The Lehmer matrix and its recursive analogue

Transcription

1 The Lehmer matrix and its recrsive analoge Emrah Kilic, Pantelimon Stănică TOBB Economics and Technology University, Mathematics Department 0660 Sogtoz, Ankara, Trkey; Naval Postgradate School, Department of Applied Mathematics 8 Dyer Rd, Monterey, CA 99, USA; pstanica@npsed Abstract This paper considers the Lehmer matrix and its recrsive analoge The determinant of Lehmer matrix is derived explicitly by both its LU and Cholesky factorizations We frther define a generalized Lehmer matrix with (i, j) entries g ij min { i+, j+ } max { i+, j+ } where n is the nth term of a binary seqence { n } We derive both the LU and Cholesky factorizations of this analogos matrix and we precisely compte the determinant Introdction DH Lehmer (see []) constrcted an n n symmetric matrix A (a ij ) i,j whose (i, j) entry is a ij { min {i, j} i/j j i, max {i, j} j/i i > j Define the second order recrrence {U n (p, q)} as follows: U n (p, q) pu n (p, q) qu n (p, q),

2 Report Docmentation Page Form Approved OMB No Pblic reporting brden for the collection of information is estimated to average hor per response, inclding the time for reviewing instrctions, searching existing data sorces, gathering and maintaining the data needed, and completing and reviewing the collection of information Send comments regarding this brden estimate or any other aspect of this collection of information, inclding sggestions for redcing this brden, to Washington Headqarters Services, Directorate for Information Operations and Reports, Jefferson Davis Highway, Site 0, Arlington VA 0-0 Respondents shold be aware that notwithstanding any other provision of law, no person shall be sbject to a penalty for failing to comply with a collection of information if it does not display a crrently valid OMB control nmber REPORT DATE 00 REPORT TYPE DATES COVERED to TITLE AND SUBTITLE The Lehmer matrix and its recrsive analoge a CONTRACT NUMBER b GRANT NUMBER c PROGRAM ELEMENT NUMBER 6 AUTHOR(S) d PROJECT NUMBER e TASK NUMBER f WORK UNIT NUMBER PERFORMING ORGANIZATION NAME(S) AND ADDRESS(ES) Naval Postgradate School,Department of Applied Mathematics,Monterey,CA,99 8 PERFORMING ORGANIZATION REPORT NUMBER 9 SPONSORING/MONITORING AGENCY NAME(S) AND ADDRESS(ES) 0 SPONSOR/MONITOR S ACRONYM(S) DISTRIBUTION/AVAILABILITY STATEMENT Approved for pblic release; distribtion nlimited SUPPLEMENTARY NOTES J of Combinat Math and CombinatCompting (00), 9-0 SPONSOR/MONITOR S REPORT NUMBER(S) ABSTRACT This paper considers the Lehmer matrix and its recrsive analoge The determinant of Lehmer matrix is derived ex- plicitly by both its LU and Cholesky factorizations We fr- ther de ne a generalized Lehmer matrix with (i; j) entries gij min fi+; j+g max fi+; j+g where n is the nth term of a binary seqence fng : We derive both the LU and Cholesky factor- izations of this analogos matrix and we precisely compte the determinant SUBJECT TERMS 6 SECURITY CLASSIFICATION OF: LIMITATION OF ABSTRACT a REPORT nclassified b ABSTRACT nclassified c THIS PAGE nclassified Same as Report (SAR) 8 NUMBER OF PAGES 9a NAME OF RESPONSIBLE PERSON Standard Form 98 (Rev 8-98) Prescribed by ANSI Std Z9-8

3 where U 0 (p, q) 0 and U (p, q) for n > As an interesting example, we mention that the set of natral nmbers can be obtained from the seqence {U n (p, q)} by taking p, q Throghot this paper, we consider the case q and we denote n U n (p, ) We now define an n n generalized Lehmer matrix, namely F n (g ij ) i,j n defined below: g ij min { i+, j+ } max { i+, j+ } i+ j+ if j i, j+ i+ if i > j where n is the nth term of the seqence { n } In this paper, we obtain the general LU factorization and other explicit formlas for both the Lehmer matrix and its recrsive analoge The Lehmer matrix is part of a family of matrices known as test matrices, which are sed to evalate the accracy of matrix inversion programs since the exact inverses are known (see [, ]) It is hoped that or generalized Lehmer matrix will add to the literatre of special matrices with known inverse The Lehmer Matrix We start by obtaining the LU factorization of the Lehmer matrix A Using the inverses of L and U, we obtain the explicit form for the inverse of A, whose inverse is well-known, ths obtaining another proof of this reslt We define the n n invertible lower trianglar matrix L (l ij ) where l ij j/i for i j and 0 otherwise Next, we define the n n invertible pper trianglar matrix U ( ij ) with ij for i ij

