a Λ q 1. Introduction

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1 International Journal of Pure and Applied Mathematics Volume 9 No 26, ISSN: -88 (printed version); ISSN: -95 (on-line version) url: doi: 272/ijpamv9i7 PAijpameu EXPLICI MOORE-PENROSE INVERSE AND GROUP INVERSE OF DOUBLY LESLIE MARIX Wiwat Wanicharpichat Department of Mathematics Faculty of Science Phitsanulok 65, HAILAND and Research Center for Academic Excellence in Mathematics Naresuan University Phitsanulok 65, HAILAND Abstract: A doubly Leslie matrix is a bordered real matrix of the form [ p a L n b n Λ q, (n,n) where a n,b n R, p,q R n, and Λ diag(s,,,s n ) is a diagonal matrix of order n he matrix L is a closed form of a doubly companion matrix, a Leslie matrix and a companion matrix his paper is discussed the explicit formula of the Moore-Penrose inverse and the group inverse of the doubly leslie matrix In general the Moore-Penrose inverse of a rectangle doubly Leslie matrix is also discussed AMS Subject Classification: 5A9, 5A2 Key Words: companion matrix, doubly companion matrix, Leslie matrix, doubly Leslie matrix, Moore-Penrose inverse, group inverse Introduction One of the most popular models of population growth is a matrix-based model, first introduced by PH Leslie In 95, he published his most famous article in Biometrika, a journal he article was entitled, On the use of matrices in Received: August 9, 26 Revised: September, 26 Published: October 8, 26 c 26 Academic Publications, Ltd url: wwwacadpubleu

2 96 W Wanicharpichat certain population mathematics [2, pp 7 2 he Leslie model describes the growth of the female portion of a population which is assumed to have a maximum lifespan he females are divided into age classes all of which span an equal number of years Using data about the average birthrates and survival probabilities of each class, the model is then able to determine the growth of the population over time, [6 A Leslie matrix arises in a discrete, age-dependent model for population growth It is a matrix of the form L r r 2 r r n r n s s n, () where r j, < s j, j,2,,n Doubly companion matrices C M n were first introduced by Butcher and Chartier in [, pp27 276, given by C α α 2 α α n α n β n β n β n 2 β2 β, (2) that is, a n n matrix C with n > is called a doubly companion matrix if its entries c ij satisfy c ij for all entries in the sub-maindiagonal of C and else c ij for i and j n We define a doubly Leslie matrix analogous as the doubly companion matrix byreplacingthesubdiagonalofthedoublycompanionmatrixbys,,,s n where s j, j,2,,n, respectively, and denoted by L, that is, a doubly

3 EXPLICI MOORE-PENROSE INVERSE AND 96 Leslie matrix is defined to be a matrix as follows: a a 2 a a n a n b n s b n b n 2 L b2 s n b, () where a j,b j R, the real numbers, j,2,,n As the Leslie matrix, we restriction only s j >, j,2,,n For convenience, we can be written the matrix L in a partitioned form as [ p a L n b n Λ q (n,n) where p a a 2 a n,q b n b n 2 and Λ diag(s,,,s n ) is a diagonal matrix of order n We also define a rectangular doubly Leslie matrix of order m n, where m n k and < k < n as follows: a a 2 a a n a n b n s b n R b n 2 () s n k b k [ p a R n b n Λ k q k (m,n) where p b (m,n) For convenience, we can be written the matrix L in a partitioned form as a b n a 2 b n 2 a n,q k b k,, (5) and Λ k [diag(s,,,s n k ) is a (n k) (n ) block matrix which the first block is a diagonal matrix diag(s,,,s n k ) and the remainder block is a zero matrix of appropriated size We abbreviate doubly Leslie matrix to DLM and rectangular doubly Leslie matrix by RDLM

