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  Advances in Linear Algebra & Matrix Theory, 2012, 2, 1-11

  doi:10.4236/alamt.2012.21001 Published Online March 2012 (http://www.SciRP.org/journal/alamt)

  

Schur Complement of con-s-k-EP Matrices

Bagyalakshmi Karuna Nithi Muthugobal

  Ramanujan Research Centre, Department of Mathematics, Government Arts College (Autonomous), Kumbakonam, India Email: bkn.math@gmail.com

  Received February 8, 2012; revised March 8, 2012; accepted March 15, 2012

  ABSTRACT Necessary and sufficient conditions for a schur complement of a con-s-k-EP matrix to be con-s-k-EP are determined.

  Further it is shown that in a con-s-k-EP matrix, every secondary sub matrix of rank “r” is con-s-k-EP . We have also

  r r

  discussed the way of expressing a matrix of rank r as a product of con-s-k-EP matrices. Necessary and sufficient condi-

  r tions for products of con-s-k-EP partitioned matrices to be con-s-k-EP are given. r r

  Keywords: con-s-k-EP Matrices; Partitioned Matrices; Schur Complements given. In this sequel, we need the following theorems.

1. Introduction Theorem 1.1 [2] Let C be the space of n × n complex matrices of order n.

  n n  Let A,B  C , then nxn

  Let C be the space of all complex n-tuples. For n S T T

  T S

  • * N AN B   

  R BR A B BA A AC , let A , A , A , A , A , A , R(A), N(A) and n n 

         

  1)

   ρ(A) denote the conjugate, transpose, conjugate transpose, for all AA {1}

  secondary transpose, conjugate secondary transpose, Moo- T T

  N AN BR BR A   B AA B    

     

  re-Penrose inverse, range space, null space and rank of A, 2)

  

  for all AA {1} respectively. A solution X of the equation AXA = A is called

   Theorem 1.2 [3]

  generalized inverses of A and is denoted by A . If

  A B AC , then the unique solution of the equations AXA   n n  * *

  Let, M , then   

  XA ] 

  XA [2] is called C D   the moore penrose inverse of A and is denoted by A . † † † † †

  = A, XAX = X, [ AX ] AX , [ 

    A matrix A is called con-s-k-EP if  A  r and

  r AA B M A CAA B M A   †    

  T T

   

  MN(A) = N (A

  VK) or R(A) = R (KVA ). Throughout this † †

     M A CA M A

     

    paper let “k” be the fixed product of disjoint transposi- tion in S = {1, 2, ···, n} and K be the associated per- T T

  n

   N AN C , N AN B ,

         

  mute- tion matrix. Let us define the function T T

  N M AN C and . N M AN B k xx , x ,  , x . A matrix A = (a ) C is ij nxn      

    k 1 k 2 k n  

         

   s-k symmetric if a a for i, j = 1, 2, ···, n. ij n k j   1,n k i   1

      † †

   

  M DA B M A

  A matrix AC is said to be con-s-k-EP if it satisfies

  nxn s †    

  Also, M    † † the condition Ax   A k ( ) x  0 or equivalently N(A)  

   D C M D M A

  T    

    = N (A

  VK). In addition to that A is con-s-k-EP KVA

  T

  is con-EP or AVK is con-EP and A is con-s-k-EP A

  

   N AN C

      T

  is con-s-k-EP. Moreover A is said to be con-s-k-EP if A is

  r T T T N AN B , N M AN C ,

         

  con-s-k-EP and of rank r. For further properties of

  N M AN B and ,  N DN B         con-s-k-EP matrices one may refer [1]. T T T T

  In this paper we derive the necessary and sufficient  

  N D N C , N M D N B ,  

       

  conditions for a schur complement of a con-s-k-EP ma- N M DN C .

     

  trix to be con-s-k-EP. Further it is shown that in a con-

  A B

    s-k-EP matrix, every secondary submatrix of rank r is

  r

   When  MA , then M and

     

      con-s-k-EP . We have also discussed the way of express- C CA B

  r

    ing a matrix of rank r as a product of con-s-k-EP matri- T T T T

  r

   

  A PA A PC

  ces. Necessary and sufficient conditions for products of

  M ,

     T T T T

  B PA B PC

    con-s-k-EP partitioned matrices to be con-s-k-EP are

  r r

  B. K. N. MUTHUGOBAL T T T T   where K and K are the permutation matrices relative to

  1

  2 where, PAABB A A A C C  .

