1 6 CHAPTER 8 COMPLEX VECTOR SPACES 5. Fnd the kernel of the lnear transformaton gven n Exercse 5. In Exercses 55 and 56, fnd the mage of v, for the ndcated composton, where and are gven by the followng matrces. A T and T A 55. T T 56. T T 57. Determne whch of the followng sets 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. 58. Determne whch of the followng sets are subspaces of the vector space of complex-valued functons (see Example ). (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. 8.5 UNITARY AND HERMITIAN MATRICES Problems nvolvng dagonalzaton of complex matrces, and the assocated egenvalue problems, requre the concept of untary and Hermtan matrces. These matrces roughly correspond to orthogonal and symmetrc real matrces. In order to defne untary and Hermtan matrces, we frst ntroduce the concept of the conjugate transpose of a complex matrx. 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, we 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 Determne A* for the matrx A 7.
2 SECTION 8.5 UNITARY AND HERMITIAN MATRICES 6 Theorem 8.8 Propertes of Conjugate Transpose Soluton A 7 7 [ ] [ ] A* A T 7 [ ] We lst several propertes of the conjugate transpose of a matrx n the followng theorem. The proofs of these propertes are straghtforward and are left for you to supply n Exercses 9 5. 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*. (ka)* = ka*. (AB)* = B*A* 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 we call such matrces untary. Defnton of a Untary Matrx A complex matrx A s called untary f A A*. EXAMPLE A Untary Matrx Show that the followng matrx s untary. A Soluton Snce AA* I we conclude that A* A. Therefore, A s a untary matrx.
3 6 CHAPTER 8 COMPLEX VECTOR SPACES In Secton 7., we showed 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 we call a set of vectors v, v,..., v m n C n (complex Eucldean space) orthonormal f the followng are true... v,,,..., m v v j, j The proof of the followng theorem s smlar to the proof of Theorem 7.8 gven n Secton 7.. 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. EXAMPLE The Row Vectors of a Untary Matrx Show that the followng complex matrx s untary by showng that ts set of row vectors form an orthonormal set n C. A Soluton We let r, r, and be defned as follows. r,, r, r 5 5, 5, 5 The length of 5 5 r s r r r r 5,. 5
4 SECTION 8.5 UNITARY AND HERMITIAN MATRICES 65 The vectors r and r can also be shown to be unt vectors. The nner product of r and s gven by r r. Smlarly, r r and r r and we can conclude that r, r, r s an orthonormal set. (Try showng that the column vectors of A also form an orthonormal set n C.) r 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. We call such a matrx 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, we can easly recognze Hermtan matrces by nspecton. To see ths, consder the matrx. A a a c c The conjugate transpose of A has the form A* A T A* A* a a b b a a b b If A s Hermtan, then A A* and we 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.
5 66 CHAPTER 8 COMPLEX VECTOR SPACES 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 th column. EXAMPLE Hermtan Matrces Whch of the followng matrces are Hermtan? (a) (c) (b) (d) Soluton (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 a b a v b. a n b n be ts correspondng egenvector. If we multply both sdes of the equaton Av v by the row vector v*, we obtan v* Av v* v v* v a b a b Furthermore, snce v* Av* v*a*(v*)* v*av a n b n.
6 SECTION 8.5 UNITARY AND HERMITIAN MATRICES 67 t follows that v*av s a Hermtan matrx. Ths mples that v*av s a real number, and we may conclude that 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, we follow the same procedure as for real matrces. EXAMPLE 5 Fndng the Egenvalues of a Hermtan Matrx Fnd the egenvalues of the followng matrx. A Soluton The characterstc polynomal of A s I A whch mples that the egenvalues of A are, 6, and. To fnd the egenvectors of a complex matrx, we use a smlar procedure 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. v v v v v v
7 68 CHAPTER 8 COMPLEX VECTOR SPACES Usng Gauss-Jordan elmnaton, or a computer or calculator, we obtan the followng egenvector correspondng to. v Egenvectors for 6 and can be found n a smlar manner. They are 6 9 6; 5 TECHNOLOGY NOTE Some computers and calculators have bult-n programs for fndng the egenvalues and correspondng egenvectors of complex matrces. For example, on the TI-85, 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 we saw n Secton 7. that real symmetrc matrces were orthogonally dagonalzable, we wll show now 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. Snce 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 tells us 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. v v Proof To prove part, let and be two egenvectors correspondng to the dstnct (and real) egenvalues and. Because Av v and Av v, we have the followng equatons 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 Therefore, v * v v * v v * v v * v snce, and we have shown that v and v are orthogonal. Part of Theorem 8. s often called the Spectral Theorem, and ts proof s omtted.
