A Note On The Exponential Of A Matrix Whose Elements Are All 1

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1 Applied Mathematics E-Notes, 8(208), c ISSN Available free at mirror sites of ame/ A Note O The Expoetial Of A Matrix Whose Elemets Are All Reza Farhadia Received 30 Jauary 208 Abstract Let J deote a matrix whose elemets are all It is well-kow that e J exp(j ) I + e J, where I is the idetity matrix of order The aim of our work is to establish a geeralizatio of this equality To prove the mai results, we will use the wellkow Helmert matrix Itroductio I liear algebra ad matrix theory there are may special ad importat matrices For example, the expoetial of a matrix is oe of them The expoetial of a complex square matrix of order such as A [a ij ] is defied as: e A exp(a ) z0 A z z! I + A + A2 2! + A3 3! + A4 4! +, () where I is the idetity matrix of order elemets are all It is well-kow that Let J deote a matrix whose e J I + e J (2) Sice J z z J for z, 2,, if oe puts J i equatio (), the proof of (2) is the immediate Now, i this paper it is show that exp (k) (J ) exp (exp ( exp (J ))) apply the expoetial of J i k times Clearly exp (k+) (J ) exp(exp (k) (J )) (3) Mathematics Subject Classificatios: 5A6, 5B0 Departmet of Statistics, Loresta Uiversity, Khorramabad, Ira 92

2 R Farhadia 93 We will prove the followig equatio: exp (k+) (J ) k ei + expk+ e where k e deotes the power tower of order k, amely () k e J, (4) e e k e e (5) k ad exp k e() deotes the iterated expoetial that is defied as: exp k e() e e e (6) Obviously, the special case of iterated expoetial is the power tower of order k, ie, k e exp k e() I subsequet sectios, the followig otatio will be used: a) R deotes the set of all real umbers b) Diag[d d 2 d ] deotes a diagoal matrix with diagoal etries d, d 2,, d c) A deotes the ivers of a matrix A d) A T deotes the traspose of a matrix A 2 Prelimiaries 2 Similarity ad Diagoalizatio I liear algebra two square matrices of order such as A ad B are said to be similar, deoted A B, if there exists a ivertible matrix T, such that T A T B (7) The matrix T is called the similarity trasformatio matrix Similar matrices have the same set of eigevalues [] Hece, it is importat if a matrix is similar to a diagoal matrix, sice the eigevalues of a diagoal matrix are its diagoal elemets I particular, we have the followig defiitio: DEFINITION ([7, page 3]) A matrix is said to be diagoalizable if it is similar to a diagoal matrix Hece, if a real square matrix A is diagoalizable, the A is similar to a diagoal matrix D Diag[d d 2 d ], ad there exists a ivertible matrix T such that T A T D or equivaletly A T D T which is the diagoalized form of A

3 94 A Note o the Expoetial of A Matrix The diagoalized form of a matrix is useful for some matrix calculatios Cosider the followig propositio which gives us a simple method to compute the expoetial of a diagoalizable matrix: PROPOSITION ([4, Propositio 23]) Let A is a diagoalizable matrix with A T D T, where D Diag[d d 2 d ] The e A T e D T, where e D Diag[e d e d2 e d ] Notice that ot all matrices ca be diagoalizable [] However, we kow that ay symmetric matrix is diagoalizable (see [, page 255]) Sice the matrix J that is the subject of the paper is symmetric, therefore the materials of this part, though brief, are very useful to prove the mai results i the ext sectio 22 The Helmert Matrix A Helmert matrix of order is a square matrix that was itroduced by H O Lacaster i 965 [5] The Helmert matrix of order is defied as: H ( ) ( ) ( ) ( ) ( ) ( ) Moreover, the first row of the Helmert matrix of order, has the followig form Ad the other i-th rows 2 i are formed by (8) [ ] (9) items [ i(i ) i(i ) i(i ) i items (i ) 0 0 i(i ) ] (0) i items Furthermore, we kow that the Helmert matrix is orthogoal [2]: H H T H T H I () I other words, H H T The Helmert matrix is usually used i statistics for the aalysis of variace (ANOVA), see [2, 6] Recetly, i 207, R Farhadia ad N Asadia showed that the Helmert matrix ca be used i stochastic processes [3]

