MATHEMATICS

512.643.8

Yu. R. Hakopian, A. N. Eloyan

The Moore - Penrose Inverse for a Certain Class of Block Matrices

(Submitted by academician A. B. Nersessyan 10/III 2005)

   1. The term semi-magic square is attributed to an  n×n real matrix having the sum of elements in each row and each column equal to an identical constant (see [1], for instance). As was shown in [2] , the Moore - Penrose inverse of a semi-magic square is semi-magic as well. In this paper we introduce more general classes of matrices (see Definitions 1.1 and 2.1 below) and establish a similar property of their Moore - Penrose inverse.
   Let  Rn be the space of real  n-dimensional column vectors and  Rm×n be the space of real  m×n matrices.
   Definition 1.1. A matrix  A = [aij] Rm×n is referred to as magic rectangle if there exist constants  r and  c such that the sum of elements in each row and each column is equal to  r and  c , respectively, i.e.
n

j=1 
aij = r ,    i = 1,2,...,m ,
(1.1)
m

i=1 
aij = c ,    j = 1,2,...,n .
(1.2)
   We will call the constants  r and  c row sum and column sum, respectively.
   Example 1.1. The matrix
A =

1
0
5
3
4
-1


is a magic rectangle with r = 6 and c = 4.   
   Let us denote by MR(m, n) the set of m × n magic rectangles. It can be readily shown that MR(m,n) is a subspace of Rm×n. Next, MR(m, n; r, c) will denote the set of m×n magic rectangles with row sum r and column sum c. Let en = [11...1]T be the n-dimensional column vectors of ones. Then the properties (1.1) and (1.2) can be written, respectively, as follows:

Aen = rem,   ATem = c en.
(1.3)
   It can be easily found the following relation between row sum, column sum and the size of the matrix:
mr = nc.
(1.4)

   Basing on the relation (1.4), we can use another notation for the set of magic rectangles with fixed row sum and column sum. Namely, simultaneously with the notation MR(m, n; r, c) we will use a notation MR[m, n : g] which implies that r = ng and c = mg.
   Obviously, for m = n a magic rectangle becomes a semi-magic square.
   2. Let us define a class of block matrices composed of magic rectangles.
   Definition 2.1 A matrix A = [aij] Rm×n represented in the block form

A =




A11
A12
...
A1q
A21
A22
...
A2q
...
...
...
...
Ap1
Ap2
...
Apq





(2.1)

with submatrices  Aij MR(mi, nj; rij, cij)  (or  Aij MR[mi, nj : gij] , in another notation), where  i = 1,2,...,p ,  j = 1,2,...,q and = m, = n, will be referred to as block magic rectangle.

   Insert diagonal matrices

M =





m1
m2
0
0
mp






Rp×p ,   N =





n1
n2
0
0
nq






Rq×q
(2.2)
and  p × q matrices
R =




r11
r12
...
r1q
r21
r22
...
r2q
...
...
...
...
rp1
rp2
...
rpq





 ,   C =




c11
c12
...
c1q
c21
c22
...
c2q
...
...
...
...
cp1
cp2
...
cpq





(2.3)
into our consideration. By
BMR(M, N ; R , C)
(2.4)

we will denote the set of block magic rectangles partitioned into blocks correspondingly to (2.2) with row sums and column sums defined by the matrices (2.3).
   In accordance with relation (1.4) and notation accepted in the previous section, we have
rij = gijnj ,    cij = gijmi
(2.5)
for  i = 1,2,...,p and  j = 1,2,...,q . Consider a matrix
G =




g11
g12
...
g1q
g21
g22
...
g2q
...
...
...
...
gp1
gp2
...
gpq





 .
(2.6)
Then the relations (2.5) can be written in the matrix form:
R = GN ,    C = MG .
(2.7)
   Therefore, we will also use another notation for the set of block magic rectangles (2.4), that is
BMR[M,N : G] .
(2.8)
   The properties (1.3) for the blocks of matrix (2.1) look as
Aijenj = rijemi ,    AijTemi = cijenj ,
(2.9)
where  i = 1,2,...,p and  j = 1,2,...,q . Let us define block matrices
Emp =





