Notes on the Matrix Exponential and Logarithm. Howard E. Haber

Transcription

1 Notes on the Matrix Exponential an Logarithm Howar E. Haber Santa Cruz Institute for Particle Physics University of California, Santa Cruz, CA 9564, USA February, 28 Abstract In these notes, we summarize some of the most important properties of the matrix exponential an the matrix logarithm. Nearly all of the results of these notes are well known an many are treate in textbooks on Lie groups. A few avance textbooks on matrix algebra also cover some of the topics of these notes. Some of the results concerning the matrix logarithm are less well known. These inclue a series expansion representation of lna(t)/ (where A(t) is a matrix that epens on a parameter t), which is erive here but oes not seem to appear explicitly in the mathematics literature. Properties of the Matrix Exponential Let A be a real or complex n n matrix. The exponential of A is efine via its Taylor series, e A = I + n= A n n!, () where I is the n n ientity matrix. The raius of convergence of the above series is infinite. Consequently, eq. () converges for all matrices A. In these notes, we iscuss a number of key results involving the matrix exponential an provie proofs of three important theorems. First, we consier some elementary properties. Property : If A, B ] AB BA =, then e A+B = e A e B = e B e A. (2) This result can be prove irectly from the efinition of the matrix exponential given by eq. (). The etails are left to the ambitious reaer. Remarkably, the converse of property is FALSE. One counterexample is sufficient. Consier the 2 2 complex matrices A = An elementary calculation yiels ( 2πi ), B = ( ). (3) 2πi e A = e B = e A+B = I, (4)

2 where I is the 2 2 ientity matrix. Hence, eq. (2) is satisfie. Nevertheless, it is a simple matter to check that AB BA, i.e., A, B]. Inee, one can use the above counterexample to construct a secon counterexample that employs only real matrices. Here, we make use of the well known isomorphism between the complex numbers an real 2 2 matrices, which is given by the mapping ( ) a b z = a+ib. (5) b a It is straightforwar to check that this isomorphism respects the multiplication law of two complex numbers. Using eq. (5), we can replace each complex number in eq. (3) with the corresponing real 2 2 matrix, A = 2π 2π, B = 2π 2π One can again check that eq. (4) is satisfie, where I is now the 4 4 ientity matrix, whereas AB BA as before. It turns out that a small moification of Property is sufficient to avoi any such counterexamples. Property 2: If e t(a+b) = e ta e tb = e tb e ta, where t (a,b) (where a < b) lies within some open interval of the real line, then it follows that A, B] =. Property 3: If S is a non-singular matrix, then for any matrix A,. exp { SAS } = Se A S. (6) The above result can beerive simply by making use of the Taylor series efinition cf. eq. ()] for the matrix exponential. Property 4: For all complex n n matrices A, ( lim I + A m = e m m) A. Property 4 can be verifie by employing the matrix logarithm, which is treate in Sections 4 an 5 of these notes. Property 5: If A(t), A/] =, then ea(t) = e A(t)A(t) 2 = A(t) e A(t).

3 This result is self evient since it replicates the well known result for orinary (commuting) functions. Note that Theorem 2 below generalizes this result in the case of A(t), A/]. Property 6: If A, A, B] ] =, then e A Be A = B +A, B]. To prove this result, we efine an compute B(t) B(t) e ta Be ta, = Ae ta Be ta e ta Be ta A = A, B(t)], 2 B(t) 2 = A 2 e ta Be ta 2Ae ta Be ta A+e ta Be ta A 2 = A, A, B(t)] ]. By assumption, A, A, B] ] =, which must also be true if one replaces A ta for any number t. Hence, it follows that A, A, B(t)] ] =, an we can conclue that 2 B(t)/ 2 =. It then follows that B(t) is a linear function of t, which can be written as ( ) B(t) B(t) = B()+t. Noting that B() = B an (B(t)/) t= = A, B], we en up with t= e ta Be ta = B +ta, B]. (7) By setting t =, we arrive at the esire result. If the ouble commutator oes not vanish, then one obtains a more general result, which is presente in Theorem below. If A, B ], thee A e B e A+B. Thegeneralresult iscallehebaker-campbell-hausorff formula, which will be prove in Theorem 4 below. Here, we shall prove a somewhat simpler version, Property 7: If A, A, B] ] = B, A, B] ] =, then To prove eq. (8), we efine a function, e A e B = exp { A+B + 2 A, B]}. (8) F(t) = e ta e tb. We shall now erive a ifferential equation for F(t). Taking the erivative of F(t) with respect to t yiels F = AetA e tb +e ta e tb B = AF(t)+e ta Be ta F(t) = { A+B +ta, B] } F(t), (9) 3