4 962 W Wanicharpichat Let M be a matrix partitioned into four blocks [ A B M C D where the submatrix C is assumed to be square and nonsingular Brezinski in [, p22 asserted that, the Schur complement of C in M, denoted by (M/C), is defined by (M/C) B AC D (6) As in (6), the Schur complement of Λ in L, denoted by (L/Λ), is a matrix or a scalar (L/Λ) ( a n b n ) ( p )Λ ( q) ( ) n a i b n i (a n +b n )+ (7) s i he author [7 asserted some basic properties of doubly Leslie matrix as in the following lemma Lemma Let L be a doubly Leslie matrix as in () with partitioned as [ p a L n b n Λ q i, (n,n) where p [ [, a a 2 a n, q bn b n 2 b and Λ diag(s,,,s n ), s j >, j,2,,n is a diagonal matrix of order n, then and, if detl then n detl ( ) ((a n n +b n )+ i a i b n i s i ) n i L (L/Λ) [ Λ q (L/Λ)Λ + ( Λ qp Λ ) p Λ ( where (L/Λ) (a n +b n )+ n i s i (8), (9) (n,n) a i b n i s i ), as in (7), and Λ diag( s,,, s n ) In the present paper we give explicit Moore-Penrose inverse and group inverse formulae for the doubly Leslie matrix and give some related topics

5 EXPLICI MOORE-PENROSE INVERSE AND 96 2 Preliminaries Let R m n denote the set of all m n matrices over the field of real numbers R he Moore-Penrose inverse of a matrix A R m n is the unique matrix X R n m satisfying the four Penrose conditions A AXA, X XAX, (AX) AX and (XA) XA and is denoted by A he group inverse of a matrix A R n n is the unique matrix X R n n satisfying A AXA, X XAX and AX XA and is denoted by A A well known characterization for the existence of A is that rank(a) rank(a 2 ), [ If A is nonsingular, then A A A Recall that A R n n is called range-symmetric if range(a) range(a ) If A is range-symmetric, then A A A system of linear equation Ax b need not possess a solution when rank(a) rank[a : b hat is b is not in the range of A he Moore-Penrose inverse is most often used to solve least squares systems It is still desirable to to find a x that is closest to a solution he residual vector is a key component to solve these systems heorem 2 ([) Let A R m n with rank(a) r, and suppose A FG is a full rank factorization of A hen F (F F) F, 2 F F I r, the r r identity matrix, G G (GG ), GG I r, 5 A G F More generally, for any m n matrix A of full row rank m, A I m A is a full rank factorization of A hen A A (AA ) () he group inverse is very useful and has applications in many fields such as singular differential and difference equations, Markov chains, and iterative methods, see for instance [

6 96 W Wanicharpichat heorem ([) Let a square matrix A have the full rank factorization A FG hen A has a group inverse if and only if GF is nonsingular In which case, A F(GF) 2 G Moore-Penrose Inverse of RDLM Penrose [5, p8 It is possible to calculate A even when A A and AA are both singular by the following methods, where A is the conjugate transpose of the matrix A Any matrix M can be partitioned in the form M [ A B C D where D CA B, (using a suitable arrangement of rows and columns) A being any non-singular submatrix whose rank is equal to that of the whole matrix It is then easily verified that, M [ A B C D [ A KA A KC B KA B KC, () where K (AA +BB ) A(A A+C C) he matrices AA +BB and A A + C C are positive definite, since A is non-singular hus the generalized inverse of any matrix can be expressed in terms of ordinary reciprocals of matrices We have the following main results Lemma If A PB, where P is a permutation matrix, then A B P (2) Proof It is straightforward to verify that B P satisfies the four Penrose conditions Clearly: PB(B P )PB PBB B PB, 2 (B P )PB(B P ) B BB P B P, [PB(B P ) [PBB P P(BB ) P PBB P PB(B P ),