     

  k and k and let “V” be the permutation matrix with

  1

  2 Theorem 1.3 [4]

  units in its secondary diagonal of order 2r × 2r parti- Let A B ,  C and UC be any nonsingular ma- n nn n tioned in such a way that trix, then, T Tv

   (2.4) 1) R A ( )  R B ( )  R UAUR UBU

  V       v T T  

  2) N A ( )  N B ( )  N UAUN UBU

      Theorem 2.5

  Let S be a matrix of the form 2.2 with

2. Schur Complements of con-s-k-EP Matrices N M C  N M A and N S/  M C   N( M D , )

       

    then the following are equivalent: In this section we consider a 2r × 2r matrix M Partitioned

   

  ν

  in the form, 1) S is a con-s-k-EP matrix with k = k k and V= .

  r

  1

  2

   

  A B

   

  ν

   

  M

   (2.1)  

  C D 2) (M/C) is a con-s-k-EP, -EP.

   

  2   T T S/ M C  is con-s-k where A, B, C and D are all square matrices. If a parti-

  N M C N M D and 

     

  tioned matrix M of the form 2.1 is con-s-k-EP, then in T T NS M C   N M A . general, the schur complement of C in M, that is (M/C) is    

    not con-s-k-EP. Here, necessary and sufficient conditions

  3) Both the matrices for (M/C) to be con-s-k-EP are obtained for the class  

  M CM C M D       

   M   C and  M   C , analogous to that   and

         

       

  M AS M C   S M C       

          of results in [5] Now we consider the matrix . are con-s-k-EP .

  r

   M A M B

      Proof:

  S

     (2.2)

  M C M B    

  Since S is con-s-k-EP with k=k k , KVS is Con-EP   r

  1

  2

   K o1 the matrix formed by the Schur complements of M over and K where K and K are permutation

  1

  2

    

  o K

A, B, C and D respectively. This is also a partitioned ma-

  2

    trix. If a partitioned matrix S of the form 2.2 is con-s-

   o

  ν

  k-EP, then in general, Schur complement of (M/C) in S, matrices associated with k and k and V .

  1

  2

    

  ν o

    that is [S/(M/C)] is not con-s-k-EP. Here, the necessary and

  

  sufficient conditions for [S/(M/C)] to be con-s-k-EP  

  I M A M C   

  Consider P    , are obtained for the class  S   M C and

     

   

  O

  I

     S   M C , analogous to that of results in [5]

     

  As an application, a decomposition of a partitioned 

  I O

   

  Q  and  matrix into a sum of con-s-k-EP matrices is obtained. r

   

  M DS M C

  I    

     

  Further it is shown that in a con-s-k-EP matrix, every

  r

  secondary sub matrix of rank r, is con-s-k-EP Through- r.

   

  OS M C   

    out this section let k = k k with.

  1

  2 L    .

   

  M C O  

     K

  1 K  (2.3) Clearly P and Q are non singular.

   

  K

  2

    Now,

  

  I O

       OS M C

   K O  O ν1 I M A M C    

     KVPQL     

      

       

  O K ν O 2 O

  

I M DS M C

  I M C O

             

       

    

   

  IM A M C M DS M CM A M C   OS M CO K          

   1 ν       

   

    

   

  

    

  

  K 2 ν O M C O

   

    M DS M CI  

     

    

  

  

   

  

K K    S M C

1   ν M C ν M D S M C 1      

      

   

    

   

  K ν M A M C M C K ν S M C    M A M C M DS M C   S M C2      2           

        

   B. K. N. MUTHUGOBAL Since,

     N M C N M A

      T T N M C N M D  .

    

      

    

      (using (M/C) is con-s-k-EP

  r ).

  Therefore

  After substituting

  ( ) T T T

           M B M C M A M C M S

  

     

    D and using

          2 2 T T M A νK M A

  νK M C M C S S

  N M C N M D νK N M C νK N M D νK

        1 1 1

          2 2 T T M B νK M B

     

          

       As the equation (at the bottom of this page). and since

              1 1 1 T 1 T T

  ρ K ν M C ρ K ν M C ρ M C νK ρ M C N M C N M C νK

      

     

     

   is follows that

    

   Hence, (M/C) is con-s-k-EP. From

          1 1

  , T T

  M D νK M D νK M C

  M C

           in

  νK M C M C S S

    

          

     

      

          

          

   

   

   

     

   

                    1 1 1 1 T T 1 T T T

  νK M C M C K ν M C K

  ν M C

M C M C

N M C N K ν M C

  N M C νK

           

       

       

  

  O M C M C νK M A

          

                 

         

                          1 2 1 2 1 2 1 2 1 2 1 2 T T 1 T T T T T T T T T T T T T M C

  νK M A νK S VK S VKL L

  M D νK M B νK

  νK O S M C M C O S M C O M D νK M B

       

  νK M C νK M C M C A

  νK S M C S M C M D νK M C

  M C B νK S M C S M C M C νK M C

     

     

     

    

     

  

  , therefore,

     

          

   Since,

            

  S M C M B M A M C M D

       

                 

  K ν S M C M A M C M D S M C S M C K

          1 1 2 2 1 2 1 2 K ν M C K

  ν M D KVPQL K ν M A

  K ν M B O K ν M A M B K ν O M C M D

  K O M A M O ν O K M C M D ν O

  KVS

     

  ν M B  

               2 2 .

