Commutators Involving Matrix Functions

Transcription

1 Eletroni Journal of Linear Algera Volume 33 Volume 33: Speial Issue for the International Conferene on Matrix Analysis and its Appliations MAT TRIAD 017 Artile Commutators Involving Matrix untions Osman KAN PhD student Süleyman Solak Nemettin Erakan University ollow this and additional works at: Part of the Algera Commons and the Other Mathematis Commons Reommended Citation KAN Osman and Solak Süleyman (018 "Commutators Involving Matrix untions" Eletroni Journal of Linear Algera Volume 33 pp DOI: This Artile is rought to you for free and open aess y Wyoming Sholars Repository It has een aepted for inlusion in Eletroni Journal of Linear Algera y an authorized editor of Wyoming Sholars Repository or more information please ontat sholom@uwyoedu

2 COMMUTATORS INVOLVING MATRIX UNCTIONS OSMAN KAN AND SÜLEYMAN SOLAK Astrat Some results are otained for matrix ommutators involving matrix exponentials ( e A B ] e A e B] and their norms Key words Matrix ommutators Matrix exponential Norm AMS sujet lassifiations 15A60 15A45 15A16 1 Introdution The ommutator of two matries A and B is defined as A B] = AB BA and plays an important role in many ranhes of mathematis mathematial physis quantum physis and quantum hemistry Most of the studies on matrix ommutators have een foused on norm inequalities in reent years 1 3 4] Bötther and Wenzel otained a nie inequality for A B M n (C (11 A B] A where stands for the roenius norm 1] Matrix funtions are used in many areas of linear algera and arise in numerous appliations in siene and engineering Square root polynomial trigonometri funtions exponential and logarithm of matries are widely used in matrix theory The matrix exponential is y far most studied matrix funtion There are many various ways in the literature to ompute it One of them is as follows: (1 exp(a = I + A + A! + + As + = for A M n (C7] Our motivation in the present study is to otain some results onerning relations etween f(a B] f(a f(b] and A B] where f(x stands for e x and to otain inequalities for the roenius norms of matrix ommutators (11 suh as s=0 A s f(a B] A and f(a f(b] k A Let us give some easy properties of matrix ommutators and matrix exponential efore we present our main results or A B M n (C we have A + B C] = A C] + B C] AB C] = A B C] + A C] B Reeived y the editors on Deemer Aepted for puliation on Novemer Handling Editor: Natalia Beiano Corresponding Author: Osman Kan Karatay Belediyesi Religious Voational Seondary Shool 4100 Karatay KonyaTURKEY (matrixinequality@gmailom Mathematis Department Nemettin Erakan University 4090Meram Konya TURKEY (ssolak4@yahooom

3 Osman Kan Süleyman Solak 114 A BC] = A B] C + B A C] A s B] = sa s 1 A B] where A A B]] = 0 e 0 = I where 0 denotes zero matrix e A e B = e A+B if AB = BA e A e B = e B e A if AB = BA e P 1 AP = P 1 e A P where P is invertile Main Results In this setion we will present our results in two susetionsthe results onerning the relations etween e A B ] e A e B] and A B] will e presented in Setion 1 and the results related to norm inequalities will e presented in Setion 1 Properties of Commutators Involving Exponential Matries Let us start with an easyto-prove lemma whih we will use in main results Lemma 1 Let A e an aritrary n-square matrix Then Proof We have e A B ] = e A B Be A hand-side of the equation we get e A B ] = e A B ] = and this onludes the proof e A B ] = (I + A + A! + + As + (B + AB + A! B + + As B + A s B] If we replae e A y its Taylor expansion (1 in the right B B (I + A + A! + + As + (B + BA + B A! + + B As + e A B ] = AB BA + A! B B A! + + As B B As + e A B ] A s B] = A nie orollary an e otained for a speial ase of A Corollary Let A e an n-square matrix satisfying A A B]] = 0 Then e A B ] = e A A B] Proof In the previous lemma it was shown that e A B ] = A s B] = sa s 1 A B] when A A B]] = 0 Comining oth results we get e A B ] = e A B ] = sa s 1 A B] A s 1 A B] (s 1! A s B] On the other hand we know that

