336_8_.qxp 4/7/8 :9 AM Page 48 8 Complex Vector Spaces 8. Complex Numbers 8. Conjugates and Dvson of Complex Numbers 8.3 Polar Form and DeMovre s Theorem 8.4 Complex Vector Spaces and Inner Products 8.5 Untary and Hermtan Matrces CHAPTER OBJECTIVES Graphcally represent complex numbers n the complex plane. Perform operatons wth complex numbers. Represent complex numbers as vectors. Use the Quadratc Formula to fnd all zeros of a quadratc polynomal. Perform operatons wth complex matrces. Fnd the determnant of a complex matrx. Fnd the conjugate, modulus, and argument of a complex number. Multply and dvde complex numbers. Fnd the nverse of a complex matrx. Determne the polar form of a complex number. Convert a complex number from standard form to polar form and from polar form to standard form. Multply and dvde complex numbers n polar form. Fnd roots and powers of complex numbers n polar form. Use DeMovre s Theorem to fnd roots of complex numbers n polar form. Recognze complex vector spaces, C n. Perform vector operatons n C n. Represent a vector n C n by a bass. Fnd the Eucldan nner product and the Eucldan norm of a vector n C n. Fnd the Eucldan dstance between two vectors n C n. Fnd the conjugate transpose A* of a complex matrx A. Determne f a matrx A s untary or Hermtan. Fnd the egenvalues and egenvectors of a Hermtan matrx. Dagonalze a Hermtan matrx. Determne f a Hermtan matrx s normal. 48
336_8_.qxp 4/7/8 :9 AM Page 48 48 Chapter 8 Complex Vector Spaces 8. Complex Numbers So far n the text, the scalar quanttes used have been real numbers. In ths chapter, you wll expand the set of scalars to nclude complex numbers. In algebra t s often necessary to solve quadratc equatons such as x 3x. The general quadratc equaton s ax bx c, and ts solutons are gven by the Quadratc Formula, x b b 4ac, a where the quantty under the radcal, b 4ac, s called the dscrmnant. If b 4ac, then the solutons are ordnary real numbers. But what can you conclude about the solutons of a quadratc equaton whose dscrmnant s negatve? For example, the equaton x 4 has a dscrmnant of b 4ac 6. From your experence wth ordnary algebra, t s clear that there s no real number whose square s 6. By wrtng 6 6 6 4, you can see that the essence of the problem s that there s no real number whose square s. To solve the problem, mathematcans nvented the magnary unt, whch has the property. In terms of ths magnary unt, you can wrte 6 4 4. The magnary unt s defned as follows. Defnton of the Imagnary Unt The number s called the magnary unt and s defned as where. REMARK: When workng wth products nvolvng square roots of negatve numbers, be sure to convert to a multple of before multplyng. For nstance, consder the followng operatons. Correct Incorrect Wth ths sngle addton of the magnary unt to the real number system, the system of complex numbers can be developed.
336_8_.qxp 4/7/8 :9 AM Page 483 Secton 8. Complex Numbers 483 Defnton of a Complex Number If a and b are real numbers, then the number a b s a complex number, where a s the real part and b s the magnary part of the number. The form a b s the standard form of a complex number. Imagnary axs 3 (, ) or The Complex Plane Fgure 8. y (3, ) or 3 + Horzontal component 3 Real axs x Some examples of complex numbers wrtten n standard form are, 4 3, and 6 6. The set of real numbers s a subset of the set of complex numbers. To see ths, note that every real number a can be wrtten as a complex number usng b. That s, for every real number, a a. A complex number s unquely determned by ts real and magnary parts. So, you can say that two complex numbers are equal f and only f ther real and magnary parts are equal. That s, f a b and c d are two complex numbers wrtten n standard form, then a b c d f and only f a c and b d. The Complex Plane Because a complex number s unquely determned by ts real and magnary parts, t s natural to assocate the number a b wth the ordered par a, b. Wth ths assocaton, you can graphcally represent complex numbers as ponts n a coordnate plane called the complex plane. Ths plane s an adaptaton of the rectangular coordnate plane. Specfcally, the horzontal axs s the real axs and the vertcal axs s the magnary axs. For nstance, Fgure 8. shows the graph of two complex numbers, 3 and. The number 3 s assocated wth the pont 3, and the number s assocated wth the pont,. Another way to represent the complex number a b s as a vector whose horzontal component s a and whose vertcal component s b. (See Fgure 8..) (Note that the use of the letter to represent the magnary unt s unrelated to the use of to represent a unt vector.) Vertcal component 4 3 Vector Representaton of a Complex Number Fgure 8.
336_8_.qxp 4/7/8 :9 AM Page 484 484 Chapter 8 Complex Vector Spaces Addton and Scalar Multplcaton of Complex Numbers Because a complex number conssts of a real part added to a multple of, the operatons of addton and multplcaton are defned n a manner consstent wth the rules for operatng wth real numbers. For nstance, to add (or subtract) two complex numbers, add (or subtract) the real and magnary parts separately. Defnton of Addton and Subtracton of Complex Numbers The sum and dfference of a b and c d are defned as follows. a b c d a c b d Sum a b c d a c b d Dfference EXAMPLE Addng and Subtractng Complex Numbers (a) (b) 4 3 4 3 4 4 5 3 3 3 3 4 REMARK: Note n part (a) of Example that the sum of two complex numbers can be a real number. Usng the vector representaton of complex numbers, you can add or subtract two complex numbers geometrcally usng the parallelogram rule for vector addton, as shown n Fgure 8.3. Imagnary axs 4 3 z + w = 5 z = 3 + 4 3 4 5 6 Real axs Imagnary axs 3 3 w = 3 + Real axs 3 4 w = 4 Addton of Complex Numbers Fgure 8.3 3 z = 3 z w = 4 Subtracton of Complex Numbers
336_8_.qxp 4/7/8 :9 AM Page 485 Secton 8. Complex Numbers 485 Many of the propertes of addton of real numbers are vald for complex numbers as well. For nstance, addton of complex numbers s both assocatve and commutatve. Moreover, to fnd the sum of three or more complex numbers, extend the defnton of addton n the natural way. For example, 3 4 3 4 3 3. To multply a complex number by a real scalar, use the defnton below. Defnton of Scalar Multplcaton If c s a real number and a b s a complex number, then the scalar multple of c and a b s defned as ca b ca cb. Geometrcally, multplcaton of a complex number by a real scalar corresponds to the multplcaton of a vector by a scalar, as shown n Fgure 8.4. EXAMPLE Operatons wth Complex Numbers (a) (b) 3 7 48 6 3 4 38 7 4 3 3 4 4 4 6 3 6 Imagnary axs Imagnary axs 4 3 3 z = 3 + z = 6 + 3 4 5 6 Real axs z = 3 z = 3 + 3 Real axs 3 Multplcaton of a Complex Number by a Real Number Fgure 8.4 Wth addton and scalar multplcaton, the set of complex numbers forms a vector space of dmenson (where the scalars are the real numbers). You are asked to verfy ths n Exercse 57.
