** Researoh partzy supported by an NSF Grant No. GP

Transcription

1 ** Researoh partzy supported by an NSF Grant No. GP and Institut de Statistique mathematique~ Universite de Geneve. UN GENERALIZED IiNERSES Ii~ A LINEAR AsSOCIATIVE ALGEBRA AND THEIR APPLICATIOOS IN THE ANALYSIS OF A CLAss OF DESIGNS* I. M. Chakravarti** Department of Statistios University of North Carolina at Chapel HiU Institute of Statistics Mimeo Series No. 766 August~ 1971

2 ON GENERALIZED INVERSES IN A LINEAR AsSOCIATIVE ALGEBRA AND THEIR APPLICATIONS IN THE ANALYSIS OF ACLASS OF DESIGNS* I. M. Chakravarti University of North Carolina at Chapel, HiH and Universit~ de Gen~ve, Suisse 1. INTRODUCTION 1.1. The association matrices B O ' B l,, B m of an m-class association scheme on v objects are defined [3] by (1.1) i = 1, 2,, m, where b i = 1 as if objects a and Bare i-th associates, = 0 otherwise. (1.2) where J is the matrix with 1 as element everywhere. Clearly, B i is a vxv symmetric matrix with all row and column sums equal to n i Further, (1. 3) The commutative and associative laws hold for the multiplication of these matrices. It has been shown [3] that the linear forms cobo+clbl+ +cmbm form a linear associative and commutative algebra with a unit element, if the coefficients co,cl,,c m belong to a field. In this article, c 's will be i considered as reals. We note that BO,Bl,,B m form a basis of this algebra. * Research supported in part by a National, Science Foundation Grant No. GP

3 2 It is also known [3] that the (m+l) x (m+l) matrices PO'" I(m+l)X(m+l)' Pl,,P m defined by (1.4)... j «Pik», i,j = 0,1,...,m, k = O,l,,m provide a regular representation of the algebra defined by the matrices BO,Bl,,B m The matrices PO,Pl,,P m are linearly independent and provide a basis for the vector space generated by the linear forms copo+clpl+ +cmpm where CO,Cl'H.,C m belong to a field. Pi's are not, necessarily, symmetric We state here, without proof, a result which we shall need later. For proof, see [3]. Lemma 1.1. The two matrices B = cobo+. +cmb m and P = copo+ +cmp m have the same minimal polynomial and hence the same distinct characteristic roots. Let zui' u = O,l,.,m denote the m+l characteristic roots of i=o,i,,m. It is known that u = 0,1,.,m, i... 0,1,.,m are all real and that the (m+l) X(m+l) matrix (1.5) u = 0, 1,., m, i... 0, 1,.., m is non-singular [3]. Further, for a suitable ordering of zui for each i, the characteristic roots of the matrix P =I~=o cip i are given by (1.6) e... u u=o,i,.,m. The distinct characteristic roots of B = I~=o cib i are, therefore, to be found among e given by (1.6). u Lemma 1.2. For fixed u = O,,m, the roots zui satisfy the relations (1. 7) = For proof, see [3].

4 3 This lemma helps us to determine the ordering of the roots zui for a given 1. For a two-class association scheme (m=2), the matrix Z of ordered characteristic roots of association matrices is given by l 1 n l Z n x-l+~ (1.8) Z = 1 -y-~-~ 2 x-i-it: 1 -y-~+~j 2 where x 2 = P12 (3 6 = X 2 + 2( The multiplicities ao' ai' a2 ' are given by (1.9) 1, a o = a l = n l +n 2 (nl-nz)+y(nl+n Z ) 2 2It: n l +n 2 (n l -n 2 )+x(n l +n Z ) a 2 = + 2 2It: 2. GENERALIZED INVERSE OF A fllatrix IN A LINEAR AsSOCIATIVE ALGEBRA 2.1. Consider a matrix B = cobo+clbl+c2b2 in the linear associative algebra generated by B O ' B l and B Z ' Then P = copo+cipl+czpz is the image of B in the regular representation by the algebra generated by PO' PI and P 2 which are 3x3 matrices. Band P have the same minimal polynomial h(x) of degree at most three. Suppose that the distinct characteristic roots of B (and hence of P) are 0, 8 1 and 8, 2 The minimal polynomial h(x) is then [5], ( 2.1) ~ Since every matrix satisfies its minimal polynomial, we have

