**
Researoh partZy supported by an NSF Grant No. GP-19568.
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
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.
1.1.
The association matrices
scheme on
v
INTRODUCTION
B ' B , ••• , B of an m-class association
O l
m
objects are defined [3] by
(1.1)
i
= 1,
2, ••• , m,
where
as i =
b
1
if objects a and
otherwise.
=0
Bare i-th associates,
(1.2)
where
J
vxv
is the matrix with
1
as element everywhere.
symmetric matrix with all row and column sums equal to
n •
i
Clearly,
B
i
is a
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, ••• ,cm belong to a field.
considered as reals.
*
We note that
BO,Bl, ••• ,B
In this article,
m
c 's
i
will be
form a basis of this algebra.
Research supported in part by a National, Science Foundation Grant No.
GP-19568.
2
It is also known [3] that the
(m+l) x (m+l)
matrices
PO'" I(m+l)X(m+l)'
Pl, ••• ,Pm defined by
...
(1.4)
«Pik
j
»,
i,j = 0,1, ...
,m,
k = O,l, ••• ,m
provide a regular representation of the algebra defined by the matrices
BO,Bl, ••• ,Bm•
The matrices
PO,Pl, ••• ,P
m
are linearly independent and provide
a basis for the vector space generated by the linear forms
where
CO,Cl'H.,Cm belong to a field.
Pi's
cOPO+clPl+ ••• +cmPm
are not, necessarily, symmetric.
1.2. We state here, without proof, a result which we shall need later.
For
proof, see [3].
Lemma 1.1. The two matrices B = cOBO+ •.• +cmBm and P
= cOPO+ ••• +cmP
m have
the same minimal polynomial and hence the same distinct characteristic roots.
Let
zui'
i=O,I, ••• ,m.
u
= O,l, .•• ,m
denote the
It is known that
real and that the
(m+l) X(m+l)
(1.5)
u
is non-singular [3].
eu ...
=
0, 1, .•• , m,
i
...
are all
0, 1, •.. , m
Further, for a suitable ordering of
P
=I~=o
ciP i
zui
for each
are given by
u=O,I, ••. ,m.
= I~=o
ciB i
are, therefore, to be
the roots
zui
satisfy the relations
B
eu given by (1.6).
Lemma 1.2. For fixed u = O, ••• ,m,
(1. 7)
i ... 0,1, ••. ,m
matrix
The distinct characteristic roots of
found among
characteristic roots of
u = 0,1, .•• ,m,
the characteristic roots of the matrix
(1.6)
m+l
=
For proof, see [3].
i,
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 char-
acteristic roots of association matrices is given by
nl
1
(1.8)
Z
=
Z
x-l+~
1
2
x-I-It:
2
1
n
-y-~-~
l
J
-y-~+~
where
x = P12
2
The multiplicities
(1.9)
ao
2.
2.1.
=
1,
6
(3
ao' aI' a 2 '
l
=
a2
=
n +n 2
(n -n )+x(n +n Z)
l
l 2
l
+
2
2It:
generated by
+ 1.
(nl-nZ)+y(nl+nZ)
2It:
BO' Bl
B = cOBO+clBl+c2B2
and
BZ'
Then
P
in the linear associative algebra
= cOPO+cIPl+cZPZ
in the regular representation by the algebra generated by
3x3
matrices.
of degree at most three.
(and hence of
P) are
Band
is the image of
PO' PI
and
P have the same minimal polynomial
B
P2
h(x)
Suppose that the distinct characteristic roots of
0, 8
1
and
8 ,
2
The minimal polynomial
[5] ,
( 2.1)
~
2(3
GENERALIZED INVERSE OF A fllATRIX IN A LINEAR AsSOCIATIVE ALGEBRA
Consider a matrix
which are
X2 +
are given by
n +n
l 2
2
a
=
Since every matrix satisfies its minimal polynomial, we have
h(x)
is then
B
4
(2.2)
We define
.
s
(2.3)
Then
BSB
(2.4)
..
B.
This implies that
S" B
is a generalized inverse of
B.
In the same
way we find that
(2.5)
.