8 SECTION 8.5 UNITARY AND HERMITIAN MATRICES 69 EXAMPLE 6 The Egenvectors of a Hermtan Matrx The egenvectors of the Hermtan matrx gven n Example 5 are mutually orthogonal because the egenvalues are dstnct. We can verfy ths by calculatng the Eucldean nner products v v, v v and v v. For example, v v The other two nner products v v and v v 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 egenvector may not be orthogonal. However, once we obtan any set of lnearly ndependent egenvectors for a gven egenvalue, we can use the Gram-Schmdt orthonormalzaton process to obtan an orthogonal set. EXAMPLE 7 Dagonalzaton of a Hermtan Matrx Fnd a untary matrx P such that P*AP s a dagonal matrx where A. Soluton The egenvectors of A are gven after Example 5. We form the matrx P by normalzng these three egenvectors and usng the results to create the columns of P. Thus, snce v,, 5 7 v, 6 9, v,, we obtan the untary matrx P, 7 78 P =
9 7 CHAPTER 8 COMPLEX VECTOR SPACES Try computng the product P*AP for the matrces A and P n Example 7 to see that you obtan * AP P 6 where, 6, and are the egenvalues of A. We have seen that Hermtan matrces are untarly dagonalzable. However, t turns out that there s a larger class of matrces, called normal matrces, whch are also untarly dagonalzable. A square complex matrx A s normal f t commutes wth ts conjugate transpose: AA* = A*A. The man theorem of normal matrces says that a complex matrx A s normal f and only f t s untarly dagonalzable. You are asked to explore normal matrces further n Exercse 59. The propertes of complex matrces descrbed n ths secton are comparable to the propertes of real matrces dscussed n Chapter 7. The followng summary ndcates the correspondence between untary and Hermtan complex matrces when compared wth orthogonal and symmetrc real matrces. Comparson of Hermtan and Symmetrc Matrces A s a symmetrc matrx (Real) A s a Hermtan matrx (Complex). Egenvalues of A are real.. Egenvalues of A are real.. Egenvectors correspondng to. Egenvectors correspondng to dstnct egenvalues are dstnct egenvalues are orthogonal. orthogonal.. There exsts an orthogonal. There exsts a untary matrx matrx P such that P such that P T AP s dagonal. P*AP s dagonal.
10 SECTION 8.5 EXERCISES 7 SECTION 8.5 EXERCISES In Exercses 8, determne the conjugate transpose of the gven matrx.. A. A. A. 5. A A A 5 6 In Exercses 9, explan why the gven matrx s not untary. 9. A. A. A. A 7. A 8. A In Exercses 8, determne whether A s untary by calculatng AA*.. A. 5. A I n 6. A A A 7 5 A 6. A 6 6 In Exercses 8, determne whether the matrx A s Hermtan.. A A A 7. A 8. A In Exercses 9. (a) verfy that A s untary by showng that ts rows are orthonormal, and (b) determne the nverse of A A. A 5 5. A [ ] 5 6 A
11 7 CHAPTER 8 COMPLEX VECTOR SPACES In Exercses 9, determne the egenvalues of the matrx A. 9. A.. A.. A. In Exercses 5 8, determne the egenvectors of the gven matrx. 5. The matrx n Exercse The matrx n Exercse. 7. The matrx n Exercse. 8. The matrx n Exercse. In Exercses 9, fnd a untary matrx P that dagonalzes the gven matrx A. 9. A.. A. A A. A 6 A A A. Let z be a complex number wth modulus. Show that the followng matrx s untary. A z z z z [ ] In Exercses 5 8, use the result of Exercse to determne a, b, and c so that A s untary A A a b c 5 b A a A a 6 b c 5 b c In Exercses 9 5, prove the gven formula, where A and B are n n complex matrces. 9. A A 5. A B A B 5. ka ka 5. AB BA 5. Let A be a matrx such that A A O. Prove that A s Hermtan. 5. Show that deta deta, where A s a matrx. In Exercses 55 56, assume that the result of Exercse 5 s true for matrces of any sze. 55. Show that deta deta. 56. Prove that f A s untary, then deta. 57. (a) Prove that every Hermtan matrx A can be wrtten as the sum A B C, where B s a real symmetrc matrx and C s real and skew-symmetrc. (b) Use part (a) to wrte the matrx A as a sum A B C, where B s a real symmetrc matrx and C s real and skew-symmetrc. (c) Prove that every n n complex matrx A can be wrtten as A B C, where B and C are Hermtan. (d) Use part (c) to wrte the complex matrx A a c as a sum A B C, where B and C are Hermtan.
