Th Matrix Exponntial (with xrciss) by D. Klain Vrsion 207.0.05 Corrctions and commnts ar wlcom. Th Matrix Exponntial For ach n n complx matrix A, dfin th xponntial of A to b th matrix A A k I + A + k! A2 + 3! A3 + () k0 It is not difficult to show that this sum convrgs for all complx matrics A of any finit dimnsion. But w will not prov this hr. If A is a matrix t, thn A t, by th Maclaurin sris formula for th function y t. Mor gnrally, if D is a diagonal matrix having diagonal ntris d, d 2,..., d n, thn w hav D I + D + D2 + 0 0 d 0 0 0 0.... + 0 d 2 0. 0... + 0 0 0 0 0 d n d 0 0 0 d 2 0.... 0 0 0 dn d 2 0 0 d 0 2 0. +... 0 d 0 0 2 Th situation is mor complicatd for matrics that ar not diagonal. Howvr, if a matrix A happns to b diagonalizabl, thr is a simpl algorithm for computing A, a consqunc of th following lmma. Lmma Lt A and P b complx n n matrics, and suppos that P is invrtibl. Thn P AP P A P Proof: Rcall that, for all intgrs m 0, w hav (P AP ) m P A m P. Th dfinition () thn yilds P AP I + P AP + (P AP ) 2 + I + P A2 AP + P P + P (I + A + A2 + )P P A P
If a matrix A is diagonalizabl, thn thr xists an invrtibl P so that A P DP, whr D is a diagonal matrix of ignvalus of A, and P is a matrix having ignvctors of A as its columns. In this cas, A P D P. Exampl: Lt A dnot th matrix A 5 2 2 Th radr can asily vrify that 4 and 3 ar ignvalus of A, with corrsponding ignvctors w and w 2. It follows that 2 A P DP 2 4 0 0 3 2 so that A 2 4 0 0 3 2 2 4 3 4 3 2 3 2 4 2 3 4 Th dfinition () immdiatly rvals many othr familiar proprtis. Th following proposition is asy to prov from th dfinition () and is lft as an xrcis. Proposition 2 Lt A b a complx squar n n matrix.. If 0 dnots th zro matrix, thn 0 I, th idntity matrix. 2. A m A A A m for all intgrs m. 3. ( A ) T (AT ) 4. If AB BA thn A B B A and A B B A. Unfortunatly not all familiar proprtis of th scalar xponntial function y t carry ovr to th matrix xponntial. For xampl, w know from calculus that s+t s t whn s and t ar numbrs. Howvr this is oftn not tru for xponntials of matrics. In othr words, it is possibl to hav n n matrics A and B such that A+B A B. S, for xampl, Exrcis 0 at th nd of this sction. Exactly whn w hav quality, A+B A B, dpnds on spcific proprtis of th matrics A and B that will b xplord latr on. Manwhil, w can at last vrify th following limitd cas: Proposition 3 Lt A b a complx squar matrix, and lt s, t C. Thn A(s+t) As At. 2
Proof: From th dfinition () w hav As At ( I + As + A2 s 2 ) ( + I + At + A2 t 2 A j s j ( A k t k ) j! k! j0 k0 + ) A j+k s j t k j0 k0 j!k! ( ) Lt n j + k, so that k n j. It now follows from ( ) that As At j0 nj A n s j t n j j!(n j)! n0 A n n! n j0 n! j!