     

    , we have by Theorem 1.1

  , by Theorem 1.1 we have

         - M A = M A M C M C ,

  that is,

         2 2 - K ν M A =K ν M A M C M C

  . Since,

      N S M C N M D   

          M D M D S M C S M C

  Also,

  

          .

  That is,

          1 1 K ν M D K

  ν M D S M C S M C

           .

   

    

      

  We choose

  ( ) ( ) ( ) ( T T

  N KVS N KVS N L N S VK   

  Therefore, by using Theorem 1.1 again we get, T T

  S VK S VKL L

   holds for every . L

  

                 

  ρ KVS ρ L

          1 2 2 1 1 2 1 T 2 T T T T T T T T T T M A M B K O O ν S VK

  M C M D O K ν O M A M C O νK

  νK O M B M D M C

  νK M A νK M D νK M B νK

       

        

   ( ) ( N KVS N L  But S is con-s-k-EP. Therefore, KVS is con-EP (By Theorem 2.11 [1]).

  Hence and ( ) (

    

     ) ).

     

     

     

      

  

  B

  )

    Thus KVS is factorized as KVS = KVPQL.

  L

  as

      O M C L S M C O

    

     

    

     

        

  • EP r.

         

       

        

    

  (2.6)

        

     

     

                   

       

             

       

     

     

  

 

           

     

  

     

 

      

   

         

   

  K ν M C KVS K

  ν M C K ν M A K ν M C K

  ν M C K ν M C

  M C M D M C M A K ν M C M C M D M C νK

  M C M A K ν M C M C νK

  S S S S S

  S S S

   

    

     

     

     

     

   

     

     

     

      1 1 1 † 1 †† 2 † † 1 1 1 2 1 2 2 1 2 † 2 † 1 2 2 1 K ν M C K ν M C

  ν M D K ν M C

   

   

   

   

   

   

   

   

   

   

   

   

   

   

    

    

   

   

   

  K ν M C M C K

  S S

  ν M C M C M D M C M D M C νK M A K

  ν M C K ν M D M C

  K ν M D M C νK

  M A K ν M C KVS KVS K ν M A M C

  K ν M A K ν M C K ν M A M C

  M D M C νK M D M C M D K ν M C

  K ν M B M C K ν M B M C M A K ν M C S S S S S S

     

  2 νK

       

      

      

   

     

     

     

    

  K ν M C K ν M A

  ν M D K ν M C K

  B. K. N. MUTHUGOBAL We get,

      

              

      

              2 2 T 2 T T

  S M C νK S M C νK S M C S M C N S M C N S M C νK

  

            

            

  By Theorem 1.1 and since

      

     

      2 T T ρ K ν S M C ρ S M C ρ S M C       

        we get,

     

    2 T N K ν S M C N S M C     

     

     

     

   

           

    

          2 2 T T M B νK M B

  νK M C M C S S

        

  

          

             2 T 2 T M C M A M C M B νK M C

  M A M C M B νK M C M C S S S S

      

  S S S S  

        

       

    

  

                        2 2 2 T 2 T T T M C

  νK M A M C M B νK M C

  νK M C M C M A M C M B νK M C M C

    2 N S M C νK N S M C

    S M C   

  ν M C K ν M C K

     

     T T N S M C N M A   

   

  

  holds, according to the assumption by applying Theorem 1.2,

    KVS is given

  by the formula

         

       

      N S M C N M D   

     

     

     

     

     

                         

                    1 1 1 1 1 2 † † 2 2 1 † † 2 2 1 2 † † † † 1 1 2 † † 1 2 K

    and

      T T N M C N M D  ,

    is con-s-k

      2 2 2 2 2 2 T 2 T T T T T T T T M A νK M A

  2

  Further

               

     

     

     

     

  νK S M C S M C N S M C N M A νK N K ν S M C

      N M C N M A  ,

  N M A νK N S M C νK N M A νK N S M C N M A

  

          

       

       

       

       

  Thus 2) holds 2) 1). Since 

   

  (b)

  Thus

     

      2 2 2 1 K ν M C K ν M C

  K ν M C K ν

  M C S S S

     

          

         † † KVS KVS KVS KVS† † † † † † KVSS VK S VKKVS

     

  KVSS VK S S KVSS S SKV

        S  is con-s-k-EP (by Theorem 2.11 [1]).