336_8_.qxp 4/7/8 :9 AM Page 486 486 Chapter 8 Complex Vector Spaces Multplcaton of Complex Numbers The operatons of addton, subtracton, and multplcaton by a real number have exact counterparts wth the correspondng vector operatons. By contrast, there s no drect counterpart for the multplcaton of two complex numbers. Defnton of Multplcaton of Complex Numbers The product of the complex numbers a b and c d s defned as a bc d ac bd ad bc. Rather than try to memorze ths defnton of the product of two complex numbers, you should smply apply the dstrbutve property, as follows. a bc d ac d bc d ac ad bc bd ac ad bc bd ac bd ad bc ac bd ad bc Ths s demonstrated n the next example. Dstrbutve property Dstrbutve property Use. Commutatve property Dstrbutve property EXAMPLE 3 Multplyng Complex Numbers (a) (b) 3 6 4 3 8 6 4 3 8 6 4 3 8 3 6 4 Technology Note Many computer software programs and graphng utltes are capable of calculatng wth complex numbers. For example, on some graphng utltes, you can express a complex number a b as an ordered par a, b. Try verfyng the result of Example 3(b) by multplyng, and 4, 3. You should obtan the ordered par,. EXAMPLE 4 Complex Zeros of a Polynomal Use the Quadratc Formula to fnd the zeros of the polynomal px x 6x 3 and verfy that px for each zero.
336_8_.qxp 4/7/8 :9 AM Page 487 Secton 8. Complex Numbers 487 SOLUTION Usng the Quadratc Formula, you have Substtutng these values of x nto the polynomal px, you have and x b b 4ac 6 36 5 a 6 6 6 4 3. p3 3 63 3 3 3 63 3 9 6 6 4 8 3 9 4 8 3 p3 3 63 3 3 3 63 3 9 6 6 4 8 3 9 4 8 3. In Example 4, the two complex numbers 3 and 3 are complex conjugates of each other (together they form a conjugate par). A well-known result from algebra states that the complex zeros of a polynomal wth real coeffcents must occur n conjugate pars. (See Revew Exercse 86.) More wll be sad about complex conjugates n Secton 8.. Complex Matrces Now that you are able to add, subtract, and multply complex numbers, you can apply these operatons to matrces whose entres are complex numbers. Such a matrx s called complex. Defnton of a Complex Matrx A matrx whose entres are complex numbers s called a complex matrx. All of the ordnary operatons wth matrces also work wth complex matrces, as demonstrated n the next two examples.
336_8_.qxp 4/7/8 :9 AM Page 488 488 Chapter 8 Complex Vector Spaces EXAMPLE 5 SOLUTION Operatons wth Complex Matrces Let A and B be the complex matrces below and B A 3 4 and determne each of the followng. (a) 3A (b) B (c) A B (d) BA (a) 3A 3 3 4 3 3 3 6 9 (b) B 4 4 3 (c) A B 3 4 3 (d) BA 3 3 4 6 4 5 4 8 7 3 9 EXAMPLE 6 Fndng the Determnant of a Complex Matrx Fnd the determnant of the matrx SOLUTION A 4 3 deta 4 3 5 3. 45 3 3 5 3 6 6 8 6 Technology Note Many computer software programs and graphng utltes are capable of performng matrx operatons on complex matrces. Try verfyng the calculaton of the determnant of the matrx n Example 6. You should obtan the same answer, 8, 6.
336_8_.qxp 4/7/8 :9 AM Page 489 Secton 8. Complex Numbers 489 SECTION 8. Exercses In Exercses 6, determne the value of the expresson.. 3. 88 3. 44 4. 3 5. 4 6. 7 In Exercses 7, plot the complex number. 7. z 6 8. z 3 9. z 5 5. z 7. z 5. z 5 In Exercses 3 and 4, use vectors to llustrate the operatons graphcally. Be sure to graph the orgnal vector. 3. u and u, where u 3 4. 3u and 3 u, where u In Exercses 5 8, determne x such that the complex numbers n each par are equal. 5. x 3, 6 3 6. x 8 x, 4 7. x 6 x, 5 6 8. x 4 x, x 3 In Exercses 9 6, fnd the sum or dfference of the complex numbers. Use vectors to llustrate your answer graphcally. 9. 6 3 3.. 5 5. 3 3. 6 4. 7 3 4 5. 6. In Exercses 7 36, fnd the product. 7. 5 5 3 8. 3 3 9. 7 7 3. 4 4 3. a b 3. a ba b 33. 3 34. 4 35. a b 3 36. In Exercses 37 4, determne the zeros of the polynomal functon. 37. px x x 5 38. px x x 39. px x 5x 6 4. px x 4x 5 4. px x 4 6 4. px x 4 x 9 In Exercses 43 46, use the gven zero to fnd all zeros of the polynomal functon. 43. px x 3 3x 4x Zero: x 44. px x 3 x x 5 Zero: x 4 45. px x 3 3x 5x 75 Zero: x 5 46. px x 3 x 9x 9 Zero: x 3 In Exercses 47 56, perform the ndcated matrx operaton usng the complex matrces A and B. A and 47. A B 48. B A 49. A 5. B 5. A 5. 4 B 53. deta B 54. detb 55. 5AB 56. BA 57. Prove that the set of complex numbers, wth the operatons of addton and scalar multplcaton (wth real scalars), s a vector space of dmenson. 58. (a) Evaluate n for n,, 3, 4, and 5. (b) Calculate. (c) Fnd a general formula for n for any postve nteger n. 59. Let A 3. B 3 3 (a) Calculate A n for n,, 3, 4, and 5. (b) Calculate A. (c) Fnd a general formula for A n for any postve nteger n.
336_8_.qxp 4/7/8 :9 AM Page 49 49 Chapter 8 Complex Vector Spaces 6. Prove that f the product of two complex numbers s zero, then at least one of the numbers must be zero. True or False? In Exercses 6 and 6, determne whether each statement s true or false. If a statement s true, gve a reason or cte an approprate statement from the text. If a statement s false, provde an example that shows the statement s not true n all cases or cte an approprate statement from the text. 6. 4 6.
336_8_.qxp 4/7/8 :9 AM Page 49 Secton 8. Conjugates and Dvson of Complex Numbers 49 8. Conjugates and Dvson of Complex Numbers In Secton 8., t was mentoned that the complex zeros of a polynomal wth real coeffcents occur n conjugate pars. For nstance, n Example 4 you saw that the zeros of px x 6x 3 are 3 and 3. In ths secton, you wll examne some addtonal propertes of complex conjugates. You wll begn wth the defnton of the conjugate of a complex number. Defnton of the Conjugate of a Complex Number The conjugate of the complex number z a b s denoted by z and s gven by z a b. From ths defnton, you can see that the conjugate of a complex number s found by changng the sgn of the magnary part of the number, as demonstrated n the next example. EXAMPLE Fndng the Conjugate of a Complex Number Complex Number (a) z 3 (b) z 4 5 (c) z (d) z 5 Conjugate z 3 z 4 5 z z 5 REMARK: In part (d) of Example, note that 5 s ts own complex conjugate. In general, t can be shown that a number s ts own complex conjugate f and only f the number s real. (See Exercse 39.) Geometrcally, two ponts n the complex plane are conjugates f and only f they are reflectons about the real (horzontal) axs, as shown n Fgure 8.5 on the next page.