P
P.
is a generalized inverse of
P-
roots of
8
1
and
8
If
B
are
of
2
It is easy to verify that the characteristic
(1/6 +1/6 ), 1/6
and 1/8
corresponding to the roots 0,
1
1
2
2
P. The same is true of the distinct characteristic roots of
B •
has the characteristic roots
teristic roots
0,
e1 2
e2 2
and
0, 8
1
and
2
8 ,
2
B
has the charac-
and the minimal polynomial of
(2.6)
Let
(2.7)
M
=
and
(2.8)
Then we have
(2.9)
BGB
=
B,
GBG
..
G,
(BG) ,
=
BG,
(GB) ,
..
GB.
2
B
is
5
G = B+
Hence
is the Moore-Penrose inverse of
..
(2.10)
e 2e 2
2
is the Moore-Penrose inverse of
P.
roots of
1/e •
2
and
are
0, 1/e
1
and
We note that the distinct characteristic
The same is true of
P+ belong to the suba1gebras generated by
30
Similarly,
1
- -=-1
B+
B ([7],[8]).
Band P
p+.
Further,
B+
respectively.
GENERALIZED INVERSE OF A PARTITIONED f"lATRIX
We consider a matrix
~--t-~
(3.1)
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 co1umn-
vectors of
K" BL.
NN'.
That is,
Note that
F is not necessarily non-
negative.
Let
B
be a generalized inverse of
be a generalized inverse of Q.
Now
B.
Define
Q. -K'B-K and let Q-
Q .. -K'B-K .. -L'BB-BL
= -L'BL.
Consider
the matrix
(3.2)
Then it is easy to verify that
FHF
(3.3)
H =F
and hence
for
F-
..
is
F,
is a generalized inverse of
F.
An
alternative expression
6
(3.4)
where
R· L'BL
and
K· BL.
The expression for a generalized inverse of
when it is non-negative Hermitian is given in [10].
For
F
F non-singular, its
inverse is given in [6].
4.
4.1.
rt>Rr-W..
EQUATIONS.
APPLICATIONS.
Consider the linear model
E(I)
(4.1)
where
LINEAR f'·'bDEL AND SowrION OF
~
- ,
•
A' lJ
is a column-vector of
n
random variables
uncorrelated and have the same variance
02•
Yl'Y2""'Yn which are
It is known ([1],[9]) that the
minimum variance unbiased linear estimator of an estimable linear function
is given by
1\
l'~
AA'~
(4.2)
If
where
•
1\
~
l'~
is a solution of the normal equations
A~.
(AA')- is a generalized inverse of AA',
then
C.
(AA')-A~
is a
solution of (4.2).
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
=
7
The representation of
p
(4.4)
C is
•
The characteristic roots of
(4.5)
Zu,
where
P
are given by
Z is as defined by (1.10) and
u'· (u ,u ,u )
O l 2
is
given by
(4.6)
Using the relation
(4.7)
for a partially balanced design, it is easy to verify that
and the other two roots
e
el and e2 must be distinct if
0
Al ~
is a root of
A
2
•
Thus
P
0,
el , e2 will also be the distinct roots of C, 0 being a simple root and el
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)
80
•
~l
(say),
81
=
the first element of
Bl~·
sum of the
para~
eters for those treatments which are first associates of the first
(4.10)
treatment,
= the
B H..= sum of the parameters
2
for those treatments which are second associates of the first
treatment.
82
first element of
8
are similarly defined in terms of
.and
~•
through (2.10), we can find a generalized inverse for
Using results (2.3)
C and
P
and use it
for solving the equations (4.8) and (4.9) respectively.
4.3.
A partially balanced weighing design based on two-association classes has
been defined [11] as an arrangement of
object occurs in
r
1..
(ii)
All
blocks such that each
2p
and every block can
such that
times and in different half blocks of the same block
times,
21
any two objects which are second associates occur together in the same
half block
1..
12
times and in different half blocks of the same block
times.
1.. 22
~
p
b
any two objects which are first associates occur together in the same
half block
Let
objects in
blocks and each block is of size
be divided into two subblocks of size
(i)
v
be the vector of parameters for the
(4.11)
E(I) =
where X
is the vector of
v
objects and the model
N'll
-'
n
random variables and
N'
= «nij »
the design
matrix.
n
(4.12)
=1
ij
= -1
= 0
if the i-th object is in the first half of the j-th block,
if the i-th object is in the second half of the j-th block,
if the i-th object does not occur in the j-th block.