4 after noting that B commutes with e Bt an employing eq. (7). By assumption, both A an B, an hence their sum, commutes with A, B]. Thus, in light of Property 4 above, it follows that the solution to eq. (9) is F(t) = exp { t(a+b)+ 2 t2 A, B] } F(). Setting t =, we ientify F() = I, where I is the ientity matrix. Finally, setting t = yiels eq. (8). Property 8: For any matrix A, etexpa = exp { TrA }. () If A is iagonalizable, then one can use Property 3, where S is chosen to iagonalize A. In this case, D = SAS = iag(λ, λ 2,..., λ n ), where the λ i are the eigenvalues of A (allowing for egeneracies among the eigenvalues if present). It then follows that ete A = i e λ i = e λ +λ λ n = exp { TrA }. However, not all matrices are iagonalizable. One can moify the above erivation by employing the Joran canonical form. But, here I prefer another technique that is applicable to all matrices whether or not they are iagonalizable. The iea is to efine a function f(t) = ete At, an then erive a ifferential equation for f(t). If δt/t, then ete A(t+δt) = et(e At e Aδt ) = ete At ete Aδt = ete At et(i +Aδt), () after expaning out e Aδt to linear orer in δt. We now consier +A δt A 2 δt... A n δt A 2 δt +A 22 δt... A 2n δt et(i +Aδt) = et A n δt A n2 δt... +A nn δ = (+A δt)(+a 22 δt) (+A nn δt)+o ( (δt) 2) = +δt(a +A A nn )+O ( (δt) 2) = +δt TrA+O ( (δt) 2). Inserting this result back into eq. () yiels ete A(t+δt) ete At δt = TrA ete At +O(δt). 4

5 Taking the limit as δt yiels the ifferential equation, The solution to this equation is eteat = TrA ete At. (2) ln ete At = t TrA, (3) where the constant of integration has been etermine by noting that (ete At ) t= = eti =. Exponentiating eq. (3), we en up with ete At = exp { t TrA }. Finally, setting t = yiels eq. (). Note that this last erivation hols for any matrix A (incluing matrices that are singular an/or are not iagonalizable). Remark: For any invertible matrix function A(t), Jacobi s formula is ( eta(t) = eta(t) Tr A (t) A(t) ). (4) Note that for A(t) = e At, eq. (4) reuces to eq. (2) erive above. Another result relate to eq. (4) is ( ) et(a+tb) = eta Tr(A B). t= 2 Theorems Involving the Matrix Exponential In this section, we state an prove four important theorems concerning the matrix exponential. The proofsoftheorems, 2 an4canbefoun insection 5. of Ref. ]. The proofof Theorem 3 is base on results given in section 6.5 of Ref. 2], where the author also notes that eq. (34) has been attribute variously to Duhamel, Dyson, Feynman an Schwinger. Theorem 3 is also quote in eq. (5.75) of Ref. 3], although the proof of this result is relegate to an exercise. Finally, we note that a number of aitional results involving the matrix exponential that make use of parameter ifferentiation techniques (similar to ones employe in this Section) with applications in mathematical physics problems have been treate in Ref. 4]. The ajoint operator a A, which is a linear operator acting on the vector space of n n matrices, is efine by a A (B) = A,B] AB BA. (5) Note that involves n neste commutators. (a A ) n (B) = A, A,A,B]] ] }{{} n (6) 5