5 4 (2.2) (2.3) We define s.. Then (2.4) BSB.. B. This implies that S" B is a generalized inverse of B. In the same way we find that (2.5) P.. is a generalized inverse of P. It is easy to verify that the characteristic roots of P- are (1/6 1 +1/6 2 ), 1/6 1 and 1/8 2 corresponding to the roots 0, 8 1 and 8 2 of P. The same is true of the distinct characteristic roots of B If B has the characteristic roots 0, 8 1 and 8 2, B 2 has the characteristic roots 0, e 2 and e 2 and the minimal polynomial of B 2 is 1 2 (2.6) Let (2.7) M = and (2.8) Then we have (2.9) BGB = B, GBG.. G, (BG), = BG, (GB),.. GB.

6 Hence G = B+ is the Moore-Penrose inverse of B ([7],[8]). Similarly, 5 (2.10) =-- e 2e is the Moore-Penrose inverse of P. We note that the distinct characteristic roots of B+ are 0, 1/e 1 and 1/e 2 The same is true of p+. Further, B+ + and P belong to the suba1gebras generated by Band P respectively. 30 GENERALIZED INVERSE OF A PARTITIONED f"latrix We consider a matrix (3.1) ~--t-~ li'l9j where N, K, K' are real matrices, and K' is the transpose of K. We assume that the column-vectors of K belong to the vectorspace generated by the co1umnvectors of NN'. That is, K" BL. Note that F is not necessarily nonnegative. Let B be a generalized inverse of B. Define Q. -K'B-K and let Q be a generalized inverse of Q. Now Q.. -K'B-K.. -L'BB-BL = -L'BL. Consider the matrix (3.2) Then it is easy to verify that (3.3) FHF.. F, and hence H = F is a generalized inverse of F. An alternative expression for F- is

7 6 (3.4) where R L'BL and K BL. The expression for a generalized inverse of F when it is non-negative Hermitian is given in [10]. For F non-singular, its inverse is given in [6]. 4. LINEAR f' 'bdel AND SowrION OF rt>rr-w.. EQUATIONS. APPLICATIONS Consider the linear model (4.1) E(I) A' lj-, where ~ is a column-vector of n random variables Yl'Y2""'Yn which are uncorrelated and have the same variance 0 2 It is known ([1],[9]) that the minimum variance unbiased linear estimator of an estimable linear function l'~ 1\ 1\ is given by l'~ where ~ is a solution of the normal equations (4.2) AA'~ A~. If (AA')- is a generalized inverse of AA', then C. (AA')-A~ is a solution of (4.2) Consider a (connected) partially balanced design based on a two-class association scheme. Then the coefficient matrix C in the normal equations for estimating the treatment effects after adjusting for the block effects [2] is (4.3) C =

8 7 The representation of C is (4.4) p The characteristic roots of P are given by (4.5) Zu, where Z is as defined by (1.10) and u' (u O,u l,u 2 ) is given by (4.6) Using the relation (4.7) for a partially balanced design, it is easy to verify that 0 is a root of P and the other two roots e l and e 2 must be distinct if Al ~ A 2 Thus 0, e, l e will also be the distinct roots of C, 0 being a simple root and 2 e e l and e2 will occur with multiplicities a l and a 2 given by (1.9). Now, it is easy to show that for solving the linear equations (4.8) CH.. = ~, it is enough if one solves (4.9) (4.10) 8 ~l 0 (say), 8 1 = the first element of Bl~ sum of the para~ eters for those treatments which are first associates of the first treatment, 8 2 = the first element of B H..= sum of the parameters 2 for those treatments which are second associates of the first treatment.

9 8.and are similarly defined in terms of ~ Using results (2.3) through (2.10), we can find a generalized inverse for C and P and use it for solving the equations (4.8) and (4.9) respectively A partially balanced weighing design based on two-association classes has been defined [11] as an arrangement of v objects in b blocks such that each object occurs in r blocks and each block is of size 2p and every block can be divided into two subblocks of size p such that (i) (ii) any two objects half block times, All any two objects half block times. which are first associates occur together in the same times and in different half blocks of the same block which are second associates occur together in the same times and in different half blocks of the same block Let ~ be the vector of parameters for the v objects and the model (4.11) E(I) = N'll -' where X is the vector of n random variables and N' = «n ij» the design matrix. (4.12) n ij = 1 if the i-th object is in the first half of the j-th block, = -1 if the i-th object is in the second half of the j-th block, = 0 if the i-th object does not occur in the j-th block. Then it is easy to verify that (4.13) NN' = = B (say) where A -A are then, and The normal equations (4.14) Bll = NX.