336_8_.qxp 4/7/8 :9 AM Page 49 49 Chapter 8 Complex Vector Spaces z = + 3 z = 3 Fgure 8.5 Imagnary axs 4 3 3 3 Real axs Imagnary axs Conjugate of a Complex Number 3 3 5 6 7 3 4 5 z = 4 5 Complex conjugates have many useful propertes. Some of these are shown n Theorem 8.. 5 4 3 z = 4 + 5 Real axs THEOREM 8. Propertes of Complex Conjugates For a complex number z a b, the followng propertes are true.. zz a b. zz 3. zz f and only f z. 4. z z PROOF To prove the frst property, let z a b. Then z a b and zz a ba b a ab ab b a b. The second and thrd propertes follow drectly from the frst. Fnally, the fourth property follows the defnton of the complex conjugate. That s, z a b a b a b z. EXAMPLE Fndng the Product of Complex Conjugates Fnd the product of z and ts complex conjugate. SOLUTION Because z, you have zz 4 5.
336_8_.qxp 4/7/8 :9 AM Page 493 Secton 8. Conjugates and Dvson of Complex Numbers 493 The Modulus of a Complex Number Because a complex number can be represented by a vector n the complex plane, t makes sense to talk about the length of a complex number. Ths length s called the modulus of the complex number. Defnton of the Modulus of a Complex Number The modulus of the complex number z a b s denoted by and s gven by z a b. z REMARK: The modulus of a complex number s also called the absolute value of the number. In fact, when z a s a real number, you have z a a. EXAMPLE 3 SOLUTION Fndng the Modulus of a Complex Number For z 3 and w 6, determne the value of each modulus. (a) (b) (c) (a) (b) z w zw z 3 3 w 6 37 (c) Because zw 36 5 6, you have zw 5 6 48. zw z w. Note that n Example 3, In Exercse 4, you are asked to prove that ths multplcatve property of the modulus always holds. Theorem 8. states that the modulus of a complex number s related to ts conjugate. THEOREM 8. The Modulus of a Complex Number For a complex number z zz. z, PROOF Let z a b, then z a b and you have zz a ba b a b z.
336_8_.qxp 4/7/8 :9 AM Page 494 494 Chapter 8 Complex Vector Spaces Dvson of Complex Numbers One of the most mportant uses of the conjugate of a complex number s n performng dvson n the complex number system. To defne dvson of complex numbers, consder z a b and w c d and assume that c and d are not both. If the quotent z x y w s to make sense, t has to be true that z wx y c dx y cx dy dx cy. But, because z a b, you can form the lnear system below. cx dy a dx cy b Solvng ths system of lnear equatons for x and y yelds ac bd x ww and y bc ad ww. Now, because zw a bc d ac bd bc ad, the defnton below s obtaned. Defnton of Dvson of Complex Numbers The quotent of the complex numbers z a b and w c d s defned as z a b w c d provded c d. ac bd bc ad c d c d w zw REMARK: If c d, then c d, and w. In other words, as s the case wth real numbers, dvson of complex numbers by zero s not defned. In practce, the quotent of two complex numbers can be found by multplyng the numerator and the denomnator by the conjugate of the denomnator, as follows. a b a b c d c d c dc d a bc d c dc d ac bd bc ad c d ac bd bc ad c d c d
336_8_.qxp 4/7/8 :9 AM Page 495 Secton 8. Conjugates and Dvson of Complex Numbers 495 EXAMPLE 4 Dvson of Complex Numbers (a) 3 4 (b) 3 4 3 43 4 9 6 5 5 Now that you can dvde complex numbers, you can fnd the (multplcatve) nverse of a complex matrx, as demonstrated n Example 5. EXAMPLE 5 Fndng the Inverse of a Complex Matrx Fnd the nverse of the matrx A 3 and verfy your soluton by showng that AA I. SOLUTION Usng the formula for the nverse of a matrx from Secton.3, you have A A 6 3 Furthermore, because A you can wrte 6 5 3 6 4 5 6 5 3 A 6 5 3 3 3 6 3 3 3 3 To verfy your soluton, multply A and A as follows. AA 3 5 6, 7 7. 5 6 5. 5 3 3 7 7
336_8_.qxp 4/7/8 :9 AM Page 496 496 Chapter 8 Complex Vector Spaces Technology Note If your computer software program or graphng utlty can perform operatons wth complex matrces, then you can verfy the result of Example 5. If you have matrx A stored on a graphng utlty, evaluate A. The last theorem n ths secton summarzes some useful propertes of complex conjugates. THEOREM 8.3 Propertes of Complex Conjugates For the complex numbers z and w, the followng propertes are true.. z w z w. z w z w 3. zw z w 4. zw zw PROOF To prove the frst property, let z a b and w c d. Then z w a c b d a c b d a b c d z w. The proof of the second property s smlar. The proofs of the other two propertes are left to you. SECTION 8. Exercses In Exercses 6, fnd the complex conjugate z and graphcally represent both z and z.. z 6 3. z 5 3. z 8 4. z 5. z 4 6. z 3 In Exercses 7, fnd the ndcated modulus, where z, w 3, and v 5. z wz z v 7. 8. 9. zw... zv wz w z zw, 3. Verfy that where z and w. 4. Verfy that zv z v z v, where z and v 3. In Exercses 5, perform the ndcated operatons. 5. 6. 6 3 7. 3 5 8. 3 4 9. 3 3. 4 5
336_8_.qxp 4/7/8 :3 AM Page 497 Secton 8. Conjugates and Dvson of Complex Numbers 497 In Exercses 4, perform the operaton and wrte the result n standard form... 5 3 3. 4. 3 3 3 4 In Exercses 5 8, use the gven zero to fnd all zeros of the polynomal functon. 5. px 3x 3 4x 8x 8 Zero: 3 6. px 4x 3 3x 34x Zero: 3 7. px x 4 3x 3 5x x Zero: 3 8. px x 3 4x 4x Zero: 3 In Exercses 9 and 3, fnd each power of the complex number z. (a) z (b) z 3 (c) z (d) z 9. 3. z z In Exercses 3 36, determne whether the complex matrx A has an nverse. If A s nvertble, fnd ts nverse and verfy that AA I. 3. 3. A A 6 3 3 3 33. 34. A A 35. 36. A A In Exercses 37 and 38, determne all values of the complex number z for whch A s sngular. (Hnt: Set deta and solve for z. ) z 5 37. A 3 38. A z 39. Prove that z z f and only f z s real. 4. Prove that for any two complex numbers z and w, each of the statements below s true. (a) zw z w (b) If w, then zw z w. 4. Descrbe the set of ponts n the complex plane that satsfes each of the statements below. (a) (b) (c) (d) True or False? In Exercses 4 and 43, determne whether each statement s true or false. If a statement s true, gve a reason or cte an approprate statement from the text. If a statement s false, provde an example that shows the statement s not true n all cases or cte an approprate statement from the text. 4. 43. There s no complex number that s equal to ts complex conjugate. 44. Descrbe the set of ponts n the complex plane that satsfes each of the statements below. (a) (b) (c) (d) z 3 z z 5 z 5 z 4 z z z > 3 45. (a) Evaluate n for n,, 3, 4, and 5. (b) Calculate and. (c) Fnd a general formula for n for any postve nteger n. 46. (a) Verfy that. (b) Fnd the two square roots of. (c) Fnd all zeros of the polynomal x 4.