6 Theorem : e A Be A = exp(a A )(B) n= n! (a A) n (B) = B +A,B]+ A,A,B]]+. (7) 2 Proof: Define B(t) e ta Be ta, (8) an compute the Taylor series of B(t) aroun the point t =. A simple computation yiels B() = B an B(t) = Ae ta Be ta e ta Be ta A = A,B(t)] = a A (B(t)). (9) Higher erivatives can also be compute. It is a simple exercise to show that: n B(t) n = (a A ) n (B(t)). (2) Theorem then follows by substituting t = in the resulting Taylor series expansion of B(t). We now introuce two auxiliary functions that are efine by their power series: These functions satisfy: f(z) = ez z = n= g(z) = lnz z = n= z n, z <, (2) (n+)! ( z) n n+, z <. (22) f(lnz)g(z) =, for z <, (23) f(z)g(e z ) =, for z <. (24) Theorem 2: e A(t) e A(t) = f(a A ) ( ) A = A A, A ] A, A, A ]] +, (25) 2! 3! where f(z) is efine via its Taylor series in eq. (2). Note that in general, A(t) oes not commute with A/. A simple example, A(t) = A+tB where A an B are inepenent of t an A,B], illustrates this point. In the special case where A(t),A/] =, eq. (25) reuces to Proof: Define e A(t) e A(t) = A, if A(t), A ] =. (26) B(s,t) e sa(t) e sa(t), (27) 6

7 an compute the Taylor series of B(s,t) aroun the point s =. It is straightforwar to verify that B(,t) = an ( ) n B(s,t) A s n = (a A(t) ) n, (28) s= for all positive integers n. Assembling the Taylor series for B(s,t) an inserting s = then yiels Theorem 2. Note that if A(t),A/] =, then ( n B(s,t)/s n ) s= = for all n 2, an we recover the result of eq. (26). There are two aitional forms of Theorem 2, which we now state for completeness. Theorem 2(a): where f(z) is efine via its Taylor series, ea(t) = e A(t) f(aa ) ( ) A, (29) f(z) = e z z = n= ( ) n (n+)! zn, z <. (3) Eq. (29) is an immeiate consequence of eq. (25) since, e A(t) ea(t) = A A, A ] + A, A, A ]] = 2! 3! f(a A ) ( ) A. Theorem 2(b): ( ) ea+tb = e A f(aa )(B), (3) t= where f(z) is efine via its Taylor series in eq. (3). Eq. (3) efines that Gâteau erivative of e A (also calle the irectional erivative of e A along the irection of B). The relation between Theorems 2(a) an 2(b) can be seen more clearly as follows. Denoting the right han sie of eq. (3) by F(A,B) an employing the efinition of the erivative, ( ) ( ea+tb = lim ǫ t= ) e A+(t+ǫ)B e A+tB ǫ t= = lim ǫ e A+ǫB e A ǫ Working consistently to first orer in ǫ an employing eq. (32) in the final step, e A(t+ǫ) e A(t) eat = lim ǫ ǫ = lim ǫ e A(t)+ǫA (t) e A(t) ǫ where A (t) A/. That is, eqs. (29) an (3) are equivalent. = F(A,B). (32) = F ( A(t),A (t) ), (33) In the present application, the Gâteau erivative exists, is a linear function of B, an is continuous in A, in which case it coincies with the Fréchet erivative8]. 7