(n j)! sj t n j A n (s + t) n n0 n! A(s+t) Stting s and t in Proposition 3, w find that A A A(+()) 0 I. In othr words, rgardlss of th matrix A, th xponntial matrix A is always invrtibl, and has invrs A. W can now prov a fundamntal thorm about matrix xponntials. Both th statmnt of this thorm and th mthod of its proof will b important for th study of diffrntial quations in th nxt sction. Thorm 4 Lt A b a complx squar matrix, and lt t b a ral scalar variabl. f(t) ta. Thn f (t) A ta. Lt Proof: Applying Proposition 3 to th limit dfinition of drivativ yilds f (t) lim h 0 A(t+h) At h Applying th dfinition () to Ah I thn givs us At (lim h 0 Ah ) I h f (t) (lim At Ah + A2 h 2 ) + At A A At. h 0 h Thorm 4 is th fundamntal tool for proving important facts about th matrix xponntial and its uss. Rcall, for xampl, that thr xist n n matrics A and B such that A B A+B. Th following thorm provids a condition for whn this idntity dos hold. 3
Thorm 5 Lt A, B b n n complx matrics. If AB BA thn A+B A B. Proof: If AB BA, it follows from th formula () that A Bt Bt A, and similarly for othr combinations of A, B, A + B, and thir xponntials. Lt g(t) (A+B)t Bt At, whr t is a ral (scalar) variabl. By Thorm 4, and th product rul for drivativs, g (t) (A + B) (A+B)t Bt At + (A+B)t ( B) Bt At + (A+B)t Bt ( A) At (A + B)g(t) Bg(t) Ag(t) 0. Hr 0 dnots th n n zro matrix. Not that it was only possibl to factor ( A) and ( B) out of th trms abov bcaus w ar assuming that AB BA. Sinc g (t) 0 for all t, it follows that g(t) is an n n matrix of constants, so g(t) C for som constant matrix C. In particular, stting t 0, w hav C g(0). But th dfinition of g(t) thn givs C g(0) (A+B)0 B0 A0 0 0 0 I, th idntity matrix. Hnc, I C g(t) (A+B)t Bt At for all t. Aftr multiplying by At Bt on both sids w hav At Bt (A+B)t. Exrciss:. If A 2 0, th zro matrix, prov that A I + A. 2. Us th dfinition () of th matrix xponntial to prov th basic proprtis listd in Proposition 2. (Do not us any of th thorms of th sction! Your proofs should us only th dfinition () and lmntary matrix algbra.) 3. Show that ci+a c A, for all numbrs c and all squar matrics A. 4. Suppos that A is a ral n n matrix and that A T A. Prov that A is an orthogonal matrix. 5. If A 2 A thn find a nic simpl formula for A, similar to th formula in th first xrcis abov. 6. Comput A for ach of th following xampls: 0 (a) A (b) A 0 0 0 (c) A a b 0 a 4
7. Comput A for ach of th following xampls: a b (a) A 0 0 (b) A a 0 b 0. If A 2 I, show that 2 A ( + ) ( I + ) A. 9. Suppos λ C and X C n is a non-zro vctor such that AX λx. Show that A X λ X. 0. Lt A and B dnot th matrics 0 A 0 0 B 0 0 0 Show by dirct computation that A+B A B.. Show that, if A is diagonalizabl, thn dt( A ) trac(a). Not: Latr it will b sn that this is tru for all squar matrics. 5
Slctd Answrs and Solutions 4. Sinc ( A ) T AT, whn A T A w hav ( A ) T A AT A A A A A 0 I 5. If A 2 A thn A I + ( )A. 6. (a) A 0 (b) A 0 (c) A a a b 0 a 7. (a) A a b a (a ) 0 (Rplac b a (a ) by in ach cas if a 0.) (b) A a 0 b a (a ) 6