336_8_3.qxp 4/7/8 :3 AM Page 498 498 Chapter 8 Complex Vector Spaces 8.3 Polar Form and DeMovre s Theorem (a, b) b r Imagnary axs a θ Complex Number: a + b Rectangular Form: (a, b) Polar Form: (r, θ) Fgure 8.6 Real axs At ths pont you can add, subtract, multply, and dvde complex numbers. However, there s stll one basc procedure that s mssng from the algebra of complex numbers. To see ths, consder the problem of fndng the square root of a complex number such as. When you use the four basc operatons (addton, subtracton, multplcaton, and dvson), there seems to be no reason to guess that. That s,. To work effectvely wth powers and roots of complex numbers, t s helpful to use a polar representaton for complex numbers, as shown n Fgure 8.6. Specfcally, f a b s a nonzero complex number, then let be the angle from the postve x-axs to the radal lne passng through the pont a, b and let r be the modulus of a b. So, a r cos, b r sn, and r a b and you have a b r cos r sn, from whch the polar form of a complex number s obtaned. Defnton of the Polar Form of a Complex Number The polar form of the nonzero complex number z a b s gven by z rcos sn where a r cos, b r sn, r a b, and tan ba. The number r s the modulus of z and s the argument of z. REMARK: The polar form of z s expressed as z cos sn, where s any angle. Because there are nfntely many choces for the argument, the polar form of a complex number s not unque. Normally, the values of that le between and are used, although on occason t s convenent to use other values. The value of that satsfes the nequalty < Prncpal argument s called the prncpal argument and s denoted by Arg( z). Two nonzero complex numbers n polar form are equal f and only f they have the same modulus and the same prncpal argument. EXAMPLE Fndng the Polar Form of a Complex Number Fnd the polar form of each of the complex numbers. (Use the prncpal argument.) (a) z (b) z 3 (c) z
336_8_3.qxp 4/7/8 :3 AM Page 499 Secton 8.3 Polar Form and DeMovre s Theorem 499 SOLUTION (a) Because a and b, then r, whch mples that r. From a r cos and b r sn, you have So, cos a r and 4 and (b) Because a and b 3, then r 3 3, whch mples that r 3. So, cos a and sn b r 3 r 3 3 and t follows that So, the polar form s (c) Because a and b, t follows that r and so z cos sn.98.. z cos 4 sn 4. z 3 cos.98 sn.98. sn b r., The polar forms derved n parts (a), (b), and (c) are depcted graphcally n Fgure 8.7. Fgure 8.7 Imagnary axs θ (a) z = cos 4 z = Real axs π [ ) + sn π ( ( )] 4 Imagnary axs 4 3 θ z = + 3 (b) z 3[cos(.98) + sn(.98)] Real axs Imagnary axs z = θ Real axs ( ) (c) z = cos π + sn π EXAMPLE Convertng from Polar to Standard Form Express the complex number n standard form. z 8 cos 3 sn 3
336_8_3.qxp 4/7/8 :3 AM Page 5 5 Chapter 8 Complex Vector Spaces SOLUTION Because cos3 and sn3 3, you can obtan the standard form z 8 cos 3 4 43. The polar form adapts ncely to multplcaton and dvson of complex numbers. Suppose you have two complex numbers n polar form z and z r cos sn r cos sn. Then the product of z and s expressed as Usng the trgonometrc denttes and you have 3 sn 3 8 z z z r r cos sn cos sn r r cos cos sn sn cos sn sn cos. cos cos cos sn sn sn sn cos cos sn z z r r cos sn. Ths establshes the frst part of the next theorem. The proof of the second part s left to you. (See Exercse 65.) THEOREM 8.4 Product and Quotent of Two Complex Numbers Gven two complex numbers n polar form z and z r cos sn r cos sn the product and quotent of the numbers are as follows. z z r r cos sn Product z z r r cos sn, z Quotent Ths theorem says that to multply two complex numbers n polar form, multply modul and add arguments. To dvde two complex numbers, dvde modul and subtract arguments. (See Fgure 8.8.)
336_8_3.qxp 4/7/8 :3 AM Page 5 Secton 8.3 Polar Form and DeMovre s Theorem 5 Imagnary axs Imagnary axs z z z z z θ + θ r r r r z θ θ Real axs r r r r z z θ θ θ θ Real axs To multply z and z : Multply modul and add arguments. To dvde z and z : Dvde modul and subtract arguments. Fgure 8.8 EXAMPLE 3 Multplyng and Dvdng n Polar Form Fnd z z and z z for the complex numbers z and z 5 cos sn 3 cos sn 4 6 SOLUTION Because you have the polar forms of and z, you can apply Theorem 8.4, as follows. z z 5 multply dvde 3 cos z 5 z 3 cos 4 4 subtract 4 add z 6 sn 6 sn 4 4 subtract add 6 5 3 6 5 cos 5 5 cos sn 6. sn REMARK: Try performng the multplcaton and dvson n Example 3 usng the standard forms z 5 5 and z 3 6 6.
336_8_3.qxp 4/7/8 :3 AM Page 5 5 Chapter 8 Complex Vector Spaces DeMovre s Theorem The fnal topc n ths secton nvolves procedures for fndng powers and roots of complex numbers. Repeated use of multplcaton n the polar form yelds Smlarly, z r cos sn z r cos sn r cos sn r cos sn z 3 r cos sn r cos sn r 3 cos 3 sn 3. z 4 r 4 cos 4 sn 4 z 5 r 5 cos 5 sn 5. Ths pattern leads to the next mportant theorem, named after the French mathematcan Abraham DeMovre (667 754). You are asked to prove ths theorem n Revew Exercse 85. THEOREM 8.5 DeMovre s Theorem If z rcos sn and n s any postve nteger, then z n r n cos n sn n. EXAMPLE 4 Rasng a Complex Number to an Integer Power Fnd 3 and wrte the result n standard form. SOLUTION Frst convert to polar form. For 3, r 3 whch mples that 3. So, and tan 3 3 3 cos sn 3 3. By DeMovre s Theorem, 3 cos sn 3 3 cos sn 3 3 496cos 8 sn 8 496 496.