4 R Farhadia 95 3 Mai results First, let us cosider the followig lemma: LEMMA Let H be the Helmert matrix of order ad β R The β β β βj H T β β PROOF First recall (8) (0) Hece J H T By multiplyig the above equatio by β the proof is the complete THEOREM Let H be the Helmert matrix of order ad D Diag[ + β ], ( ) ( ) ( ) ( ) ( ) ( ) where, β R The H T D H I + βj (2) PROOF Startig from the left side of (2), we advace the proof: H T D

5 96 A Note o the Expoetial of A Matrix β β +β β ( ) ( ) ( ) ( ) ( ) ( ) (+) (+) (+) (+) β β β β H T + β β Usig Lemma i (3), we obtai ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) + β β β β + β β (3) H T D H T + βj H T (4) Sice the Helmert matrix is orthogoal (see ()), by multiplyig both sides of equatio (4) by the Helmert matrix from the right, we have I I {}}{ H T D H (H T + βj H T )H H T H +βj H T H I + βj The theorem is proved

6 R Farhadia 97 COROLLARY Let H be the Helmert matrix of order The with similarity trasformatio matrix H J Diag[ 0 0], PROOF Usig Theorem with 0 ad β ad equatio (7), the proof is immediate COROLLARY 2 The matrix e J is a diagoalizable matrix as e J H T D H, where H deotes the Helmert matrix of order ad D Diag[e ] PROOF Recall equatio (2) of e J The the corollary is a immediate cosequece of Theorem (with ad β e ), equatio (7) ad Defiitio Now, i the ext theorem we shall prove (4) THEOREM 2 exp (k+) (J ) k ei + expk+ e () k e J PROOF By Corollary 2, we kow that e 0 0 exp(j ) H T H (3), (7), Defiitio, Propositio, Corollary 2 ad Theorem give e e 0 0 exp (2) (J ) exp(exp(j )) H T 0 e e H e ei + ee J (5) (3), (7), Defiitio, Propositio, (5) ad Theorem give e ee 0 0 exp (3) (J ) exp(exp (2) (J )) H T 0 e e e e H

7 98 A Note o the Expoetial of A Matrix e e I + eee e e J (6) (3), (7), Defiitio, Propositio, (6) ad Theorem give e eee 0 0 exp (4) (J ) exp(exp (3) (J )) H T 0 e ee e ee e ee I + eee e e ee J H I geeral, sice (3) is a recurrece relatio, repeatig the above actios (use the equatio (7), Defiitio, Propositio ad Theorem for exp (r) (J ) where r 5, 6,, k, k + ), we have ee e e exp (k+) (J ) e e e ee e I + J (7) ee Usig (5) ad (6) (with exp k+ e () e e ) i (7) we obtai The theorem is proved exp (k+) (J ) k ei + expk+ e () k e J We ca also establish a similar geeralizatio for the matrix J I fact sice J is a orthogoal projectio matrix oto the space spaed by the vector of it has a sigle eigevalue of which is the dimesio of the space o which it projects ad a eigevalue of zero with multiplicity Hece, J is a idempotet matrix Followig [, page 248], if A is a idempotet matrix, the exp(a ) I + (e )A Therefore exp ( ) J I + e J Now, let us cosider the followig theorem to geeralize the equality exp ( ) J I + e J i order to get exp ( ) (k+) J THEOREM 3 exp (k+) ( ) J k k+ e k e ei + J PROOF Sice exp ( J ) I + e J, so by Theorem, we have exp ( J ) H T D H, where D Diag[e ] ad H is the Helmert matrix of order The the proof is similar to the proof of the former theorem The theorem is proved Ackowledgmet The author wishes to thak the editor ad the aoymous referee for their helpful commets

8 R Farhadia 99 Refereces [] K M Abadir ad J R Magus, Matrix Algebra, Cambridge Uiversity Press, New York, 2005 [2] B R Clarke, Liear Models: The Theory ad Applicatio of Aalysis of Variace, Joh Wiley & Sos, New Jersey, 2008 [3] R Farhadia ad N Asadia, O the Helmert matrix ad applicatio i stochastic processes, It J Math Comput Sci, 2(207), 07 5 [4] B C Hall, Lie Groups, Lie Algebras, ad Represetatios, Graduate Texts i Mathematics, Spriger, Switzerlad, 205 [5] H O Lacaster, The Helmert matrices, Amer Math Mothly, 72(965), 4 2 [6] G A F Seber, A Matrix Hadbook for Statisticia, Joh Wiley & Sos, New Jersey, 2007 [7] X Zha, Matrix theory, America Mathematical Society, Providece, Rhode Islad, 203