12 CHAPTER 8 REVIEW EXERCISES Determne whch of the followng sets are subspaces of the vector space of n n complex matrces. (a) The set of n n Hermtan matrces. (b) The set of n n untary matrces. (c) The set of n n normal matrces. 59. (a) Prove that every Hermtan matrx s normal. (b) Prove that every untary matrx s normal. (c) Fnd a matrx that s Hermtan, but not untary. (d) Fnd a matrx that s untary, but not Hermtan. (e) Fnd a matrx that s normal, but nether Hermtan nor untary. (f) Fnd the egenvalues and correspondng egenvectors of your matrx from part (e). (g) Show that the complex matrx s not dagonalzable. Is ths matrx normal? CHAPTER 8 REVIEW EXERCISES In Exercses 6, perform the gven operaton.. Fnd u z : u, z. Fnd u z : u, z 8. Fnd uz : u, z. Fnd uz : u, z u 5. Fnd : u 6, z z 6. Fnd u z : u 7, z In Exercses 7, fnd all zeros of the gven polynomal. 7. x x 8 8. x x 7 9. x x. x x x In Exercses, perform the gven operaton usng A and B.. A B. B. deta B. BA In Exercses 5, perform the gven operaton usng, v, and z. 5. z 6. v 7. w 8. vz 9. wv. zw In Exercses, perform the ndcated operaton... w ( ( 5.. ( )( ) In Exercses 5 and 6, fnd A (f t exsts). 5. A In Exercses 7, determne the polar form of the complex number In Exercses, fnd the standard form of the gven complex number... A 5 5 cos 6 sn 6 cos 5 5 sn. 6 cos sn. In Exercses 5 8, perform the ndcated operaton. Leave the result n polar form. 5. cos sn cos 6 sn 7 cos sn 6
13 7 CHAPTER 8 COMPLEX VECTOR SPACES [cos() sn ()] 8. 7[cos() sn ()] In Exercses 9, fnd the ndcated power of the gven number and express the result n polar form sn In Exercses 6, express the gven roots n standard form.. Square roots: 5 cos sn. Cube roots: 7 cos 6 sn 5. Cube roots: 6. Fourth roots: 6 cos sn In Exercses 7 and 8, determne the conjugate transpose of the gven matrx. 7. cos 9[cos() sn ()] 6[cos() sn ()] cos sn A 5 8. A 6 7 In Exercses 9 5, fnd the ndcated vector usng u,, v,, and w,. 9. 7u v 5. w v 5. u v w 5. u w In Exercses 5 and 5, determne the Eucldean norm of the gven vector. 5. v 5, 5. v, 5, In Exercses 55 and 56, fnd the Eucldean dstance between the gven vectors. 55. v,, u, cos sn 6 5 cos sn 56. v,,, u,, In Exercses 57 6, determne whether the gven matrx s untary In Exercses 6 and 6, determne whether the gven matrx s Hermtan In Exercses 6 and 6, fnd the egenvalues and correspondng egenvectors of the gven matrx Prove that f A s an nvertble matrx, then A* s also nvertble. 66. Determne all complex numbers z such that z z 67. Prove that f the product of two complex numbers s zero, then one of the numbers must be zero. 68. (a) Fnd the determnant of the followng Hermtan matrx. (b) Prove that the determnant of any Hermtan matrx s real. 69. Let A and B be Hermtan matrces. Prove that AB BA f and only f AB s Hermtan. 7. Let u be a unt vector n C n. Defne H I uu. Prove that H s an n n Hermtan and untary matrx.
14 CHAPTER 8 PROJECTS Use mathematcal nducton to prove DeMovre s Theorem. 7. Prove that f z s a zero of a polynomal equaton wth real coeffcents, then the conjugate of z s also a zero. 7. Show that f z z and z z are both nonzero real numbers, then z and z are both real numbers. 7. Prove that f z and w are complex numbers, then z w z w. 75. Prove that for all vectors u and v n a complex nner product space, [ u, v u v u v u v ] u v. CHAPTER 8 PROJECTS Populaton Growth and Dynamcal Systems - II In the projects for Chapter 7, you were asked to model the populaton of two speces usng a system of dfferental equatons of the form y t ay t by t y t cy t dy t. The constants a, b, c, and d depend on the partcular speces beng studed. In Chapter 7, we looked at an example of a predator prey relatonshp, n whch a.5, b.6, c., and d.. Suppose we now consder a slghtly dfferent model. y t.6y t.8y t, y 6 y t.8y t.6y t, y. Use the dagonalzaton technque to fnd the general solutons y t and y t at any tme t >. Although the egenvalues and egenvectors of the matrx.6.8 A.8.6 are complex, the same prncples apply, and you can obtan complex exponental solutons.. Convert your complex solutons to real solutons by observng that f s a (complex) egenvalue of A wth (complex) egenvector v, then the real and magnary parts of et v form a lnearly ndependent par of (real) solutons. You wll need to use the formula e cos sn.. Use the ntal condtons to fnd the explct form of the (real) solutons to the orgnal equatons.. If you have access to a computer or graphng calculator, graph the solutons obtaned n part () over the doman t. At what moment are the two populatons equal? 5. Interpret the soluton n terms of the long-term populaton trend for the two speces. Does one speces ultmately dsappear? Why or why not? Contrast ths soluton to that obtaned for the model n Chapter If you have access to a computer or graphng calculator that can numercally solve dfferental equatons, use t to graph the solutons to the orgnal system of equatons. Does ths numercal approxmaton appear to be accurate? a b