336_8_3.qxp 4/7/8 :3 AM Page 53 Secton 8.3 Polar Form and DeMovre s Theorem 53 Recall that a consequence of the Fundamental Theorem of Algebra s that a polynomal of degree n has n zeros n the complex number system. So, a polynomal such as px x 6 has sx zeros, and n ths case you can fnd the sx zeros by factorng and usng the Quadratc Formula. x 6 x 3 x 3 x x x x x x Consequently, the zeros are x ±, x ± 3, and x ± 3. Each of these numbers s called a sxth root of. In general, the nth root of a complex number s defned as follows. Defnton of the nth Root of a Complex Number The complex number w a b s an nth root of the complex number z f z w n a b n. DeMovre s Theorem s useful n determnng roots of complex numbers. To see how ths s done, let w be an nth root of z, where w scos sn and z rcos cos. Then, by DeMovre s Theorem you have w n s n cos n sn n, and because w n z, t follows that s n cos n sn n rcos sn. Now, because the rght and left sdes of ths equaton represent equal complex numbers, you can equate modul to obtan s n r, whch mples that s r, n and equate prncpal arguments to conclude that and must dffer by a multple of. Note that r s a postve real number and so s r n s also a postve real number. Consequently, for some nteger k, n whch mples that k n k,. n Fnally, substtutng ths value of nto the polar form of w produces the result stated n the next theorem. THEOREM 8.6 The nth Roots of a Complex Number For any postve nteger n, the complex number z rcos sn has exactly n dstnct roots. These n roots are gven by nr k cos n sn where k,,,..., n. k n
336_8_3.qxp 4/7/8 :3 AM Page 54 54 Chapter 8 Complex Vector Spaces n r Fgure 8.9 Imagnary axs π n π n The nth Roots of a Complex Number Real axs REMARK: Note that when k exceeds n, the roots begn to repeat. For nstance, f k n, the angle s n n n whch yelds the same values for the sne and cosne as k. The formula for the nth roots of a complex number has a nce geometrc nterpretaton, as shown n Fgure 8.9. Note that because the nth roots all have the same modulus (length) nr, they wll le on a crcle of radus nr wth center at the orgn. Furthermore, the n roots are equally spaced around the crcle, because successve nth roots have arguments that dffer by n. You have already found the sxth roots of by factorng and the Quadratc Formula. Try solvng the same problem usng Theorem 8.6 to see f you get the roots shown n Fgure 8.. When Theorem 8.6 s appled to the real number, the nth roots have a specal name the nth roots of unty. + 3 Imagnary axs + 3 Real axs 3 3 The Sxth Roots of Unty Fgure 8. EXAMPLE 5 SOLUTION Fndng the nth Roots of a Complex Number Determne the fourth roots of. In polar form, you can wrte as cos sn so that r and. Then, by applyng Theorem 8.6, you have 4 4 cos k 4 4 sn k 4 4 k cos 8 sn k 8.
336_8_3.qxp 4/7/8 :3 AM Page 55 Secton 8.3 Polar Form and DeMovre s Theorem 55 Settng k,,, and 3, you obtan the four roots as shown n Fgure 8.. z cos sn 8 8 z cos 5 5 sn 8 8 z 3 cos 9 9 sn 8 8 z 4 cos 3 3 sn 8 8 REMARK: In Fgure 8., note that when each of the four angles 8, 58, 98, and 38 s multpled by 4, the result s of the form k. cos 5π 8 + sn 5π 8 Imagnary axs cos π 8 + sn Real axs π 8 cos 9π 8 Fgure 8. + sn 9π 8 cos 3π + sn 3π 8 8 SECTION 8.3 Exercses In Exercses 4, express the complex number n polar form.. Imagnary. axs Real axs Imagnary axs 3 + 3 Real axs 3. Imagnary 4. Imagnary axs axs 6 6 5 4 3 3 3 Real axs 3 3 Real axs
336_8_3.qxp 4/7/8 :3 AM Page 56 56 Chapter 8 Complex Vector Spaces In Exercses 5 6, represent the complex number graphcally, and gve the polar form of the number. (Use the prncpal argument.) 5. 6. 7. 3 5 8. 3 9. 6.. 7. 4 3. 3 3 4. 5. 6. 5 In Exercses 7 6, represent the complex number graphcally, and gve the standard form of the number. 7. cos sn 8. 3 5 5 9. cos sn. 3 3. 3.75 cos sn. 8 cos sn 4 6 3. 4 cos 3 3 sn 4. 5. 7cos sn 6. 6cos sn In Exercses 7 34, perform the ndcated operaton and leave the result n polar form. 7. 8. 9. 3. 3. 3. 33. 34. 3 cos 3 4 cos.5cos sn.5cos sn 3 cos sn sn 3 sn 3 cos3 sn3 4[cos56 sn56] cos53 sn53 cos sn 4 3 4 cos 6 cos 3 3 cos3 sn3 3cos6 sn6 9cos34 sn34 5cos4 sn4 sn 6 4 5 cos 3 3 sn 4 4 3 4 6 cos 5 5 sn 6 6 sn 7 cos 4 6 4 cos sn 3 3 sn 7 4 6 In Exercses 35 44, use DeMovre s Theorem to fnd the ndcated powers of the complex number. Express the result n standard form. 35. 4 36. 6 37. 38. 39. 4. 5 cos sn 9 5 4. 3 5 5 4 cos sn 4. cos sn 4 4 6 6 3 3 3 7 cos 43. sn 44. In Exercses 45 56, (a) use DeMovre s Theorem to fnd the ndcated roots, (b) represent each of the roots graphcally, and (c) express each of the roots n standard form. 45. Square roots: 6 cos sn 3 46. Square roots: 47. Fourth roots: 48. Ffth roots: 8 9 cos sn 3 3 6 cos 4 4 sn 3 3 3 cos 5 5 sn 6 6 49. Square roots: 5 5. Fourth roots: 65 5. Cube roots: 5 3 5. Cube roots: 4 53. Cube roots: 8 54. Fourth roots: 55. Fourth roots: 56. Cube roots: In Exercses 57 64, fnd all the solutons to the equaton and represent your solutons graphcally. 57. x 4 58. x 4 6 59. x 3 6. x 3 7 6. x 5 43 6. x 4 8 63. x 3 64 64. x 4 65. When provded wth two complex numbers z and z r cos sn r cos sn, wth z, prove that z z r r cos sn. 3 5 5 3 cos 9 3 sn 3 4
336_8_3.qxp 4/7/8 :3 AM Page 57 Secton 8.3 Polar Form and DeMovre s Theorem 57 66. Show that the complex conjugate of z rcos sn s z rcos sn. 67. Use the polar forms of z and z n Exercse 66 to fnd each of the followng. (a) zz (b) zz, z 68. Show that the negatve of z rcos sn s z rcos sn. 69. Wrtng (a) Let z rcos sn cos 6 sn Sketch z, z, and z n the complex plane. (b) What s the geometrc effect of multplyng a complex number z by? What s the geometrc effect of dvdng z by? 7. Calculus Recall that the Maclaurn seres for e x, sn x, and cos x are e x x x x3! 3! x 4 4!... sn x x x 3 3! x5 5! x 7 7!... cos x x! x 4 x6 4! 6!.... 6. (a) Substtute x n the seres for and show that e cos sn. (b) Show that any complex number z a b expressed n polar form as z re. (c) Prove that f z re, then z re. (d) Prove the amazng formula e. can be True or False? In Exercses 7 and 7, determne whether each statement s true or false. If a statement s true, gve a reason or cte an approprate statement from the text. If a statement s false, provde an example that shows the statement s not true n all cases or cte an approprate statement from the text. 7. Although the square of the complex number b s gven by b b, the absolute value of the complex number z a b s defned as a b a b. 7. Geometrcally, the nth roots of any complex number z are all equally spaced around the unt crcle centered at the orgn. e x
