean Norm In Exercises 59 and 60, determine the Euclidean norm of the vector. 59. v 共3 5i, 2i兲 60. v 共3i, 1 5i, 3 2i兲 In Exercises 61 and 62, find the Euclidean distance between the vectors. 61. v 共2 i, i兲, u 共i, 2 i兲 62. v 共2 i, 1 2i, 3i兲, u 共4 2i, 3 2i, 4兲 冤 9 76. 2 i 2 2i 0 1 i Finding the Conjugate Transpose In Exercises 67–70, determine the conjugate transpose of the matrix. 1 4i 3 i 67. A 3i 2i 冤 2i 68. A 冤 1 2i 冤 冤 冥 2i 2 2i 冥 2i 3 2i 2i 5 69. A 2 2i 3i 2 70. A i 1 1i 2 2i 1i i 0 2i 3 2i i 1 2i 冥 冥 Identifying Unitary Matrices In Exercises 71–74, determine whether the matrix is unitary. i 1 2i 1i 4 4 冪2 冪2 71. 72. 冪2 i 1 i 冪2 冪2 冪3 冪3 1 1 0 冪2 冪2 1 0 73. 74. 0 i 0 i i 1i 1 i 0 2 2 冤 冤 冤 冥 冤 冥 冥 冥 In Exercises 75 and 76, determine whether the matrix is Hermitian. 1 1 i 2 i 75. 1 i 3 i 2i i 4 Identifying Hermitian Matrices 冤 冥 冥 In Exercises 77 and 78, find the eigenvalues and corresponding eigenvectors of the matrix. Finding Eigenvalues and Eigenvectors 77. 冤 4 2i 2i 0 冥 Finding the Euclidean Distance Finding Complex Inner Products In Exercises 63–66, use the inner product 具u, v典 ⴝ u1v1 ⴙ 2u2v2 to find 具u, v典. 63. u 共i, 3i兲 and v 共2i, 2i兲 64. u 共i, 2 i兲 and v 共5 i, i兲 65. u 共1 i, 1 2i兲 and v 共2 i, 1 2i兲 66. u 共2 2i, 1兲 and v 共3 4i, 2兲 2 1 i 3 431 冤 2 0 78. 0 3 i 0 i 0 2 冥 79. Proof Prove that if A is an invertible matrix, then A* is also invertible. 80. Determine all complex numbers z such that z z. 81. Proof Prove that if z is a zero of a polynomial equation with real coefficients, then the conjugate of z is also a zero. 82. (a) Find the determinant of the Hermitian matrix 冤 冥 2i 3i 0 1i . 1i 0 (b) Proof Prove that the determinant of any Hermitian matrix is real. 83. Proof Let A and B be Hermitian matrices. Prove that AB BA if and only if AB is Hermitian. 84. Proof Let u be a unit vector in C n. Define H I 2uu*. Prove that H is an n n Hermitian and unitary matrix. 85. Proof Use mathematical induction to prove DeMoivre’s Theorem. 86. Proof Show that if z1 z2 and z1z2 are both nonzero real numbers, then z1 and z2 are both real numbers. 87. Proof Prove that if z and w are complex numbers, then zw z w. 88. Proof Prove that for all vectors u and v in a complex inner product space, 1 具u, v典 4 关 储u v储2 储u v储2 i储u iv储2 i储u iv储2兴. ⱍ 3 2i 3i ⱍ ⱍⱍ ⱍ ⱍ True or False? In Exercises 89 and 90, determine whether each statement is true or false. If a statement is true, give a reason or cite an appropriate statement from the text. If a statement is false, provide an example that shows the statement is not true in all cases or cite an appropriate statement from the text. 89. A square complex matrix A is called normal if it commutes with its conjugate transpose so that AA* A*A. 90. A square complex matrix A is called Hermitian if A A*. 9781133110873_08REV.qxp 432 8 3/10/12 Chapter 8 6:53 AM Page 432 Complex Vector Spaces Project 1 Population Growth and Dynamical Systems (II) In the projects for Chapter 7, you were asked to model the populations of two species using a system of differential equations of the form y 1 共t兲 ay1共t兲 by2共t兲 y 2 共t兲 cy1共t兲 dy2共t兲. The constants a, b, c, and d depend on the particular species being studied. In Chapter 7, you looked at an example of a predator-prey relationship, in which a 0.5, b 0.6, c 0.4, and d 3.0. Now consider a slightly different model. y 1 共t兲 0.6y1共t兲 0.8y2共t兲, y1共0兲 36 y 2 共t兲 0.8y1共t兲 0.6y2共t兲, y2共0兲 121 1. Use the diagonalization technique to find the general solutions y1共t兲 and y2共t兲 at any time t > 0. Although the eigenvalues and eigenvectors of the matrix A 0.6 冤0.8 冥 0.8 0.6 are complex, the same principles apply, and you can obtain complex exponential solutions. 2. Convert the complex solutions to real solutions by observing that if a bi is a (complex) eigenvalue of A with (complex) eigenvector v, then the real and imaginary parts of etv form a linearly independent pair of (real) solutions. Use the formula ei cos i sin . 3. Use the initial conditions to find the explicit form of the (real) solutions of the original equations. 4. If you have access to a graphing utility or software program, graph the solutions obtained in part 3 over the domain 0 t 3. At what moment are the two populations equal? 5. Interpret the solution in terms of the long-term population trend for the two species. Does one species ultimately disappear? Why or why not? Contrast this solution to that obtained for the model in Chapter 7. 6. If you have access to a graphing utility or software program that can numerically solve differential equations, use it to graph the solutions of the original system of equations. Does this numerical approximation appear to be accurate? Supri Suharjoto/Shutterstock.com