4 115 Matrix Commutators Involving Matrix untions e A B ] = e A A B] and this onludes the proof Theorem 3 Let A and B e omplex n-square matries Then e A e B] = st=1 A s B t ] t! Proof As e A e B] = e A e B e B e A replaing e A and e B y their Taylor expansions (1 in the right hand-side of the equation we get e A e B e B e A = (I + A + (I A! + + B + B + (I + B + B + (I + A + A!!! + e A e B e B e A = AB BA + A B BA + AB B A + A B B A +!!!! A e A e B e B e A B ] ] A B A B ] = A B] !!!! e A e B] A s B t ] = t! and this onludes the proof st=1 We an otain nie orollaries related to this theorem y using some speial A and B Corollary 4 Let A and B e n-square matries suh that A = 0 B = 0 Then e A e B] = A B] Proof This result is an immediate onsequene of Theorem 3 Beause A = 0 A n = 0 for n for any square matrix the Taylor expansion of e A is equal to I + A Thus e A e B] = e A e B e B e A e A e B] = (I + A(I + B (I + B(I + A e A e B] = AB BA = A B] Corollary 5 Let A and B e n-square matries suh that A = I B = I Then e A e B] = A B] (s 1!(t 1! Proof In this ase it is ovious that A n = A B n = B for odd n and A n = I B n = I for even n eause A = I and B = I So the proof an easily e onluded using the Taylor expansions of e A and e B Our next two results are related to e A e B] where A and B oth ommute with their ommutator There are some studies related to exponential of matries whih ommute with their ommutator in the literature One of them is as follows

5 Osman Kan Süleyman Solak 116 Lemma 6 6] If A B M n (C ommute with their ommutator then e A e B = e A+B+ 1 AB] Theorem 7 Let A and B e omplex n-square matries whih ommute with their ommutator then e A e B] ( A B] = e A+B sinh Proof Let A and B ommute with their ommutator then Sine A B]= B A] e B e A = e A+B 1 AB] e A e B = e A+B+ 1 AB] e A e B = e A+B+ 1 AB] e B e A = e A+B 1 AB] e A e B e B e A = e A+B+ 1 AB] e A+B 1 AB] It is lear that A + B ommutes with ± 1 A B] eause A and B ommute with their ommutator So e A+B± 1 AB] = e A+B e ± 1 AB] e A e B] = e A+B e 1 AB] e A+B e 1 AB] Sine e 1 AB] e 1 AB] an e written as sinh e A e B] = e A+B ( e 1 AB] e 1 AB] ( AB] in view of sinh(x = ex e x e A e B] = e A+B sinh ( AB] Theorem 8 Let the omplex n-square matries A B ommute with their ommutator Then e A e B] = s 1 st=1 r=0 A s r 1 B t 1 A r (t 1! A B] Proof We have A s B t] = A s 1 A B t] + A s 1 B t] A = A s 1 A B t] + ( A s A B t] + A s B t] A A = A s 1 A B t] A 0 + A s A B t] A + A s B t] A If we ontinue reduing the power of A in the ommutator we get A s B t] s 1 = A s r 1 A B t] A r st=1 r=0 On the other hand we know that A B t ] = t A B] B t 1 when B A B]] = 0 So A s B t] s 1 = ta s r 1 B t 1 A r A B] st=1 r=0

6 117 Matrix Commutators Involving Matrix untions eause of A and B ommute with their ommutator Applying this equality to e A e B] we get e A e B] = st=1 A s B t ] t! This onludes the proof e A e B] = s 1 st=1 r=0 A s r 1 B t 1 A r (t 1! A B] Norm Inequalities In this setion we will present our results related to the norm inequalities of A B M (C suh as f(a B] A and f(a f(b] k A in the ases where their eigenvalues are real and omplex numers respetively Lemma 9 Putzer s Spetral ormula for matries If the matrix A has real distint eigenvalues λ 1 and λ then e At = e λ1t I + eλ1t e λt λ 1 λ (A λ 1 I If the matrix A has one doule real eigenvalue λ 1 then e At = e λ1t I + te λ1t (A λ 1 I If the matrix A has omplex eigenvalues λ 1 =λ =a + i > 0 then e At = e at os(ti + eat sin(t (A ai Theorem 10 Let the matrix A have real distint eigenvalues λ 1 and λ Then the following norm inequality holds where = ( e λ 1 e λ λ 1 λ e A B ] A Proof Let the matrix A have real distint eigenvalues Then e At B ] = (e λ1t I + eλ1t e λt (A λ 1 I B B (e λ1t I + eλ1t e λt (A λ 1 I λ 1 λ λ 1 λ e At B ] = e λ1t B + eλ1t e λt λ 1 λ (A λ 1 I B (e λ1t B + eλ1t e λt B (A λ 1 I λ 1 λ e At B ] = eλ1t e λt ((A λ 1 I B B (A λ 1 I λ 1 λ e At B ] = eλ1t e λt λ 1 λ A B] e At B ] = e λ1t e λt λ 1 λ A B]