  Thus 1) holds 2)  3)

        2 2 2 K ν M C K ν M A

  K ν M C S

       

     

     

  . We have,

    1 K ν M C

    1 K ν M C

  ,

   M A M A M C M C KVS KVS

  reduces to the form As the Equation (b) below.

  Since,

    M C is con-s-k

  1

   is con-EP.

  r

  Therefore we have

          1 1 1 1 K ν M C K v M C K v M C K v M C

              

      Similarly, since

    M C S

      is con-s-k

  2

    is con-EP if and only if

  and

           and

  ν M C K ν M C KVS KVS

   

  (a)

        

     

     

     

1 1 2 2 K

  K ν M C K ν

   

  M C S S

   

   

    

   

         

   

         

    

    2 K ν M C

       

  S

      are con-EP. Therefore,

        1 2 M C K M A M C K S ν ν

      

     

   

     

   

        

     

     

     

1 1 2 2 K ν M C K ν M C

  KVS KVS K ν M C K

  ν M C S S

   

           † †

          M D M D M C M C S S

  B. K. N. MUTHUGOBAL According to Theorem 1.1 the assumptions N(M/C)

     

                  2 2 2 1 1 K ν M B K ν M C

  K ν M C K ν M C

  K ν M D S   

    

  Therefore

         

     2 2 2 1 1 K

  

  ν M C K ν M B

  K ν M A K ν M C K ν

   M D S

     

   

  ν M C K ν M D K ν M B K ν M C S

  Hence

    M C

   that is,

      and

  ν M C K ν M C

  ν M A K ν M A K

      2 2 1 1 K

     

     

     

  ν M D K ν M D K ν M C K ν M C S S     

    is invariant for every choice of

      1 1 2 2 K

     

         

  

  N(M/A) and       T T

  M C M D M N N S    C

      

  • EP
  • EP

      and

  and

  ν M A K ν S M C S M C

  νK K ν 2 M A KK ν M C M C M A S S

          

          

          M A M C M C M S S     

     

  A

              1 1 1 1 1 K ν M D K

  That is

  ν M C M C νK K ν M D K ν M C M C M D

   

         M D M C M C M D  ,

    KVS

  

  KVS reduces to the form, As the Equation (a) below.

  Again using

                2 2 2 2 K

  ν M C K v M D  .

       

      

      2 2 2 2 K

     

       

  Further using

        

           M A M C M D M B M C S

   

  M C S

  ν M D K ν M C K

  ν M B M C M D K ν M B

             2 2 K

   

     

      1 1 1 1 K

    2 1 1 2 2 K ν M A K

                 

  ν M A K ν M C K ν M C K v M A S S     

  • EP an
  • EP and

  • EP and
  • s-k-EP with k = k k where T

       

  1

  2

  2 1 1 1

  1

  2

  1

  1 1 1

  2

  1 S  

     

    

     

    

     

  1

     

        

  1

  1

  1

  1 K  

      

       

      

      

      

      

  1

      

   

       

         

      

      

      

      

     

  1

  1

  1

  2 ,

  2 1 1 1

  M A M B

      

      

  1

  ,

     

  1

  2

  1

  1 , 1 1

  2

  1 M C M D

      

     

     ,

    

          M A M B S M C M D

  1

  1

  M

  B M A S

      

        

      

    

      

  ,  KVS is con-EP  S is con-

  A M B M C M A

         with

  

      N M C N M A  and

           1 M C M M D M C  

  

  

      

   

  1

  1

  2

  1 M A

      

    ,

   

  1

  2 1 1

  M B

      

    

     

  2 KVS   

  1 V  

      

      

       

      

      

      

      

  1

  1

  1

  1 KV  

      

       

      

      

  1

      

  Now

  KVS is symmetric of rank 3 s-k-EP.

  1 1

  2

  1

  1

  2

  1

  1

  2 1 1 1

  1

  1

      

    1 0 1 1 1 1 0 1 1 1 1 0

  0 1 1 1

    .

   

    M C is con-s-k

  1

    S M C  

    is

  2

  rther con-s-k -EP fu

     T N M C N

   T M D and

     T N S M C N M   

  ce

   D

  This proves the equivalen of 2) and 3). The proof is co

     

  .7

  atrix of the form (2.2) with mplete.

  Theorem 2

  Let S be a m

      T T M C N M D  and N

      T N M C N M S   

    , then the following are equivalent.