8 Theorem 3: e A(t) = This integral representation is an alternative version of Theorem 2. Proof: Consier sa A e e ( s)a s. (34) ( e sa e ( s)b) = Ae sa e ( s)b +e sa e ( s)b B s = e sa (B A)e ( s)b. (35) Integrate eq. (35) from s = to s =. Using eq. (35), it follows that: ( e sa e ( s)b) = e sa e ( s)b = e A e B. (36) s e A e B = se sa (B A)e ( s)b. (37) In eq. (37), we can replace B A+hB, where h is an infinitesimal quantity: Taking the limit as h, e A e (A+hB) = h lim h h se sa Be ( s)(a+hb). (38) e (A+hB) e A] = se sa Be ( s)a. (39) Finally, we note that the efinition of the erivative can be use to write: Using e A(t+h) e A(t) e A(t) = lim. (4) h h A(t+h) = A(t)+h A +O(h2 ), (4) it follows that: ( exp A(t)+h A )] exp A(t)] e A(t) = lim. (42) h h Thus, we can use the result of eq. (39) with B = A/ to obtain e A(t) = which is the result quote in Theorem 3. sa A e e ( s)a s, (43) 8

9 As in the case of Theorem 2, there are two aitional forms of Theorem 3, which we now state for completeness. Theorem 3(a): ea(t) = ( s)a A e esa s. (44) This follows immeiately from eq. (34) by taking A A an s s. In light of eqs. (32) an (33), it follows that, Theorem 3(b): ( ) ea+tb = t= e ( s)a Be sa s. (45) Secon proof of Theorems 2 an 2(b): One can now erive Theorem 2 irectly from Theorem 3. First, we multiply eq. (34) by e A(t) an change the integration variable, s s. Then, we employ eq. (7) to obtain, e A(t) e A(t) = = = n= sa A e e sa s = exp sa A ] ( A (n+)! (a A) n expa sa ] ) s = n= ( ) A = f(a A ) ( ) A s n! (a A) n ( ) A s n s which coincies with Theorem 2. Likewise, starting from eq. (45) an making use of eq. (7), it follows that, ( ) t= e A+tB = e A e sa Be sa s = e A s exp ] a sa (B) = e A s exp ] sa A (B) = e A = e A n= which coincies with Theorem 2(b). ( ) A, (46) ( ) n (a A ) n (B) n! n= s n s ( ) n (n+)! (a A) n (B) = e A f(aa )(B), (47) 9

10 Theorem 4: The Baker-Campbell-Hausorff (BCH) formula ln ( e A e B) = B + gexp(ta A )exp(a B )](A), (48) where g(z) is efine via its Taylor series in eq. (22). Since g(z) is onlyefine for z <, it follows that the BCH formula for ln ( e A e B) converges provie that e A e B I <, where I is the ientity matrix an is a suitably efine matrix norm. Expaning the BCH formula, using the Taylor series efinition of g(z), yiels: e A e B = exp ( A+B + 2 A,B]+ 2 A,A,B]]+ 2 B,B,A]]+...), (49) assuming that the resulting series is convergent. An example where the BCH series oes not converge occurs for the following elements of SL(2,R): ( ) ( )] ( )] e λ M = e λ = exp λ exp π, (5) where λ is any nonzero real number. It is easy to prove 2 that no matrix C exists such that M = expc. Nevertheless, the BCH formula is guarantee to converge in a neighborhoo of the ientity of any Lie group. Proof of the BCH formula: Define or equivalently, C(t) = ln(e ta e B ). (5) Using Theorem, it follows that for any complex n n matrix H, e C(t) = e ta e B. (52) exp a C(t) ] (H) = e C(t) He C(t) = e ta e B He ta e B = e ta exp(a B )(H)]e ta 2 The characteristic equation for any 2 2 matrix A is given by = exp(a ta )exp(a B )(H). (53) λ 2 (Tr A)λ+et A =. Hence, the eigenvalues of any 2 2 traceless matrix A sl(2,r) that is, A is an element of the Lie algebra of SL(2,R)] are given by λ ± = ±( et A) /2. Then, { Tr e A 2cosh et A /2, if et A, = exp(λ + )+exp(λ ) = 2cos et A /2, if et A >. Thus, if et A, then Tr e A 2, an if et A >, then 2 Tr e A < 2. It followsthat for any A sl(2,r), Tr e A 2. For the matrix M efine in eq. (5), Tr M = 2coshλ < 2 for any nonzero real λ. Hence, no matrix C exists such that M = expc. Further etails can be foun in section 3.4 of Ref. 5] an in section.5(b) of Ref. 6].