7 Osman Kan Süleyman Solak 118 If we omine this with (11 we get e At B ] e λ1t e λt λ 1 λ A The proof an e onluded y setting t = 1 e A B ] e λ1 e λ λ 1 λ A Theorem 11 Let the matrix A have a doule real eigenvalue λ inequality holds where = e λ e A B ] A Then the following norm Proof We have e At B ] = ( e λt I + te λt (A λi B B ( e λt I + te λt (A λi e At B ] = ( e λt B + te λt (A λi B ( e λ1t B + te λt (A λi e At B ] = te λt ((A λi B B (A λi If we omine this with (11 we get e At B ] = te λt A B] e At B ] = e λt t A B] e At B ] e λt t A The proof an e onluded y setting t = 1 e A B ] e }{{ λ A } Theorem 1 Let A B M (C and A have omplex eigenvalues λ 1 =λ =a + i > 0 Then the following norm inequality holds where = e a sin( e A B ] A Proof Let the matrix A have omplex eigenvalues λ 1 =λ =a + i > 0 Then e At B ] ( ( = e at os(ti + eat sin(t (A ai B B e at os(ti + eat sin(t (A ai e At B ] = e at os(tb + eat sin(t e At B ] = eat sin(t (A ai B Be at os(t eat sin(t B (A ai (A ai B] = eat sin(t A B]

8 119 Matrix Commutators Involving Matrix untions e At B ] = e at sin(t A B] If we omine this with (11 we get e At B ] e at sin(t A The proof an e onluded y setting t = 1 e A B ] e a sin( A Theorem 13 Let the matries A and B have real distint eigenvalues Then the following norm inequality holds e A e B] k A where k = ( ( e λ 1 e λ e µ 1 e µ λ 1 λ µ 1 µ λ s and µ s are eigenvalues of A and B respetively Proof We have e At e Bs = (e λ1t I + eλ1t e λt (A λ 1 I (e µ1s I + eµ1s e µs (B µ 1 I λ 1 λ µ 1 µ Let eλ 1 t e λ s λ 1 λ = x and eµ 1s e µ s µ 1 µ = y where x and y are real numers Then we get e At e Bs = e λ1t+µ1s + e λ1t y (B µ 1 I + e µ1s x (A λ 1 I + xy (A λ 1 I (B µ 1 I e Bs e At = e λ1t+µ1s + e µ1s x (A λ 1 I + e λ1t y (B µ 1 I + xy (B µ 1 I (A λ 1 I e At e Bs] = xy ((A λ 1 I (B µ 1 I (B µ 1 I (A λ 1 I If we omine this with (11 we get e At e Bs] = xy (A λ 1 I (B µ 1 I] = xy A B] e At e Bs] = xy A B] e At e Bs] xy A e At e Bs] ( e λ 1t e λt ( e µ 1s e µs A λ 1 λ µ 1 µ k The proof an e onluded y setting s t = 1 e A e B] ( e λ 1 ( e λ e µ 1 e µ A λ 1 λ µ 1 µ k

9 Osman Kan Süleyman Solak 10 Theorem 14 Let the matries A and B have real doule eigenvalues λ and µ respetively Then the following norm inequality holds where k = e λ+µ Proof Aording to the hypothesis we have e A e B] k A e At e Bs = ( e λt I + te λt (A λi (e µs I + se µs (B µi e At e Bs = e λt+µs I + se λt+µs (B µi + te λt+µs (A λi + ste λt+µs (A λi (B µi e Bs e At = e µs+λt I + te µs+λt (A λi + se µs+λt+ (B µi + ste µs+λt (B µi (A λi e At e Bs] = e At e Bs e Bs e At = ste λt+µs ((A λi (B µi (B µi (A λi e At e Bs] = ste λt+µs (A λi (B µi] If we omine this with (11 we get e At e Bs] = ste λt+µs A B] e At e Bs] = st e λt+µs A B] e At e Bs] e λt+µs st A k The proof follows y setting s t = 1 e A e B] e }{{ λ+µ A } k Theorem 15 Let the matries A and B have omplex eigenvalues λ 1 =λ =a + i > 0 and µ 1 =µ = + id d > 0 respetively Then the following norm inequality holds where k = e a+ sin( sin(d d e A e B] k A Proof Expressing e At and e Bs using Putzer s Spetral ormula for matries with omplex eigenvalues and then omputing e At e Bs e Bs e At as in the previous theorems the result follows Aknowledgment The authors are grateful to the editors of ELA for their useful omments REERENCES 1] A Botther and D Wenzel The roenius norm and the ommutator Linear Algera and Its Appliations 49: ] L Laszlo Proof of Bötther and Wenzels onjeture on ommutator norms for 3-y-3 matries Linear Algera and Its Appliations 4: ] SW Vong and XQ Jin Proof of Botther and Wenzels onjeture Operators and Matries :

10 11 Matrix Commutators Involving Matrix untions 4] Y-D Wu and X-Q Liu A Short Note on the roenius Norm of the Commutator Mathematial Notes 87: ] Niholas J Higham untions of Matries Manhester Institute for Mathematial Sienes Shool of Mathematis e-print ISSN ] Niholas J Higham untions of Matries: Theory and Computation Soiety for Industrial and Applied Mathematis 008 7] R Horn and C Johnson Matrix Analysis Camridge University Press ] Grnt B Gustafson Applied Differential Equations and Linear Algera Univ of Utah Salt Lake City USA Leture Notes retrieved 9 Novemer 018 from gustafso/s019/80/hapter11-systems/