  A

  1) S is con

  1 2 1 K 2 K K

     and

  ν

    

    is con-s-k-EP if and only if

     

  V ν

  Further

  B. K. N. MUTHUGOBAL is con-EP if and only if

   1 K νM C and

    2 K ν M C are con-EP.

        M C M A M C S

    

     

     is con-s-k-EP if and only if

    M C is con-s-k

  1

    M C S  

    is con-s-k

  2 -EP.

      M C N M A

        M C M D S M C

  and

  N     T

  C N M D

    Also T

  N M S         1 1 2 K

  ν M C K ν M D K

  ν S M C

       

     

    is con-EP if and only if and

    2 K ν S M C

      and con-EP.

  Therefore,

     

     

  2)

  1  1 1

  e matrix -EP. 3) Th

        M C M A M C S

       

     

    is con-s-k- EP.

  Rem

  ons taken on S in Theorem 2.6 and Theo-

  ark 2.9

  The conditi rem 2.7 are essential. This is illustrated in the following example.

  Let

  A B MC D

       

  A

    is con-s- k

      

  ,  1 1

  0 1

  B

      

    ,

  1 1 0 1

  C

      

    ,

  1 0 1 1

  D

      

  2

    M C S  

  1

  • EP. Further and

    M C is con-s-k

  1

    M C S  

    is con-s-k

  • EP. Further

  2

      N M C NM A and

      N M C N M D S   

    3) Both the matrices

        M C M A M C S

       

     

    and

        M C M D S M C

      are con-s-k-EP.

       

  Proof

  follows from Theorem 2.5 and from the fact that

  S

  he This is con-s-k-EP  S

  T is con-s-k-EP.

  In particular, w n

      T M D M A  , we got the fol-

  lo

          T M     T A .

  uiva

  N M C N M S   

    Then the following are eq lent.

  1) S is a con-s-k-EP matrix.

  2) (M/C) is con-s-k d

2.8 Let S = wing. Corollary

  Now

  S

   

  1 1 2

  M D

      

    

          M A M B S M C M D

      

    0 1 2 1 1

  1

  1

  1

  1

  1 1 1 2

        

     

     

     

      

  

  T

  2.12 However this fails if we omit the condi con-s-k-EP.

  For example, tion that S is  

         

       

      

    

  1

  1

    ,

  1 M C    

  1 K  

       

  1  

      

  , 1 1 0 1

  D

      

    1 1 1 0 1

  1 1 1 1 1 0 1

  M

       

           

    

       

       

   

  1

   

  0 1 , , ,

  1 A B C D M A

     

    ,

  

   

  2

  1 M B   

     

    ,

   

  1

  1

  1

       

    

  ν M D

    is not con-EP.

  Therefore S is not con-s-k-E Here P.

    1

  1

  1

  1

  1 K

  ν M C

      

    

    is con-EP.

    M C  is con-s-k-EP.

        1 1 T K ν M D K

   ,

      

        1 1 T K

  ν M D K , T M D ν

  

    1 1 T M D νK A νK

   ,

      ν M C ν M A

   ,

      T T ν M C ν M D and  .

  Hence

    S M C  

    is independent of the choice of

    M

   C .

       

       

         

       

    

       

       

       

   

  1

  1

  1

  1 V  

       

         

    

       

       

    

   

  1 1 1 2

  1

  1

  1

  1

  1

  1

  2 KVS   

     

       

      

   

    

  1

  1 B  

  B. K. N. MUTHUGOBAL

   M C  is con-s-k

  3 3

  3 S M C  

      

     

  Hence

    2

  0 3 3 3

  K ν S M C

     

    

      is con-EP, that is

    S

  2 -EP.

   

  , Also

     1

  1 1

  K ν M

    

   

  

  1 2 1 1

  1

  2 M C C     

     

     is con-EP.

   

    

  1

   

  , C  

    

  ,  

  1 A  

  1 1

   

  D

  

  C

  , where

     

  A B M

  1

    

  1

  2

  1 M D   

      

  ,

    1

  1

  2

  1

  1

  1

  3 M C

  

  is con-EP

  • Moreo

  M A S M C M C

     

  M C

  

  M D S M C

     is not con-s-k-EP.

  Thus the Theorem 2.5 a s well as the corollary 2.8 fail.

  Re

  x of the form 2.2 and k = k

  1

  k

  2

  w valent. nd the Theorem 2.7 a

  marks 2.10

  We conclude from Theorem 2.5 and Theorem 2.7 that for a con-s-k-EP matri here 1 2 k k

  K

     and

      

  N

    Let

     

  N M C N M D N S M C N M A

  , T T T

         

  

  2.11 N

  ν ν ν

      

  S M C N M D

  , N M C M A

         

     the following are equi

     

  

       

  3 K

   N M

  1

  2

  V

  1

  A      

  Further . T

      

  N S M C N M

  S M D      T

    and

      N N M D  

   . But N M

      T T D N M C

  and

     C N M A

  Therefore,

     

    is not con-EP.