4 i j and 0 otherwise For example, when n, we get L 0 0 and U The following reslt holds Theorem For n > 0, the LU factorization of Lehmer matrix is given by A LU where L and U were defined previosly Proof We split the proof into three cases Case : i j By t k (k ) t, then a ii l ik ki k l ik ki k Case : i > j Ths a ij Case : j > i Then a ij l ik kj k k k ij l ik kj k k k ij k k (k ) i ki l ik kj k ij k k j i k l ik kj k ij k k i j, k k k (k ) i kj k (k ) i kj k i

5 which completes the proof We display an example below: As a conseqence of Theorem, we obtain an explicit vale of the determinant of the Lehmer matrix in the following corollary Corollary For n > 0, det A (n)! n (n!) Proof The proof follows from the LU factorization of matrix A by considering det A det U n i i i by The nth Catalan nmber is given in terms of binomial coefficients Ths we may note that C n ( ) n (n)! n + n (n + )!n! det A (n + ) n C n n! We contine or analysis by determining the L L T (named after Cholesky) factorization of the Lehmer matrix, where L is a lower trianglar matrix The Cholesky factorization was obtained for a different kind of matrix defined sing binary seqences by the second athor in []

6 Theorem The Cholesky factorization of the Lehmer matrix is given by A L L T where L (f ij ) is a lower trianglar matrix with f ij all i j Proof If i > j, then a ij f ir f jr r ij If i j, then a ii f ir f jr r (r ) j i r fir r i r Finally, if i < j, then a ij f ir f jr r which proves the theorem fir r j i r r r i ( r r (r ) i i f ir f jr ij r i ) j (r ) i j, As an example, for n and p (the Fibonacci seqence case), we have r for 9

7 By Theorem, we find that, since A L L T, we have that det (A) n i f ii n t i (n)!, that is, Corollary i n (n!) The Inverse of the Lehmer Matrix Now we find an explicit formla for the inverse of the Lehmer matrix For this prpose, we se its LU factorization as A U L We first derive the inverses of the matrices L and U Lemma Let L (t ij ) denote the inverse of L Then if i j, t ij j i if i j +, 0 otherwise, Proof The proof can be easily checked from the prodct L L Lemma Let U (w ij ) denote the inverse of U Then i i if i j w ij i(i+) i+ if i + j, 0 otherwise, Proof The proof follows from the prodct U U The inverse of the Lehmer matrix is fond in the following theorem Theorem For n > 0, let A (b ij ), then b ij i i if i j < n n n if i j n, i(i+) i+ if i j, 0 otherwise, 6

8 Proof Since A U L, sing the previos two lemmas, we obtain for i n, b ii w ik t ki w ii + w i,i+ t i+,i k i i + i (i + ) i + i (i + ) i i + i i + When i j n, it is easy to see that b nn w nn i j +, then b i+,i w i+,k t ki w i+,i+ t i+,i k (i + ) i + ( ) i i (i + ) i + i + i i n n If The last case j i + can be similarly done, and the proof is complete Therefore we recover the known fact that the inverse of the Lehmer matrix is a symmetric tridiagonal matrix We give the following example as a conseqence of the above theorem: for n, 0 0 A

9 We also give a relation between the terms of inverse of the Lehmer matrix and trianglar nmbers Recall that the nth trianglar nmber T n is defined as the sm of the first n natral nmbers, that is, T n n n(n+) We can re-write A (b ij ) as b ij T i i+ for i j, and b ii i i Recrsive Analoge of the Lehmer Matrix In this section we investigate the same qestions for or generalized recrsive analoge of the Lehmer matrix F n defined in the first section, namely, F n (g ij ): g ij min { i+, j+ } max { i+, j+ } where n is the nth term of the seqence { n } i+ j+ if j i, j+ i+ if i > j For example, when n and p, the matrix F takes the following form: F In order to give the LU factorization of the matrix F n, we define two trianglar matrices Define the n n nit lower trianglar matrix L (c ij ) with c ij j+ i+ for all i j and ij 0 for all i < j 8