336_8_4.qxp 4/7/8 :3 AM Page 59 Secton 8.4 Complex Vector Spaces and Inner Products 59 8.4 Complex Vector Spaces and Inner Products All the vector spaces you have studed so far n the text have been real vector spaces because the scalars have been real numbers. A complex vector space s one n whch the scalars are complex numbers. So, f v, v,..., v m are vectors n a complex vector space, then a lnear combnaton s of the form c v c v where the scalars c are complex numbers. The complex verson of R n, c,..., c s the complex vector space C n m consstng of ordered n-tuples of complex numbers. So, a vector n C n has the form It s also convenent to represent vectors n v a b a b. a n b n. c m v m v a b, a b,..., a n b n. C n by column matrces of the form As wth R n, the operatons of addton and scalar multplcaton n C n are performed component by component. EXAMPLE Vector Operatons n C n Let v, 3 and u, 4 be vectors n the complex vector space C. Determne each vector. (a) v u (b) v (c) 3v 5 u SOLUTION (a) In column matrx form, the sum v u s (b) Because 5 and 3 7, you have (c) v u 3 4 3 7. v, 3 5, 7. 3v 5 u 3, 3 5, 4 3 6, 9 3 9 7, 4,
336_8_4.qxp 4/7/8 :3 AM Page 5 5 Chapter 8 Complex Vector Spaces Many of the propertes of R n are shared by C n. For nstance, the scalar multplcatve dentty s the scalar and the addtve dentty n C n s,,,...,. The standard bass for C n s smply e,,,..., e,,,...,.. e n,,,..., whch s the standard bass for R n. Because ths bass contans n vectors, t follows that the dmenson of C n s n. Other bases exst; n fact, any lnearly ndependent set of n vectors n C n can be used, as demonstrated n Example. EXAMPLE Verfyng a Bass Show that S,,,,,,,, s a bass for C 3. SOLUTION Because C 3 has a dmenson of 3, the set v, v, v 3 wll be a bass f t s lnearly ndependent. To check for lnear ndependence, set a lnear combnaton of the vectors n S equal to, as follows. c v c v c 3 v 3,, c,, c, c,,, c 3,, c c, c, c 3,, Ths mples that c c v v v 3 c c 3. So, c c c 3, and you can conclude that v, v, v 3 s lnearly ndependent. EXAMPLE 3 Representng a Vector n C n by a Bass Use the bass S n Example to represent the vector v,,.
336_8_4.qxp 4/7/8 :3 AM Page 5 Secton 8.4 Complex Vector Spaces and Inner Products 5 SOLUTION By wrtng you can obtan whch mples that c, So, v c v c v c 3 v 3 c c, c, c 3,,, c c c c c 3, and v v v v 3. c 3. Try verfyng that ths lnear combnaton yelds,,. Other than C n, there are several addtonal examples of complex vector spaces. For nstance, the set of m n complex matrces wth matrx addton and scalar multplcaton forms a complex vector space. Example 4 descrbes a complex vector space n whch the vectors are functons. EXAMPLE 4 The Space of Complex-Valued Functons Consder the set S of complex-valued functons of the form fx f x f x where f and f are real-valued functons of a real varable. The set of complex numbers form the scalars for S, and vector addton s defned by fx gx f x f x g (x g x f x g x f x g x. It can be shown that S, scalar multplcaton, and vector addton form a complex vector space. For nstance, to show that S s closed under scalar multplcaton, let c a b be a complex number. Then cfx a bf x f x af x bf x bf x af x s n S.
336_8_4.qxp 4/7/8 :3 AM Page 5 5 Chapter 8 Complex Vector Spaces The defnton of the Eucldean nner product n C n s smlar to the standard dot product n R n, except that here the second factor n each term s a complex conjugate. Defnton of the Eucldean Inner Product n C n Let u and v be vectors n C n. The Eucldean nner product of u and v s gven by u v u v u v u n v n. REMARK: Note that f u and v happen to be real, then ths defnton agrees wth the standard nner (or dot) product n R n. EXAMPLE 5 Fndng the Eucldean Inner Product n C 3 Determne the Eucldean nner product of the vectors u,, 4 5 and v,,. SOLUTION u v u v u v u 3 v 3 4 5 3 Several propertes of the Eucldean nner product C n are stated n the followng theorem. THEOREM 8.7 Propertes of the Eucldean Inner Product Let u, v, and w be vectors n C n and let k be a complex number. Then the followng propertes are true.. u v v u. u v w u w v w 3. ku v ku v 4. u kv ku v 5. u u 6. u u f and only f u. PROOF The proof of the frst property s shown below, and the proofs of the remanng propertes have been left to you. Let u u, u,..., u n and v v, v,..., v n.
336_8_4.qxp 4/7/8 :3 AM Page 53 Secton 8.4 Complex Vector Spaces and Inner Products 53 Then v u v u v u... v n u n v u v u... v n u n v u v u... v n u n u v u v u v. You wll now use the Eucldean nner product n C n to defne the Eucldean norm (or length) of a vector n and the Eucldean dstance between two vectors n. C n u n v n C n Defntons of the Eucldean Norm and Dstance n C n The Eucldean norm (or length) of u n C n s denoted by u and s u u u. The Eucldean dstance between u and v s du, v u v. The Eucldean norm and dstance may be expressed n terms of components as u u u... u n du, v u v u v... u n v n. EXAMPLE 6 Fndng the Eucldean Norm and Dstance n C n Determne the norms of the vectors u,, 4 5 and v,, and fnd the dstance between u and v. SOLUTION The norms of u and v are expressed as follows. u u u u 3 4 5 5 4 46 v v v v 3 5 7
336_8_4.qxp 4/7/8 :3 AM Page 54 54 Chapter 8 Complex Vector Spaces The dstance between u and v s expressed as du, v u v,, 4 5 4 5 5 4 47. Complex Inner Product Spaces The Eucldean nner product s the most commonly used nner product n C n. On occason, however, t s useful to consder other nner products. To generalze the noton of an nner product, use the propertes lsted n Theorem 8.7. Defnton of a Complex Inner Product Let u and v be vectors n a complex vector space. A functon that assocates u and v wth the complex number u, v s called a complex nner product f t satsfes the followng propertes.. u, v v, u. u v, w u, w v, w 3. ku, v ku, v 4. u, u and u, u f and only f u. A complex vector space wth a complex nner product s called a complex nner product space or untary space. EXAMPLE 7 SOLUTION A Complex Inner Product Space Let u u and be vectors n the complex space C, u v v, v. Show that the functon defned by s a complex nner product. Verfy the four propertes of a complex nner product as follows... 3. 4. u, v u v u v v, u v u v u u v u v u, v u v, w u v w u v w u w u w v w v w u, w v, w ku, v ku v ku v ku v u v k u, v u, u u u u u u u Moreover, u, u f and only f u u. Because all propertes hold, u, v s a complex nner product.