11 Hence, the following operator equation is vali: exp a C(t) ] = exp(taa )exp(a B ), (54) after noting that exp(a ta ) = exp(ta A ). Next, we use Theorem 2 to write: e C(t) ( ) C e C(t) = f(a C(t) ). (55) However, we can compute the left-han sie of eq. (55) irectly: e C(t) e C(t) = e ta e B e B e ta = e ta e ta = A, (56) since B is inepenent of t, an ta commutes with (ta). Combining the results of eqs. (55) an (56), ( ) C A = f(a C(t) ). (57) Multiplying both sies of eq. (57) by g(expa C(t) ) an using eq. (24) yiels: C = g(expa C(t))(A). (58) Employing the operator equation, eq. (54), one may rewrite eq. (58) as: C = g(exp(ta A)exp(a B ))(A), (59) which is a ifferential equation for C(t). Integrating from t = to t = T, one easily solves for C. The en result is C(T) = B + T g(exp(ta A )exp(a B ))(A), (6) where the constant of integration, B, has been obtaine by setting T =. Finally, setting T = in eq. (6) yiels the BCH formula. Finally, we shall use eq. (48) to obtain the terms exhibite in eq. (49). In light of the series efinition of g(z) given in eq. (22), we nee to compute an I exp(ta A )exp(a B ) = I (I +ta A + 2 t2 a 2 A)(I +a B + 2 a2 B) = a B ta A ta A a B 2 a2 B 2 t2 a 2 A, (6) I exp(taa )exp(a B ) ] 2 = a 2 B +ta A a B +ta B a A +t 2 a 2 A, (62) after ropping cubic terms an higher. Hence, using eq. (22), g(exp(ta A )exp(a B )) = I 2 a B 2 ta A 6 ta Aa B + 3 ta Ba A + 2 a2 B + 2 t2 a 2 A. (63)

12 Noting that a A (A) = A,A] =, it follows that to cubic orer, B + g(exp(ta A )exp(a B ))(A) = B +A B,A] ] ] 2 2 A,B,A] + 2 B,B,A] which confirms the result of eq. (49). Theorem 5: The Zassenhaus formula = A+B + 2 A,B]+ 2 A,A,B] ] + 2 B,B,A] ], The Zassenhaus formula for matrix exponentials is sometimes referre to as the ual of the Baker-Campbell Hausorff formula. It provies an expression for exp(a + B) as an infinite prouce of matrix exponentials. It is convenient to insert a parameter t into the argument of the exponential. Then, the Zassenhaus formula is given by 7], exp { t(a+b) } = e ta e tb exp { 2 t2 A,B ]} exp { 6 t3( 2 B,A,B] ] + A,A,B] ])}, (65) where the exponents of higher orer in t involve neste commutators. A technique for eriving the expansions exhibite in eqs. (49) an (65) can be foun in Refs. 4]. 3 Properties of the Matrix Logarithm The matrix logarithm shoul be an inverse function to the matrix exponential. However, in light of the fact that the complex logarithm is a multi-value function, the concept of the matrix logarithm is not as straightforwar as was the case of the matrix exponential. Let A be a complex n n matrix with no real negative or zero eigenvalues. Then, there is a unique logarithm, enote by lna, all of whose eigenvalues lie in the strip, π < Imz < π of the complex z-plane. We refer to lna as the principal logarithm of A, which is efine on the cut complex plane, where the cut runs from the origin along the negative real axis. If A is a real matrix (subject to the conitions just state), then its principal logarithm is real. 3 For an n n complex matrix A, we can efine lna via its Taylor series expansion, uner the assumption that the series converges. The matrix logarithms is then efine as, lna = (64) ( ) m+(a I)m, (66) m m= whenever the series converges, where I is the n n ientity matrix. The series converges whenever A I <, where inicates a suitable matrix norm. 4 If the matrix A satisfies (A I) m = for all integers m > N (where N is some fixe positive integer), then 3 For further etails, see Sections.5.7 of Ref. 8]. 4 One possible choice is the Hilbert-Schmi norm, which is efine as X = Tr(X X) ] /2, where the positive square root is chosen. 2