10 For example, when n, the matrix takes the form: L Before defining an pper trianglar matrix for the LU factorization of the matrix F n, we need to introdce a new seqence {t n } by the following relation: t n (p ) n + n, that is, t n n+ n, n >, where n is defined as before j+ Define the n n pper trianglar matrix U (d ij ) with d j for j n, d ij ( i+ i+ )t i i+ j+ for < i j n From the definition of the seqence {t n }, we rewrite the matrix U with d j j+ for j n, d ij i+ i i+ j+ for < i j n For example, when n, the matrix takes the form: U Theorem For n > 0, the factorization of matrix F n (g ij ) is given by F n L U, 9

11 where U and L were defined previosly Proof Let L U (h ij ) We consider two cases, i > j and i j For the first case, we write h ij c im d mj m c i d j + m i+ j+ + If i j, then similarly h ij c im d mj m ( m+ i+ i+ j+ ( ) m+ m) m+ j+ ( m+ m) m ( + i+ j+ i+ j+ ) j+ g ij j+ i+ c im d mj m c i d j + m i+ j+ + i+ j+ g ij, and the claim is shown c im d mj m ( m+ i+ i+ j+ ( ) m+ m) m+ j+ ( m+ m) m Now we can find the vale of det (F n ) by considering its LU factorization Corollary For n > 0, det (F n ) n ( i+ i i i+ ) 0

12 As a special cases of the matrix F n, we take the matrix F 0 n obtained sing the Fibonacci seqence, that is, F n+ F n +F n, F 0 0, F The determinant of this matrix becomes det ( Fn 0 ) F n!f n+! (F n+!), where F n! is the Fibonomial factorial, that is, F n! F F F n Next we give the Cholesky factorization of the generalized Lehmer matrix F n For this prpose we define a lower trianglar matrix L (m ij ) with m i, i+ for i n, m ij i+ j+ j for < j i n and 0 otherwise When n, the matrix L takes the form: L The proof of the next theorem is analogos to the proof of Theorem, so it will be omitted Theorem The Cholesky factorization of the recrsive analoge of the Lehmer matrix is given by F n L L T where L is the lower trianglar matrix defined previosly The Inverse of the Generalized Lehmer Matrix Here we give the inverse of the recrsive analoge of the Lehmer matrix Fn by considering its LU factorization Before this, we give the inverses of the matrices L and U in the following lemmas, stated withot proofs, as they are immediate

13 Lemma Let U (ŵ ij ) denote the inverse of U Then if i j i+ if < i j, ŵ ij i i+ i+ i+ if i + j, i+ i+ 0 otherwise, Lemma Let L (ˆt ) ij denote the inverse of L Then if i j, ˆt ij i i+ if i j +, 0 otherwise, Ths the inverse of the matrix F n is fond in the following theorem Theorem 6 For n > 0, let F n (q ij ), then q n+, q nn, q i,i+ q i+,i i+ i+ for i n, q n+ n ii i+ i+ i+( i+ i ) for i n and 0 otherwise ( i+ i )( i+ i+) Proof Since Fn U L, the proof follows from the previos two lemmas and from matrix mltiplication For example, for n, F 0 ( ) ( 0 0 ) 0 0 ( ) ( ) 0 6 Frther comment With a bit more care, one can certainly remove the constraint q on the seqence U n, and prove similar reslts like in the present paper for the corresponding generalized Lehmer matrix

14 Acknowledgments The athors wold like to thank the referee for the very carefl and constrctive comments References [] M Marcs, Basic theorems in matrix theory, Nat Br Standards Appl Math Ser (960), - [] M Newman and J Todd, The evalation of matrix inversion programs, J Society Indstrial and Appl Math 6 (98), 66 6 [] P Stănică, Cholesky factorizations of matrices associated with r- order recrrent seqences, Elect J Combinat Nmber Theory () (00), #A6 (Also pblished in Proceedings of a Conf in Honor of Tom Brown, Topics in Combinatorial Nmber Theory, DIMATIA, Volme 6, ITI Series)