em1
0
...
0
0
em2
...
0
...
...
...
...
0
0
...
emp






Rm×p , Enq =





en1
0
...
0
0
en2
...
0
...
...
...
...
0
0
...
enq






Rn×q
with blocks of the size  mi × 1 ,  i = 1,2,...,p (in the matrix  Emp) and  nj × 1 ,  j = 1,2,...,q (in the matrix  Enq). By a straightforward verification it can be easily shown that relations (2.9) are equivalent to the following ones:
AEnq = EmpR ,    ATEmp = EnqCT .
   3. Remind, that the Moore-Penrose inverse  A+ Rn×m of a matrix  A Rm×n is uniquely determined by the properties
a)
AA+A = A ,
b)
A+AA+ = A+ ,
c)
(A+A)T = A+A ,
d)
(AA+)T = AA+ 
(see [3] , for instance).
   The main result of the paper is formulated as follows.
   Theorem 3.1  If  A BMR(M, N ; R , C) then  A+ BMR(N, M ; 
) , where
= N-1/2(M1/2RN-1/2)+M1/2 ,
(3.1)
= N1/2(M-1/2CN1/2)+M-1/2 .
(3.2)
   Let us give an equivalent formulation of Theorem 3.1 , connected with the notation of type (2.8) for a set of block magic rectangles.
   Theorem 3.1A If  A BMR[M, N : Gthen  A+ BMR[N, M :
] , where
= N-1/2(M1/2GN1/2)+M-1/2 .
(3.3)
   Example 3.1. Consider a block magic rectangle
A =






1
0
5
1
1
0
3
4
-1
1
1
0
1
0
2
4
2
2
1
2
0
1
5
2
1
0
2
3
3
2
1
2
0
4
2
2







R6×6          (p = 2 , q = 3) .
According to (2.2),(2.3),(2.5),(2.6), we have
M =

2
0
0
4


 ,   N =


3
0
0
0
2
0
0
0
1



and
R =

6
2
0
3
6
2


 ,   C =

4
2
0
4
12
8


 ,   G =

2
1
0
1
3
2


 .
   The Moore-Penrose inverse of the matrix  A calculated using MATLAB is
A+ =






-0.2411
0.4256
0.3212
-0.1788
0.4879
-0.6788
0.2589
-0.0744
-0.2788
0.1212
-0.4121
0.5212
0.2589
-0.0744
-0.0788
0.0212
-0.1121
0.1212
-0.0267
-0.0267
0.0864
-0.1136
-0.0136
0.1864
-0.0267
-0.0267
-0.0136
0.1864
0.0864
-0.1136
-0.0583
-0.0583
0.0340
0.0340
0.0340
0.0340







 .
For this matrix
=



0.1845
-0.0485
-0.0534
0.1456
-0.1165
0.1359




 ,    =



0.2767
-0.0364
-0.0534
0.0728
-0.0583
0.0340




 ,

=



0.0922
-0.0121
-0.0267
0.0364
-0.0583
0.0340




 .    
   It is of interest to examine some particular cases of block magic rectangles for which the general formula, describing the Moore-Penrose inverse, is considerably simplified.
   Case 1. Suppose that in block representation (2.1)  p = q = 1 , i.e. let  A MR(m, n ; r, c) . Then, by formulae (3.1) and (3.2), we get
= n-1/2(m1/2rn-1/2)+m1/2 = n-1/2n1/2r+m-1/2m1/2 = r+ ,
= n1/2(m-1/2c n1/2)+m-1/2 = n1/2n-1/2c+m1/2m-1/2 = c+ .
   Recall that for a scalar  a , which can be considered as  1 × 1 matrix,  a+ = 1/a , if  a 0 , and  a+ = 0 , if  a = 0 . So, we arrive at the following statement.
   Theorem 3.2 If  A MR(m, n ; r , c) then  A+ MR(n, m ; r+ , c+) .
   Note, that the last proposition has been obtained recently in [4].
   We can also give an equivalent formulation of Theorem 3.2 . Let  A MR[m, n : g] . According to (3.3), we find
=n-1/2(m1/2gn1/2)+m-1/2 = n-1/2n-1/2g+m-1/2m-1/2 = 1
mn
 g+ .
   Thus, the result can be stated as follows.
   Theorem 3.2A If  A MR[m, n : gthen  A+ MR[n, m : (mn)-1 g+] .
   Case 2. Let in block form (2.1)  m1 = m2 = =  mp k and  n1 = n2 = =  nq l . It is clear, that  k = m/p and  l = n/q . Thereby, according to (2.2), we have  M = kIp and  N = lIq where  Ip and  Iq are identity matrices of  pth and  qth order, respectively. In this case our formulae (3.1) and (3.2) become extremely simple, i.e.

Thus, we get
   Theorem 3.3  If  A BMR(kIp, lIq ; R , C) then  A+ BMR(lIq, kIp ; R+ , C+) .
   Next, from (3.3) we obtain

   Consequently, the result may be formulated as follows.
   Theorem 3.3A  If  A BMR[kIp, lIq : Gthen  A+ BMR[lIq, kIp : (kl)-1 G+] .

   Yerevan State University

References

    1. Weiner L. M. - Amer. Math. Monthly 1955. V. 62. P. 237-239.
    2. Schmidt K., Trenkler G. - Int. J. Math. Educ. Sci. Technol. 2001. V. 32. No. 4. P. 624-629.
    3. Strang G. Linear Algebra and its Applications. Academic Press. 1976.
    4. Hakopian Yu. R, Eloyan A. N., Khachatryan D. E. - Algebra, Geometry & their Applications. Seminar Proceedings. Yerevan State University. 2004. V. 3-4. P. 30-34.