7 EXPLICI MOORE-PENROSE INVERSE AND 965 [(B P )PB [B P PB [B B B B B (P P)B (B P )PB Lemma 5 For an m n R-matrix N of rank r < min(m,n), and N partitioned in the form [ A B N C D where C is r r nonsingular hen [ N C KD C KC D KA D KC where K (CC + DD ) C(C C + A A) he matrices CC + DD and C C +A A are positive definite, since C is non-singular [ I Proof Let P r be a permutation matrix Premultiplying the matrix N by P I m r (m m) [ C D PN A B Since P is a unitary matrix and by (2) We have herefore and As in (), we have [ C D A B (PN) (PN) N P N [ C D A B [ C D A B N P P [ C KC C KA D KC D KA where K (CC +DD ) C(C C+A A) he matrices CC +DD and C C+A Aarepositivedefinite, sincec isnon-singular hereforecc +DD and C C +A A are also non-singular matrices We have [ C KC C KA [ C KA C KC N he proof is complete D KC D KA P, D KA D KC

8 966 W Wanicharpichat as heorem 6 Let L be a doubly Leslie matrix as in () with partitioned [ p a L n b n Λ q, (n,n) where p [ [, a a 2 a n, q bn b n 2 b and Λ diag(s,,,s n ), s j >, j,2,,n is a diagonal matrix of order n, then L [ ΛKp ΛKΛ q Kp q KΛ where K (Λ 2 +qq ) Λ(Λ 2 +pp ) Proof If detl then L L which appeared in (9) In general [ [ A B C L KA C KC C D D KA D KC [ Λ K( p ) Λ KΛ ( q) K( p ) ( q) KΛ [ ΛKp ΛKΛ q Kp q, KΛ where K (ΛΛ +( q)( q) ) Λ(Λ Λ+( p ) ( p )) (Λ 2 +qq ) Λ(Λ 2 +pp ), Corollary 7 Let R be a rectangle doubly Leslie matrix as in (5) with partitioned as [ p a R n b n Λ k q k where m n k,, (m,n) p [ a a 2 a n, q [ bn b n 2 b k, and Λ k [diag(s,,,s n k ) is a (n k) (n ) block matrix, then [ L Λk Kp Λ k KΛ k q k Kp q k KΛ k where K (Λ k Λ k +q kq k ) Λ k (Λ k Λ k +pp )

9 EXPLICI MOORE-PENROSE INVERSE AND 967 Proof he proof is an analogous as in heorem 6 Let s consider some examples Example L [ p : a n b n Λ q where p [ 2 [,, q 2 and Λ diag(,2,), is a diagonal matrix of order, then [ L ΛKp ΛKΛ q Kp q, KΛ where K (Λ 2 +qq ) Λ(Λ 2 +pp ) First we calculate qq and pp qq [ 2 2, 2 2 pp 2 [ , 2 and we have (Λ 2 +qq ) (Λ 2 +pp ) K (Λ 2 +qq ) Λ(Λ 2 +pp ) , ,

10 968 W Wanicharpichat Finally, ΛKp ΛKΛ 8 8 2, q Kp [ 2 q KΛ [ 2 [ , , 2 herefore L [ ΛKp ΛKΛ q Kp q KΛ his matrix is satisfies the four Penrose conditions Example For a full row rank rectangle doubly Leslie matrix of order R 2 2

11 EXPLICI MOORE-PENROSE INVERSE AND 969 From (), R R (RR ) his matrix is also satisfies the four Penrose conditions 2 2 Group Inverse of DLM As in [, p67 we have the following useful result heorem 8 Let A be a square singular matrix, ranka ranka 2, and R(A) be the range of A If the system has a solution, it is uniquely given by Ax b, x R(A) x A b Proof Suppose that x R(A) where R(A) is the range of A here is a vector y such that Ay x Let a solution x be written as x Ay for some y We have Ax AAy A 2 y, then A 2 y b Since ranka ranka 2, there is a unique A such that AA A A, A AA A, and AA A A