13 A I is calle nilpotent an A is calle unipotent. If A is unipotent, then the series given by eq. (66) terminates, an lna is well efine inepenently of the value of A I. For later use, we also note that if A I <, then I A is non-singular, an (I A) can be expresse as an infinite geometric series, (I A) = A m. (67) One can also efine the matrix logarithm by employing the Gregory series, 5 lna = 2 m= m= ] (I A)(I +A) 2m+, (68) 2m+ which converges uner the assumption that all eigenvalues of A possess a positive real part. In particular, eq. (68) converges for any Hermitian positive efinite matrix A. Hence, the region of convergence of the series in eq. (68) is consierably larger than the corresponing region of convergence of eq. (66). Before iscussing a number of key results involving the matrix logarithm, we first list some elementary properties without proofs. Many of the proofs can be foun in Chapter 2.3 of Ref. 9]. A number of properties of the matrix logarithm not treate in Ref. 9] are iscusse in Ref. 8]. Property : For all A with A I <, exp(lna) = A. Property 2: For all A with A < ln2, ln(e A ) = A. Note that although A < ln2 implies that e A I <, the converse is not necessarily true. This means that it is possible that ln(e A ) A espite the fact that the series that efines ln(e A ) via eq. (66) converges. For example, if A = 2πiI, then e A = e 2πi I = I an e A I =, whereas A = 2π > ln2. In this case, ln(e A ) = A. A slightly stronger version of property 2 quote in Ref. 8] states that for any n n complex matrix, ln(e A ) = A if an only if Imλ i < π for every eigenvalue λ i of A. One can exten the efinition of the matrix logarithm given in eq. (66) by aopting the following integral efinition given in Chapter of Ref. 8] an previously erive in Ref. ]. If A is a complex n n matrix with no real negative or zero eigenvalues, 6 then lna = (A I) s(a I)+I ] s. (69) A erivation of eq. (69) can be foun on pp of Ref. ]. It is straightforwar to check that if A I <, then one can expan the integran of eq. (69) in a Taylor series in s cf. eq. (67)]. Integrating over s term by term then yiels eq. (66). Of course, eq. (69) applies to a much broaer class of matrices, A. 5 See, e.g. Section.3 of Ref. 8]. 6 The absence of zero eigenvalues implies that A is an invertible matrix. 3

14 Property 3: Employing the extene efinition of the matrix logarithm given in eq. (69), if A is a complex n n matrix with no real negative or zero eigenvalues, then exp(lna) = A. To prove Property 3, we follow the suggestion of Problem 9 on pp of Ref. 2] an efine a matrix value function f of a complex variable z, f(z) = z(a I) sz(a I)+I ] s. Itisstraightforwaroshowthatf(z)isanalyticinacomplexneighborhoooftherealinterval between z = an z =. In a neighborhoo of the origin, one can verify by expaning in z an ropping terms of O(z 2 ) that expf(z) = I +z(a I). (7) Using the analyticity of f(z), we can insert z = in eq. (7) to conclue that exp(lna) = expf() = A. Property 4: If A is a complex n n matrix with no real negative or zero eigenvalues an p, then ln(a p ) = plna. In particular, ln(a ) = lna an ln(a /2 ) = 2 lna. Property 5: If A(t) is a complex n n matrix with no real negative or zero eigenvalues that epens on a parameter t, an A commutes with A/, then lna(t) = A A = A A. Property 6: If A is a complex n n matrix with no real negative or zero eigenvalues an S is a non-singular matrix, then ln(sas ) = S(lnA)S. (7) Property 7:Suppose that X any arecomplex n n complex matrices such that XY = YX. Moreover, if argλ j +argµ j < π, for every eigenvalue λ j of X an the corresponing eigenvalue µ j of Y, then ln(xy) = lnx +lny. Note that if X an Y o not commute, then the corresponing formula for ln(xy) is quite complicate. Inee, if the matrices X an Y are sufficiently close to I, so that exp(lnx) = X an ln(e X ) = X (an similarly for Y), then we can apply eq. (49) with A = lnx an B = lny to obtain, ln(xy) = lnx +lny + 2 lnx,lny ] +. 4