336_8_4.qxp 4/7/8 :3 AM Page 55 Secton 8.4 Complex Vector Spaces and Inner Products 55 SECTION 8.4 Exercses In Exercses 8, perform the ndcated operaton usng u, 3, v, 3, and w 4, 6.. 3u. 4w 3. w 4. v 3w 5. u v 6. 6 3v w 7. u v w 8. v 3 w u In Exercses 9, determne whether S s a bass for C n. 9. S,,,. S,,,. S,,,,,,,,. S,,,,,,,, In Exercses 3 6, express v as a lnear combnaton of each of the followng bass vectors. (a),,,,,,,, (b),,,,,,,, 3. v,, 4. v,, 3 5. v,, 6. v,, In Exercses 7 4, determne the Eucldean norm of v. 7. v, 8. v, 9. v 36,. v 3, 3. v,,. v,, 3. v,, 3, 4. v,,, 4 In Exercses 5 3, determne the Eucldean dstance between u and v. 5. u,, v, 6. u, 4,, v, 4, 7. u,, 3, v,, 8. u,,, v,, 9. u,, v, 3. u,,,, v,,, In Exercses 3 34, determne whether the set of vectors s lnearly ndependent or lnearly dependent. 3.,,, 3.,,,,,,,, 33.,,,,,,,, 34.,,,,,,,, In Exercses 35 38, determne whether the functon s a complex nner product, where u u, u and v v, v. 35. u, v u u v 36. u, v u v u v 37. u, v 4u v 6u v 38. u, v u v u v In Exercses 39 4, use the nner product u, v u v u v to fnd u, v. 39. u, and v, 4 4. u 3, and v, 4. u, and v 3, 3 4. u 4, 3 and v 3, 43. Let v,, and v,,. If v 3 z, z, z 3 and the set v s not a bass for C 3, v, v 3, what does ths mply about z, z, and z 3? 44. Let v,, and v,,. Determne a vector v 3 such that v s a bass for C 3, v, v 3. In Exercses 45 49, prove the property, where u, v, and w are vectors n C n and k s a complex number. u v w u w v w 45. 46. 47. u kv ku v 48. u u 49. u u f and only f u. 5. Wrtng Let u, v be a complex nner product and let k be a complex number. How are u, v and u, kv related? In Exercses 5 and 5, use the nner product u, v u v u v u v u v where u u u u u and v v v v v to fnd u, v. 5. u v 5. v u 3 ku v ku v
336_8_4.qxp 4/7/8 :3 AM Page 56 56 Chapter 8 Complex Vector Spaces In Exercses 53 and 54, determne the lnear transformaton T : C m C n that has the gven characterstcs. 53. T,,, T,, 54. T,,, T,, In Exercses 55 58, the lnear transformaton T : C m C n s shown by Tv Av. Fnd the mage of v and the premage of w. 55. A 56. A 57. 58. A A 59. Fnd the kernel of the lnear transformaton from Exercse 55. 6. Fnd the kernel of the lnear transformaton from Exercse 56. In Exercses 6 and 6, fnd the mage of v, for the ndcated composton, where and are the matrces below. T 6. T T 6. T T,, and v,, v 3,, T v, v 5, T T w w 3 w w 63. Determne whch of the sets below are subspaces of the vector space of complex matrces. (a) The set of symmetrc matrces. (b) The set of matrces A satsfyng A T A. (c) The set of matrces n whch all entres are real. (d) The set of dagonal matrces. 64. Determne whch of the sets below are subspaces of the vector space of complex-valued functons (see Example 4). (a) The set of all functons f satsfyng f. (b) The set of all functons f satsfyng f. (c) The set of all functons f satsfyng f f. True or False? In Exercses 65 and 66, determne whether each statement s true or false. If a statement s true, gve a reason or cte an approprate statement from the text. If a statement s false, provde an example that shows the statement s not true n all cases or cte an approprate statement from the text. 65. Usng the Eucldean nner product of u and v n C n, u v u v u v... u n v n. 66. The Eucldean form of u n C n denoted by u s u u.
336_8_5.qxp 4/7/8 :3 AM Page 57 Secton 8.5 Untary and Hermtan Matrces 57 8.5 Untary and Hermtan Matrces Problems nvolvng dagonalzaton of complex matrces and the assocated egenvalue problems requre the concepts of untary and Hermtan matrces. These matrces roughly correspond to orthogonal and symmetrc real matrces. In order to defne untary and Hermtan matrces, the concept of the conjugate transpose of a complex matrx must frst be ntroduced. Defnton of the Conjugate Transpose of a Complex Matrx The conjugate transpose of a complex matrx A, denoted by A*, s gven by A* A T where the entres of A are the complex conjugates of the correspondng entres of A. Note that f A s a matrx wth real entres, then A* A T. To fnd the conjugate transpose of a matrx, frst calculate the complex conjugate of each entry and then take the transpose of the matrx, as shown n the followng example. EXAMPLE Fndng the Conjugate Transpose of a Complex Matrx SOLUTION Determne A* for the matrx A 3 7 3 7 A A* A T 3 7 4. 4 3 7 4 4 Several propertes of the conjugate transpose of a matrx are lsted n the followng theorem. The proofs of these propertes are straghtforward and are left for you to supply n Exercses 55 58. THEOREM 8.8 Propertes of the Conjugate Transpose If A and B are complex matrces and k s a complex number, then the followng propertes are true.. A** A. A B* A* B* 3. ka* ka* 4. AB* B*A*
336_8_5.qxp 4/7/8 :3 AM Page 58 58 Chapter 8 Complex Vector Spaces Untary Matrces Recall that a real matrx A s orthogonal f and only f A A T. In the complex system, matrces havng the property that A A* are more useful, and such matrces are called untary. Defnton of Untary Matrx A complex matrx A s untary f A A*. EXAMPLE A Untary Matrx Show that the matrx A s untary. A SOLUTION Because AA* 4 4 4 I, you can conclude that A* A. So, A s a untary matrx. In Secton 7.3, you saw that a real matrx s orthogonal f and only f ts row (or column) vectors form an orthonormal set. For complex matrces, ths property characterzes matrces that are untary. Note that a set of vectors v, v,..., v m n C n (a complex Eucldean space) s called orthonormal f the statements below are true.. v,,,..., m. v v j, j The proof of the next theorem s smlar to the proof of Theorem 7.8 presented n Secton 7.3. THEOREM 8.9 Untary Matrces An n n complex matrx A s untary f and only f ts row (or column) vectors form an orthonormal set n C n.