Linar Systms of Ordinary Diffrntial Equations Suppos that y f(x) is a diffrntiabl function of a ral (scalar) variabl x, and that y ky, whr k is a (scalar) constant. In calculus this diffrntial quation is solvd by sparation of variabls: y y y k y dx k dx so that ln y kx + c, and y c kx, for som constant c R. Stting x 0 w find that y 0 f(0) c, and conclud that y y 0 kx. (2) Instad, lt us solv th sam diffrntial quation y ky in a slightly diffrnt way. Lt F (x) kx y. Diffrntiating both sids, w hav F (x) k kx y + kx y k kx y + kx ky 0, whr th scond idntity uss th assumption that y ky. Sinc F (x) 0 for all x, th function F (x) must b a constant, F (x) a, for som a R. Stting x 0, w find that a F (0) k0 y(0) y 0, whr w again lt y 0 dnot y(0). W conclud that y kx y 0 as bfor. Morovr, this mthod provs that (2) dscribs all solutions to y ky. Th scond point of viw will prov valuabl for solving a mor complicatd linar systm of ordinary diffrntial quations (ODEs). For xampl, suppos Y (t) is a diffrntiabl vctor-valud function: y (t) Y y 2 (t) satisfying th diffrntial quations y 5y + y 2 y 2 2y + 2y 2 3 and initial condition Y 0 Y (0). In othr words, whr A dnots th matrix Y (t) 5 2 2. 5 2 2 Y AY, To solv this systm of ODEs, st F (t) At Y, whr At is dfind using th matrix xponntial formula () of th prvious sction. Diffrntiating (using th product rul) and applying Thorm 4 thn yilds F (t) A At Y + At Y A At Y + At AY 0, 7
whr th scond idntity uss th assumption that Y AY. Sinc F (t) 0 (th zro vctor), for all t, th function F must b qual to a constant vctor v; that is, F (t) v for all t. Evaluating at t 0 givs v F (0) A0 Y (0) Y 0, whr w dnot th valu Y (0) by th symbol Y 0. In othr words, Y 0 v F (t) At Y, for all valus of t. Hnc, Y At Y 0 At 3, and th diffrntial quation is solvd! Assuming, of cours, that w hav a formula for At. 5 In th prvious sction w obsrvd that th ignvalus of th matrix A ar 2 2 4 and 3, with corrsponding ignvctors w and w 2. Thrfor, for 2 all scalar valus t, so that It follows that so that Y (t) At P DtP At P Dt P Y (t) At Y 0 At 3 y (t) y 2 (t) 2 4t 5 3t 2 4t + 0 3t 2 2 2 4t 4t 0 0 3t 4t 0 0 3t 2 2 4t 0 0 3t + 3t 5 0 2 2 2 2 4t. 3, 5 3t 2 Mor gnrally, if Y (t) AY (t), is a linar systm of ordinary diffrntial quations, thn th argumnts abov imply that Y At Y 0 If, in addition, w can diagonaliz A, so that λ 0 0 A P DP 0 λ 2 0 P.... 0 0 0 λ n P
thn λ t 0 0 At P Dt P 0 λ 2t 0 P.... P 0 0 0 λnt and Y (t) P Dt P Y 0. If th columns of P ar th ignvctors v,..., v n of A, whr ach Av i λ i v i, thn λ t 0 0 c Y (t) P Dt P 0 λ 2t 0 c 2 Y 0 v v 2 v n.... 0. 0 0 λnt whr Hnc, c c 2. P Y 0. (3) c n c Y (t) λt v λ2t v 2 λnt c 2 v n. c n Ths argumnts ar summarizd as follows. c λ t v + c 2 λ 2t v 2 + + c n λnt v n. Thorm 6 Suppos that Y (t) : R R n (or C n ) is a diffrntiabl function of t such that Y (t) AY (t), and initial valu Y (0) Y 0. whr A is a diagonalizabl matrix, having ignvalus λ,..., λ n and corrsponding ignvctors v,..., v n. Thn Y (t) c λ t v + c 2 λ 2t v 2 + + c n λnt v n. (4) If P is th matrix having columns v,..., v n thn th constants c i ar givn by th idntity (3). If on is givn a diffrnt initial valu of Y, say Y (t 0 ) at tim t 0, thn th quation (4) still holds, whr c c 2. Dt 0 P Y (t 0 ). c n For xrciss on diffrntial quations, plas consult th txtbook. c n 9