15 4 Theorems involving the Matrix Logarithm Before consiering the theorems of interest, we prove the following lemma. Lemma: If B is a non-singular matrix that epens on a parameter t, then B (t) = B B B. (72) Proof: eq. (72) is easily erive by taking the erivative of the equation B B = I. It follows that = ( ) (B B) = B B +B B. (73) Multiplying on the right of eq. (73) by B yiels B +B B B =, which immeiately yiels eq. (72). A secon form of eq. (72) employs the Gâteau (or equivalently the Fréchet) erivative. In light of eqs. (32) an (33) it follows that, ( ) (A+tB) = A BA. t= Theorem 6: lna(t) = s sa+( s)i ] A ] sa+( s)i. (74) We provie here two ifferent proofs of Theorem 6. This first proof was erive by my own analysis, although I expect that others must have prouce a similar erivation. The secon proof was inspire by a memo by Stephen L. Aler, which is given in Ref. 3]. Proof : Employing the integral representation of lna given in eq. (69), it follows that A lna = s(a I)+I ] s+(a I) ] s(a I)+I s. (75) We now make use of eq. (72) to evaluate the integran of the secon integral on the right han sie of eq. (75), which yiels A lna = s(a I)+I ] s (A I) s(a I)+I ] s A s(a I)+I] (76) 5

16 We can rewrite eq. (76) as follows, lna = which simplifies to ] ] A ] s(a I)+I s(a I)+I s(a I)+I s lna = s(a I)s(A I)+I ] A s(a I)+I], (77) Thus, we have establishe eq. (74). ] A ] s(a I)+I s(a I)+I s. Proof 2: Start with the following formula, { (A+uI ) ln(a+b) lna = u ( A+B +ui ) }, (78) Using the efinition of the erivative, ln(a(t+h) lna(t) ln A(t)+hA/+O(h 2 ) ] lna(t) lna(t) = lim = lim h h h h Denoting B = ha/ an making use of eq. (78), lna(t) = lim h h u { (A+uI ) ( A+hA/+uI ) }, (79) For infinitesimal h, we have ( A+hA/+uI ) = (A+uI)(I +h(a+ui) A/) ] Inserting this result into eq. (79) yiels lna(t) = = (I +h(a+ui) A/) (A+uI) = (I h(a+ui) A/)(A+uI) +O(h 2 ) = (A+uI) h(a+ui) A/(A+uI) +O(h 2 ). (8) u(a+ui) A (A+uI). (8) Finally, if we change variables using u = ( s)/s, it follows that lna(t) = which is the result quote in eq. (74). s sa+( s)i ] A ] sa+( s)i. (82) 6.

17 A secon form of Theorem 6 employs the Gâteau (or equivalently the Fréchet) erivative. In light of eqs. (32) an (33) it follows that, Theorem 6(a): ( ) ln(a+tb) = t= Eq. (83) was obtaine previously in eq. (3.3) of Ref. 4]. Theorem 7: 7 A(t) lna(t) = n= n+ (A a A ) n s sa+( s)i ] B sa+( s)i ]. (83) ( ) A = A + 2 A A, A ]+ 3 A 2 A, A, A ]] + (84) Proof: A matrix inverse has the following integral representation, B = e sb s, (85) if the eigenvalues of B lie in the region, Rez >, of the complex z-plane. If we perform a formal ifferentiation of eq. (85) with respect to B, it follows that B n = x m e sb s. (86) n! Thus, starting with eq. (8), we shall employ eq. (85) to write, Inserting eq. (87) into eq. (8) yiels, lna(t) = = = w (A+uI) = u v w v e v(a+ui) v. (87) we v(a+ui)a e w(a+ui) we (v+w)a e waa e wa e (v+w)u u v v +w e (v+w)a e waa e wa. (88) Let us change integration variables by replacing v with x = v + w, an then interchange the orer of integration, lna(t) = x w x e xa e waa x x e wa = x e xa we waa e wa. (89) 7 I have not seen Theorem 7 anywhere in the literature, although it is ifficult to believe that such an expression has never been erive elsewhere. 7