336_8_5.qxp 4/7/8 :3 AM Page 59 Secton 8.5 Untary and Hermtan Matrces 59 EXAMPLE 3 The Row Vectors of a Untary Matrx Show that the complex matrx A s untary by showng that ts set of row vectors forms an orthonormal set n C 3. A 3 5 5 SOLUTION Let r, r, and be defned as follows. r 3, 5 r 3 5, 3 4 3, 5 5 The length of r 3 r,, r s r r r 3 3 5 3, 3 4 4 4. 3 4 3 5 The vectors r and r 3 can also be shown to be unt vectors. The nner product of r and r s r r 3 3 3 3 3 3. 3 3 3 3 Smlarly, r r 3 and r r 3. So, you can conclude that r, r, r 3 s an orthonormal set. Try showng that the column vectors of A also form an orthonormal set n C 3.
336_8_5.qxp 4/7/8 :3 AM Page 5 5 Chapter 8 Complex Vector Spaces Hermtan Matrces A real matrx s called symmetrc f t s equal to ts own transpose. In the complex system, the more useful type of matrx s one that s equal to ts own conjugate transpose. Such a matrx s called Hermtan after the French mathematcan Charles Hermte (8 9). Defnton of a Hermtan Matrx A square matrx A s Hermtan f A A*. As wth symmetrc matrces, you can easly recognze Hermtan matrces by nspecton. To see ths, consder the matrx A. A a a c c The conjugate transpose of A has the form A* A T a a b b a a b b If A s Hermtan, then A A*. So, you can conclude that A must be of the form A a b b b b d d c c d d c c d d. b b d. Smlar results can be obtaned for Hermtan matrces of order n n. In other words, a square matrx A s Hermtan f and only f the followng two condtons are met.. The entres on the man dagonal of A are real.. The entry a j n the th row and the jth column s the complex conjugate of the entry a j n the jth row and the th column. EXAMPLE 4 Hermtan Matrces Whch matrces are Hermtan? (a) 3 3 (b) 3 3 (c) (d) 3 3 3 3 4 3 4
336_8_5.qxp 4/7/8 :3 AM Page 5 Secton 8.5 Untary and Hermtan Matrces 5 SOLUTION (a) Ths matrx s not Hermtan because t has an magnary entry on ts man dagonal. (b) Ths matrx s symmetrc but not Hermtan because the entry n the frst row and second column s not the complex conjugate of the entry n the second row and frst column. (c) Ths matrx s Hermtan. (d) Ths matrx s Hermtan because all real symmetrc matrces are Hermtan. One of the most mportant characterstcs of Hermtan matrces s that ther egenvalues are real. Ths s formally stated n the next theorem. THEOREM 8. The Egenvalues of a Hermtan Matrx If A s a Hermtan matrx, then ts egenvalues are real numbers. PROOF Let be an egenvalue of A and let a b a v b. a n b n be ts correspondng egenvector. If both sdes of the equaton Av v are multpled by the row vector v*, then v*av v*v v*v a b a b... a n b n. Furthermore, because v*av* v*a*v** v*av, t follows that v* Av s a Hermtan matrx. Ths mples that v* Av s a real number, so s real. REMARK: Note that ths theorem mples that the egenvalues of a real symmetrc matrx are real, as stated n Theorem 7.7. To fnd the egenvalues of complex matrces, follow the same procedure as for real matrces.
336_8_5.qxp 4/7/8 :3 AM Page 5 5 Chapter 8 Complex Vector Spaces EXAMPLE 5 Fndng the Egenvalues of a Hermtan Matrx Fnd the egenvalues of the matrx A. 3 3 A 3 SOLUTION The characterstc polynomal of A s Ths mples that the egenvalues of A are, 6, and. To fnd the egenvectors of a complex matrx, use a procedure smlar to that used for a real matrx. For nstance, n Example 5, the egenvector correspondng to the egenvalue s obtaned by solvng the followng equaton. I A 3 3 4 3 3 3 3 3 3 3 3 3 6 5 9 3 3 9 9 3 3 6 6. Usng Gauss-Jordan elmnaton, or a computer software program or graphng utlty, obtan the egenvector correspondng to, whch s shown below. v 3 3 3 Egenvectors for 6 and 3 can be found n a smlar manner. They are and respectvely. 6 9 3 3 5, v v v 3 v v v 3 3 3
336_8_5.qxp 4/7/8 :3 AM Page 53 Secton 8.5 Untary and Hermtan Matrces 53 Technology Note Some computer software programs and graphng utltes have bult-n programs for fndng the egenvalues and correspondng egenvectors of complex matrces. For example, on the TI-86, the egvl key on the matrx math menu calculates the egenvalues of the matrx A, and the egvc key gves the correspondng egenvectors. Just as you saw n Secton 7.3 that real symmetrc matrces are orthogonally dagonalzable, you wll now see that Hermtan matrces are untarly dagonalzable. A square matrx A s untarly dagonalzable f there exsts a untary matrx P such that P AP s a dagonal matrx. Because P s untary, P P*, so an equvalent statement s that A s untarly dagonalzable f there exsts a untary matrx P such that P* AP s a dagonal matrx. The next theorem states that Hermtan matrces are untarly dagonalzable. THEOREM 8. Hermtan Matrces and Dagonalzaton If A s an n n Hermtan matrx, then. egenvectors correspondng to dstnct egenvalues are orthogonal.. A s untarly dagonalzable. PROOF To prove part, let and be two egenvectors correspondng to the dstnct (and real) egenvalues and. Because Av v and Av v, you have the equatons shown below for the matrx product Av * v. Av * *A* * * v v v v v v Av v * v Av * v * v * v v v v * v So, v *v v *v v v v *v v *v because, and ths shows that v and v are orthogonal. Part of Theorem 8. s often called the Spectral Theorem, and ts proof s left to you. EXAMPLE 6 The Egenvectors of a Hermtan Matrx The egenvectors of the Hermtan matrx shown n Example 5 are mutually orthogonal because the egenvalues are dstnct. You can verfy ths by calculatng the Eucldean nner products v v, v v 3, and v v 3. For example,
336_8_5.qxp 4/7/8 :3 AM Page 54 54 Chapter 8 Complex Vector Spaces v v 6 9 3 6 9 3 6 9 8 3. The other two nner products v v 3 and v v 3 can be shown to equal zero n a smlar manner. The three egenvectors n Example 6 are mutually orthogonal because they correspond to dstnct egenvalues of the Hermtan matrx A. Two or more egenvectors correspondng to the same egenvalue may not be orthogonal. Once any set of lnearly ndependent egenvectors s obtaned for an egenvalue, however, the Gram-Schmdt orthonormalzaton process can be used to fnd an orthogonal set. EXAMPLE 7 Dagonalzaton of a Hermtan Matrx Fnd a untary matrx P such that P* AP s a dagonal matrx where 3 3 A 3. SOLUTION The egenvectors of A are shown after Example 5. Form the matrx P by normalzng these three egenvectors and usng the results to create the columns of P. So, because v,, 5 7 v, 6 9, 3 44 7 69 78 v 3 3,, 5 5 5 4, the untary matrx P s obtaned. P 7 7 7 Try computng the product P* AP for the matrces A and P n Example 7 to see that you obtan * AP P 78 6 9 78 3 78 where, 6, and are the egenvalues of A. 6 3 4 4 5 4