       

      

    

       

      

    

  1

      

         

    1 K ν M C

       

    M C  is con-s- EP.

  ver k

  2 1 1 1 1 3 3

  

  B. K. N. MUTHUGOBAL Let S be of the form 2.2 with   r .

  ρ S   ρ M C  

   S M C   M BM A M C M D

          

    Th en S is con-s-k-EP and K and V are of the form 2.3

  r

  

  2

  1     and 2.4 if and only if (M/C) is con-s-k -EP and

  1 r M B  , M A  ,

     

      † † T

  1

  1     M A M CM C M D .

  νK νK     1     2

   

  1 1  1 

  1    1  

  Proof

   

  M D , M C    

       1 2 

  1

  1

  2      Let S be of the form 2.2 and let k= k k with

  1

  2

   k   ν1   1 

     S M C

   

    

  K  and  then ν

     

  k 2 ν

      

  1

  1  

   K K1 ν M C   1 ν M D  

  1

  1   KVS .

     K   is not con-EP. 2   ν S M C   

   

  K K 2 ν M A   2 ν M B  

   1    

  Since ρ Sρ M Cr ,   S M C  n-s -EP.    

  2  

    is not co -k T T  

  ρ KVS ρ K ν M C r by [ 6]   1  

   

  Also, N S M C    N M D . But

     

    T T  

  N M C N M A , N M C N M D and         N S M C    N M D .

     

   

  KVS K ν M C 1     Thus, 2. 12 holds while 2.11 fails.

  R emark 2.13 K      S M C   . 2 ν S M C    

      It is clear by Remark 2. 10 that for a con-s-k-EP mar- † By Theorem 1.1 these relation equivalent to trix S, formula 2.6 gives KVS if and only if either

   

  

  K ν M A K ν M A M C , 2   2    2. 11 or 2.12 holds. Corollary 2.14

  KK M D and 1   ν M D ν M C M C 1     

  Let S be a matrix of the form 2.2 with K and V are of

  the forms 2.3 and 2.4 respe ctively, for which KVS is

  KK M D 2   ν M B ν M A M C 2     

   

  given by the formula then S is con-s-k-EP if and only if Let us consider the matrices both (M/C) and  S M C  EP.

   

      and con-s-k-

  I M A M C   

  P

    

  Pr oof

   

  I

    This follows from Theorem 2.5 and usin g Rem ark

  2.13. Now we proceed to prove the most important  

   

  I M C M D    

  Q and L

        Theorem.

   

  M C I  

     

  Theore m 2.15 † † K    

        1 ν I M A M C

  I M C M D       

  KVPLQ

         

     

  K M C 2 ν I  

  I

           

  

  K  

   νM A M C M C 1 I M C M D

          

     

       

  K 2 ν M C

  I

       

    † † †

  

  

K M A M C M C M A M C M C M C M C

   ν             1 

    

  K 

   2 νM C M C M C M D

        

    

  

  K K M D 1   ν M C ν M C M C 1     

   

   † †

  

  K K M D 2 ν M A M C M C 2 ν M A M C         

   

   K K1   ν M C ν M D 1  

   

     K   M A M B1 ν     

  K K 2   ν M A ν M B 2  

      

  

  K M C M D 2 ν    

       

  

  KVS

  it follows that

    

  ν M C M C M D  † 1 1 T T K

  and

     

        1 1 T T K

  ν M C K ν M C M C M C  From

  B. K. N. MUTHUGOBAL Thus KVS can be factorized as KVS = KVPLQ. Since

        1 T N M C N K

  ν M C

     

     

        1 T N M C N M C νK M C   

  is con-s-k-EP.

  Since (M/C) is con-s-k

     [since LVK = KVL].

  VK=Q. Therefore,         T T T KVS KVPLKVP VK KVP LKV KVP KVP KVL KVP

  T

  We have KVP

  KVP = (KVQ) T .

      2 1 T T K ν M C K

  • EP

  KVPLQ, one choice of  

     

  r

  When (M/A) is non singlular, KV(M/A) is automati- cally con-EP

  Mark 2.16

  

     

         † † 1 2 T M A M C νK M C M D νK

   

    

  ν M C M D T

         † † 1 2 M A M C νK K

  

     

         † † 2 1 T K ν M A M C M C M D νK

   (By theorem 2.11 [1])

    

  r

     

    

   

  M D

  M D M C M C M C K ν M D M C M C M C νK M D M C νK K ν M C

  ν M A M C M D M C K ν M C M C

                     2 † † 1 † † 1 † † 1 1 1 T T T T T T T T T T T K

       

             

          

  Now,

  it follows that.