10 It is easy to verify that B will have three distinct characteristic roots 9 not zero. Thus using results of Section 2, we can find a generalized inverse B given by (2.3) and a Moore-Penrose inverse B+ given by (2.8). We can use anyone of B- and B+ to get a solution of the equations (4.14) For the partially balanced weighing design of 4.3., sometimes one has restrictions on the parameters. The model is (4.15) E(z.) = N'l:!.. with the restrictions K'l:!.. =.!!!. In this situation, the equations to be solved [9] are (4.16) Let F denote the matrix of coefficients in (4.16). Then if K belongs to the vectorspace generated by the column vectors of NN' = B, a generalized inverse of F is given by H as defined in (3.2) and (3.4). If K is such that F is non-singular, its regular inverse is given [6] by (4.17) B+ : H(K'H)-J -----=----, [ JH'K) ~'I 0 where B+ is the Moore-Penrose inverse of B and H is such that H'B = 0 and H'K is non-singular. Such an H always exists iff F is non-singular [6]

11 10 So~IRE On considere l'algebre lineaire associative engendree par les matrices d'association (v,v) BO,Bl,,B m d'un schema d'association a m classes. Cette algebre est isomorphe 3 l'algebre engendree par les matrices (m+l, m+l) PO,Pl,,P m Dans cet isomorphisme une matrice B L~=O cib i et son image P = L~=O cip i ont Ie meme polyn8me minimal. A l'aide de celuici on obtient une inverse generalisee et l'inverse de Moore-Penrose de P et de B. On donne une formule pour Ie calcui d'une inverse generalisee d'une matrice symetrique partitionnee, non necessairement non-negative. Enfin ces inverses generalisees sont utilisees pour resoudre les equations normales et pour faire l'estimation dans les plans d'experiences construits sur des schemas d'association a 2 classes. In the regular representation of the linear association algebra generated by the vxv association matrices BO,B,,B of an m-class association l m scheme, in terms of the (m+l) x(m+l) matrices PO,Pl,,P m, a matrix B = L~=O cib i and its image P = L~=O cip i have the same minimal polynomial. A generalized inverse and the Moore-Penrose inverse of P and B have been derived using their minimal polynomial. An expression for a generalized inverse of a symmetric partitioned matrix, not neae88arizy non-negative, is given. Applications of these generalized inverses for solving normal equations and estimation in designs based on two-class association schemes are discussed

12 11 REFERENCES [1] R. C. Bose, "Least square aspects of analysis of variance", Institute of Statistics, Mimeo Series 9, University of North Carolina, Chapel Hill. [2] R. C. Bose and T. Shimamoto, "Classification and analysis of partially balanced incomplete block designs with two associate classes", J. Amer. Stat. Asson.~ 47 (1952), [3] R. C. Bose and D. H. Mesner, "On linear associative algebras corresponding to association schemes of partially balanced designs", Ann. Math. Stat.~ JO (1959), [4] W. S. Connor and W. H. C1atworthy, IISome theorems for partially balanced designs", Ann. Math. Stat.~ 25 (1954), [5] F. R. Gantmacher, Th~orie des Matrices~ Tome 1, Dunod, Paris (1966). {6] A. J. Goldman and M Zelen, "Weak generalized inverses and minimum variance linear unbiased estimation", J. Res. Nat. Bur. Stand. ~ 6BB (1964), [7] R. Penrose, "A generalized inverse for matrices", Prac. Camb. PhiZ. Soc.~ 51 (1955), [ 8] R. Rado, "Note on generalized inverses of matrices", Proc. Canib. PhiZ. Soc.~ 52 (1956), [9] C. R. Rao, Linear Statistical, Inference and Its AppZications~ John Wiley, New York (1965). [10] C. A. Rohde, "Generalized inverses of partitioned matrices", Jour. SIAM~ 1J (1965), [11] K. V. Suryanarayana, "Contributions to partially balanced weighing designs", (1969), Inst. Stat., Mimeo Series 621, University of North Carolina, Chapel Hill