12 97 W Wanicharpichat herefore x Ay AA Ay A 2 A y A A 2 y A Ax A b Let L be a doubly Leslie matrix as in () with partitioned as [ p a L n b n Λ q (n,n) If detl then L L which was shown in (9) We interested in study the only case rank(l) n By the definition of DLM the rank of L is at least n Since equivalence matrix has the same rank, we reduce the matrix L to a reduced echelon form as follows: s s n a a 2 a s n s n b n s a a 2 a n a n b n s b n s 2 b2 s n b b n 2 b s n a n b n a s b n a 2 b n 2 a n 2 s n 2 b 2 a n s n b We see that rank(l) n if and only if if and only if a n b n a s b n a 2 b n 2 a n 2 s n 2 b 2 a n s n b a n b n a s b n + a 2 b n a n 2 s n 2 b 2 + a n s n b

13 EXPLICI MOORE-PENROSE INVERSE AND 97 We factor L to full rank factorization as follows: a a 2 a n s b n s L FG s 2 b n 2 s b n s n [ p [ Λ In q, where p a a 2 a n, q b n s b n 2 b s n Also, by direct computation, we have GF, and Λ diag(s,,,s n ) a a 2 a a n 2 a n b n s n s s s b n n 2 s b n n s n b2 n 2 s s n 2 b n s n s : M he matrix GF : M is a doubly Leslie matrix of order (n ) (n ) [ p s M a n b n n s Λ q 2 where p a a 2 a n 2, q 2 b n 2 s n b n s n s b s n s n, and Λ diag(s,,,s n 2 ), (n,n ) is a diagonal matrix of order n By (9), we have [ M (M/Λ ) Λ q 2 (M/Λ )Λ + ( Λ q 2p ) Λ p Λ, ()

14 972 W Wanicharpichat ( s where (M/Λ ) (a n +b n n Λ diag( s,,, s n 2 ) From heorem, we have L F(GF) 2 G [ p Λ s )+s n n 2 i a i b n i s i s i+ ), as in (7), and [ ((M/Λ ) Λ q 2 (M/Λ )Λ + ( Λ q 2p Λ [ In q p Λ [ (M/Λ ) 2 p Λ [ Λ q 2 (M/Λ )Λ + ( Λ q 2p ) Λ p Λ ) ) 2 2 [ In q Let s consider the same example Example L where p [ 2 [,, q 2 and Λ diag(,2,), is a diagonal matrix of order Since det(l), we have rank(l) rank(l 2 ), we know that the unique L exists Now L FG and GF ,

15 EXPLICI MOORE-PENROSE INVERSE AND 97 Finally (GF) (GF) , L F(GF) 2 G his matrix is satisfies the three conditions for group inverse 5 Conclusion In this paper, we mainly study about the explicit formula of Moore-Penrose inverse and group inverse of doubly Leslie matrix Acknowledgments he author is very grateful to the anonymous referees for their comments and suggestions, which inspired the improvement of the manuscript his work was supported by Naresuan University References [ A Ben-Israel, NE Greville, Generalized Inverses: heory and Applications, Second Edition, Springer-Verlag, New York (2) [2 N Bacaër, A Short History of Mathematical Population Dynamics, Springer, New York (2)

16 97 W Wanicharpichat [ C Brezinski, Other manifestations of the Schur complement, Linear Algebra Appl, (988), 2-27 [ JC Butcher, P Chartier, he effective order of singly-implicit Runge-Kutta methods, Numerical Algorithms, 2 (999), [5 R Penrose, On best approximate solutions of linear matrix equations, Proc Cambridge Philos Soc, 52 (955), 7-9 [6 D Poole, Linear Algebra: A Modern Introduction, 2nd Ed, homson Learning, London (26) [7 W Wanicharpichat, Explicit minimum polynomial, eigenvector and inverse formula of doubly Leslie Matrix, J Appl Math & Informatics,, No-s: - (25), 27-26

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