  ν M C M C M D

      2 1 T T K ν M A K

  and (M/A) is con-s-k-EP

  and Theorem 2.15 llowing.

     

    

    

    

    

     

     

  K M C M C M D

  K ν M D M C M C K ν

  ν M 2 2 C M C M C M C M D ν M D K ν M B

  ν M A K ν M C K

      1 1 1 1 † † 1 T T T T 1 T T T T K ν M C K

         

                     

     

             

    

  Corollary 2.17

  and

  Let S be of the form 2.2 with C non singular and reduces to the fo

   

  [ ] ρ S ρ M C  . Then S is con-s-k-EP with K = k

  1

  k

  2

         2 T M A M C ν 1 ν

   

  ν K ν

   .

  M C

    

  

  M D νK

     

  From

  ( ) T

  M C is con-s-k-EP r .

  N KVS N KVS

    

    

    T N S N S VK

    

     S  is con-s-k-EP (Theorem 2.11 of [1]). Since

    ρ S  , S is con-s-k-EP r r .

      1 1 KVS Q P VK KVS

    

   

    

     is con-EP

  

  

  M C

  

    

     By Theorem 1.1

  N KVS N KVS  

  ( ) T

     T  

   

  Therefore (Theorem 2.11 of [1])

  ρ M Cr  

    .

  Since

  1

  r

  . We have k

  1 v(M/C) is

  con-EP

  r .

  By Theorem 1.3 Conversely, let us assume that S is con-s-k-EP

     

  r .

  Since S is con-s-k-EP

  N L N L VK      T

  r

  N KVL N KVL            T T

  , KVS is con-EP

  r

  . Since KVS =

  N KVP KVL KVP N KVP KVL KVP

  ( ) T

VS KVS KVS

   T

   1

  M C  

  ν M C K ν M C M C

                 

        As the equation (at the bottom of this page). or conversely,

  

          1 1 1 K K ν 2 2 ν M C M D P VK K ν M A K ν M B

        

  Q

  M C

         . T KVS K

  ν M A K ν  

     2 2 K

     

       

  K ν M A K ν M B

  ν M C K ν M D M B

                  1 1 1 1 2 2 T T K

  That is,

  K ν M C K ν M D

  • EP nce [M/C] is arbitrary it follows that every secondary submatrix of rank r is con-s-k-EP

  (

  2

  S = S

  1

  2

  and con-s-k

  1

  r

  . The converse is ob- vious. The condition composed into complementary sum and S of con-s- re given. S

  1

  and S

  and called comp

  r

        1 2 .

  ρ S ρ S ρ S

   

  Theorem 2.21

  Let S be of the form 2.2 with

       

  , ρ S ρ M C ρ S M C    

   

           

  Si s under which a partitioned matrix is de k-EP matrices a le- mentary summands of S if

  r

  . That is (M/C) is

      1

  is con-EP

  ν M C

  r     1 K

  m 2.15 that is con-EP . Now we conclude from Theore

    T KVS

  By [4]

  . ρ KVS ρ K ν M C r      

  • S

  • EP and
  • EP matrices such that

     

   with

  S M C M B M A M C M D    

  ν K ν M A K M B

  ν M C K ν M D KVS P KVS P

              1 1 2 T 2 T K

  , M C r       tation matrix P such that, then there exists a permu-

  ρ KVS ρ K ν

  [ ]

    1

  such that be any Principal submatrix of KVS

    1 K ν M C

   

  1

    where and K is of the form 2.3 and V is of the form 2.4. If (M/C) is con-s-k

  2

    S M C  

    is con-s-k

  • EP and

           † † 1 2 T

    1 1 T N M C vK N M A M C M C vK  and

  M D S M A

   

     

    .

  Tak

          

  M C NM A M C M A       

               

    

          

           

      

   

                  

               1 † † † † 1 † † S M C M A M C M D M A M M A M C M D M A M C M C M A M C M D M A M C M C M A M C M M A M C M D

               † † 1

  C M D M C M C M C M D M C M C M C M D  

    ing into account that 2 I M C M C

     

  M A M C νK M C M D νK

  

   and

            † † 2 1 T M D S M C

  νK S M C M C νK

           then S can be decomposed into complementary sum- m

  Let us consider the matrices, ands of con-s-k-EP matrices.