18 We can now employ the result of Theorem cf. eq. (7)] to obtain e waa e wa = Inserting this result into eq. (89), we obtain, lna(t) = = n= n= n= w n n! (a A) n x n! x e xa (a A ) n n! Finally, using eq. (86), we en up with lna(t) = If we expan out the series, we fin A(t) ( ) A. ( ) A x w n w { } ( ) A x n e xa x (a A ) n. (9) n+ n= lna(t) = A + 2 A n+ A n (a A ) n Note that if A,A/] =, then eq. (9) reuces to, 8 ( ) A. (9) A, A ]+ 3 A 2 A, A, A ]] + lna(t) = A A = A A, (92) which coincies with Property 5 given in the previous section. One can rewrite eq. (9) in a more compact form by efining the function, h(x) x ln( x) = n= x n n+. (93) It then follows that 9 A(t) ( ) lna(t) = h( ) A A a A. (94) 8 In performing the last step in eq. (92), we note that if A,A/] =, then A = A A A = A A A. Then, multiplying the above equation on the right by A justifies the final step in eq. (92). 9 In obtaining eq. (94), we mae use of the fact that A commutes with aa. In more etail, A aa(b) aa(a B) = A (AB BA) (AA B A BA) =. 8

19 A secon form of Theorem 7 employs the Gâteau (or equivalently the Fréchet) erivative. In light of eqs. (32) an (33) it follows that, Theorem 7a: ( ) ln(a+bt) = A h ( ) A a A (B), (95) t= where the function h is efine by its Taylor series given in eq. (93). In particular, ( ) ln(a+bt) = A B + 2 A 2 A,B]+ 3 A 3 A,A,B]]+ References t= ] Willar Miller Jr., Symmetry Groups an Their Applications (Acaemic Press, New York, 972). 2] Rajenra Bhatia, Positive Definite Matrices (Princeton University Press, Princeton, NJ, 27). 3] Weak Interactions an Moern Particle Theory, by Howar Georgi (Dover Publications, Mineola, NY, 29). 4] R.M. Wilcox, J. Math. Phys. 8, 962 (967). 5] Anrew Baker, Matrix Groups: An Introuction to Lie Group Theory, by Anrew Baker (Springer-Verlag, Lonon, UK, 22). 6] J.F. Cornwell, Group Theory in Physics, Volume 2 (Acaemic Press, Lonon, UK, 984). 7] F. Casas, A. Muruab, an M. Nainic, Comput. Phys. Commun. 83, 2386 (22). 8] Nicholas J. Higham, Functions of Matrices (Society for Inustrial an Applie Mathematics (SIAM), Philaelphia, PA, 28. 9] Brian Hall, Lie Groups, Lie Algebras, an Representations (Secon Eition), (Springer International Publishing, Cham, Switzerlan, 25). ] A. Wouk, J. Math. Anal. an Appl., 3 (965). ] Willi-Hans Stieb, Problems an Solutions in Introuctory an Avance Matrix Calculus (Worl Scientific Publishing Company, Singapore, 26). 2] Jacques Faraut, Analysis on Lie Groups (Cambrige University Press, Cambrige, UK, 28). 3] Stephen L. Aler, Taylor Expansion an Derivative Formulas for Matrix Logarithms, Memos. 4] L. Dieci, B. Morini an A. Papini, Siam J. Matrix Anal. Appl. 7, 57 (996). 9