Then it is easy to verify that
(4.13)
NN'
=
=
A -A
where
22
are then,
(4.14)
Bll
=
NX.
12
and
B
(say)
The normal equations
9
It is easy to verify that
Thus using results of Section 2, we can find a generalized inverse
not zero.
B
B will have three distinct characteristic roots
given by (2.3) and a Moore-Penrose inverse
use anyone of
4.4.
B-
B+
and
B+
given by (2.8).
We can
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.
(4.15)
E(z.)
=
The model is
with the restrictions
N'l:!..
K'l:!.. = .!!!.
In this situation, the equations to be solved [9] are
(4.16)
F denote the matrix of coefficients in (4.16).
Let
to the vectorspace generated by the column vectors of
inverse of
F is given by
NN'
Then if K belongs
= B,
a generalized
H as defined in (3.2) and (3.4).
K is such that F is non-singular, its regular inverse is given [6]
If
by
B+
(4.17)
-----=----,---------
[
JH'K)
where
and
[6] •
•
: H(K'H)-J
B+
H'K
~'I
0
is the Moore-Penrose inverse of
is non-singular.
Such an
B and
H is such that
H always exists iff
H'B
=0
F is non-singular
10
So~IRE
On considere l'algebre lineaire associative engendree par les matrices
d'association
Cette algebre est isomorphe
(m+l, m+l) PO,Pl, ••• ,Pm•
et son image
a
(v,v) BO,Bl, ••• ,Bm d'un schema d'association
P
= L~=O
3 l'algebre engendree par les matrices
B· L~=O ciB i
Dans cet isomorphisme une matrice
ciP i
m classes.
ont Ie meme polyn8me minimal.
A l'aide de celui-
ci on obtient une inverse generalisee et l'inverse de Moore-Penrose de
de
B.
P
et
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
a2
classes.
In the regular representation of the linear association algebra generated
by the
vxv
association matrices
scheme, in terms of the
B = L~=O ciB i
BO,B , ••• ,B
of an m-class association
l
m
(m+l) x (m+l)
matrices
PO,Pl, ••• ,Pm,
a matrix
P = L~=O ciP i have the same minimal polynomial.
A generalized inverse and the Moore-Penrose inverse of P and B have been
and its image
derived using their minimal polynomial.
An expression for a generalized in-
verse 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 •
•
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,
[3]
R. C. Bose and D. H. Mesner,
[4]
W. S. Connor and W. H. C1atworthy, IISome theorems for partially balanced
designs", Ann. Math. Stat.~ 25 (1954), 100-112.
[5]
F. R. Gantmacher,
{6]
A. J. Goldman and M Zelen, "Weak generalized inverses and minimum variance linear unbiased estimation", J. Res. Nat. Bur. Stand. ~ 6BB
(1964), 151-172.
[7]
R. Penrose,
"Classification and analysis of partially
balanced incomplete block designs with two associate classes",
J. Amer. Stat. Asson.~ 47 (1952), 151-184.
"On linear associative algebras corresponding to association schemes of partially balanced designs",
Ann. Math. Stat.~ JO (1959), 22-38.
Soc.~
[ 8]
R. Rado,
[9]
C. R. Rao,
Th~orie
des
Matrices~
Tome 1, Dunod, Paris (1966).
"A generalized inverse for matrices",
51 (1955), 406-413.
Prac. Camb. PhiZ.
"Note on generalized inverses of matrices",
Soc.~ 52 (1956), 600-601.
Linear Statistical, Inference and Its
Proc. Canib. PhiZ.
AppZications~
John Wiley, New York (1965).
•
[10]
C. A. Rohde, "Generalized inverses of partitioned matrices", Jour.
SIAM~ 1J (1965), 1033-1035.
[11]
K. V. Suryanarayana, "Contributions to partially balanced weighing
designs", (1969), Inst. Stat., Mimeo Series 621, University of
North Carolina, Chapel Hill •