Th Matrix Exponntial (with xrciss) by Dan Klain Vrsion 28928 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 + k 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 + d + d 2 + d n d 2 d 2 + d d 2 d n 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, 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 PDP, 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 d 2
2 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 PDP 4 2 2 3 so that A 2 4 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 dnots th zro matrix, thn 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 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 Proof From th dfinition () w hav As At (I + As + A2 s 2 ( j A j s j j! ) ( k ) + (I + At + A2 t 2 ) A k t k k! ) + j k A j+k s j t k j!k! ( )
3 Lt n j + k, so that k n j It now follows from ( ) that As At j nj A n s j t n j j!(n j)! A n n n! n j n! j!(n j)! sj t n j A n (s + t) n n n! A(s+t) Stting s and t in Proposition 3, w find that A A A(+()) 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 Lt f (t) ta Thn f (t) A ta Proof Applying Proposition 3 to th limit dfinition of drivativ yilds f (t) A(t+h) At lim At (lim Ah ) I h h h h Applying th dfinition () to Ah I thn givs us f (t) (lim At Ah + A2 h 2 ) + At A A At h 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 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) Hr 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) 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, w hav C g() But th dfinition of g(t) thn givs C g() (A+B) B A I,
4 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, 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 (i Prov that, if B A, thn B T B I) 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: (a) A (b) A (c) A a b a 7 Comput A for ach of th following xampls: a b (a) A (b) A a b 8 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 Lt A and B dnot th matrics A Show by dirct computation that A+B A B B Th trac of a squar n n matrix A is dfind to b th sum of its diagonal ntris: trac(a) a + a 22 + + a nn 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 I 5 If A 2 A thn A I + ( )A 6 (a) A (b) A (c) A a a b a 7 (a) A a b a (a ) (Rplac b a (a ) by in ach cas if a ) (b) A a 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 w find that y f () c, and conclud that (2) y y kx 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, whr th scond idntity uss th assumption that y ky Sinc F (x) for all x, th function F(x) must b a constant, F(x) a, for som a R Stting x, w find that a F() k y() y, whr w again lt y dnot y() W conclud that y kx y 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 Y() In othr words, 8 Y 5 (t) Y AY, 2 2 whr A dnots th matrix 5 2 2 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, whr th scond idntity uss th assumption that Y AY Sinc F (t) (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 givs v F() A Y() Y, whr w dnot th valu Y() by th symbol Y In othr words, Y v F(t) At Y,
7 for all valus of t Hnc, Y At Y At 3 8, 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 2 2 ar 4 and 3, with corrsponding ignvctors w and w 2 Thrfor, 2 for all scalar valus t, At PDtP so that It follows that so that Y(t) At P Dt P Y(t) At Y At 3 8 y (t) y 2 (t) Mor gnrally, if 2 4t 5 3t 2 4t + 3t 2 2 2 4t 4t 3t 4t 3t 2 2 Y (t) AY(t), 4t 3t + 3t 5 2 2 2 2 4t 3 8, 5 3t is a linar systm of ordinary diffrntial quations, thn th argumnts abov imply that Y At Y If, in addition, w can diagonaliz A, so that λ A PDP λ 2 P λ n thn P λ t At P Dt P λ 2t P λ nt P and Y(t) P Dt P Y If th columns of P ar th ignvctors v,, v n of A, whr ach Av i λ i v i, thn λ t c Y(t) P Dt P Y v v 2 v n λ 2t c 2 λ nt c n 2
8 whr (3) Hnc, c c n c 2 P Y c Y(t) λt v λ2t v 2 λnt v n c 2 c λ t v + c 2 λ 2t v 2 + + c n λ nt v n Ths argumnts ar summarizd as follows 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() Y whr A is a diagonalizabl matrix, having ignvalus λ,, λ n and corrsponding ignvctors v,, v n Thn (4) Y(t) c λ t v + c 2 λ 2t v 2 + + c n λ nt v n 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 ) at tim t, thn th quation (4) still holds, whr c c 2 Dt P Y(t ) For xrciss on diffrntial quations, plas consult th txtbook c n c n