  Proof                  1

  M C M C M C M D S M A M C M C M

    

  A M C M D

    

       and

      

         

     

  )

  S M C

  matr

    

  I M C M C

  r

  r

  1

  1

  1

  1

  1

  1

  1

  1

  1

  1

  1

  However,

  1

  1

  1

  1

  1

  1

  1 K V M C  

       

     

    1

  KVS is not con-EP Therefore S is not Con-s-k-EP.

        

  2 But [ ] [ ] ρ KVS ρ S

  B. K. N. MUTHUGOBAL

  Remark 2.18

  When k(i) = i, we have K

  1

  = K

  2

  = I, then the Theorem 2.15 reduces to the result for con-s-EP matrices. When KV = I then Theorem 2.15 reduces to Theorem 3 of [5].

  Remark 2.19

  Theorem 2.15 fails if we relax the condition on the rank of S. For example, let us consider the matrix S and K given in Remark 2.10,

    .

  . M A   

     

    1

  1,

  ρ K V M C ρ M C

   

     

    1

  ( ) ( )

  ρ KVS ρ K ν M A ρ S ρ

             

          is con-EP.

  ma n-EP

  with K = K K , where

  , M

      1 1 0 M C M D νK

  

      . 2 1 0

    

    Thus the theorem fails.

  Corollary 2.20

  Les S be a 2r x 2r matrix of rank r. Thus S is con-s-k-EP

  r

  1

    

  2 1 2 K K

        and V =

  ν ν

    

      every secondary sub matrix of S of rank r is con-s-k-EP

  r .

  trix then KVS is an co ix by Theorem 2.11 [1]. Let

  Proof

  Suppose S is con-s-k-EP

     

     

  Therefore (M/C) is con-s-k

     

  1

    1

  1

  1

  1

  1

  1

  2 M C

  

    

  νK

     

  ,

     1 1

  1

  1

  1

  1

  1

  2 A M C

  M C M D D

  • EP and

            

  S M C M A K ν I M C M C S M C

    1 1 T T T T T

    

  I M C M C νK

                  M A

   1  

     

       

      

      

           

     

        

     

      

        

               

     

       

     

      

  S M C M A K ν K ν M C M C

  I M C M C K ν S M C v A K ν M C M C K ν

  I M C M C S M C v A

  I M C M C S M C M A νK S M C v A νK

      

     

  ( T T T T T T T T

  [6]

  of Mathem

  N 1974. hugobal,

  thematics, Vol. 26, No. 1

  1 Generaliza eme Moore-Pe

  [5]

  Cambridge Philosophica pp. 17-19.

  R. P Matr

    1971. en “O ximate Solutions of Linear

  On Sums of con-s-k-EP Matrix Thai Journal atics, in Press, 2012.

  [8] S. Krishnamoorthy, K. Gunasekaran and B. K. N. Mut-

  A. B. Isral and T. . E. Greviue, “Generalized Inverses Theory and Applications,” Wiley and Sons, New York,

  D. H. Carlson, E. Haynesworth and T. H. Markham, “A tion of the Schur Compl nt by Means of the nrose Inverse,” SIAM Journal on Applied Ma- , 1974, pp. 169-175. [7]

  Mathematica Hungarica, Vol. 16, No. 3, 985, pp. 193- 200.

      

  A. R. Meenakshi, “On Schur Complements in an EP Ma- trix, Periodica, Mathematica Hungarica,” Periodica

  2, No. 1, 1969, pp. 87-103.

  EP r Matrices,” Linear Algebra and Its Applications, Vol.

  [4] T. S. Baskett and I. J. Katz, “Theorems on Products of

  l Society, Vol. 52, No. 1, 1959,

  C. R. Rao and S. K. Mitra, “Generalized Inverse of Ma- trices and Its Applications,” Wiley and Sons, New York, [3] rose, n Best Appro ix Equations,” Mathematical Proceedings of the

  [2]

  Sciences and Engineering Applications, Vol. 5, No. 1, 2011, pp. 353-364.

  [1] S. Krishnamoorthy, K. Gunasekaran and B. K. N. Mut- hugobal, “con-s-k-EP Matries,” Journal of Mathematical

  REFERENCES

     

    

  I M C M C M D S M C νK

                                             2 † † 1 1 1 1 1 1

  B. K. N. MUTHUGOBAL We obtain by [6] that

  1

  I M C

  S M C N

  , N M C M D   

         

  Theorem 1 of [6], gives

       

  ρ S ρ M C ρ S M C

  ,

       

  Since

  1 -EP.

  is con-s-k

  We have by Theorem 2.15, that is

    

   

  νK M A M

  S 1 † † 1 M A M C M C M C νK M A M C M

   

  νK M C M C M C M D νK

    † † 1 1 † † 2 T T C M C C νK M C M D

          

                  

        

  1

  . ρ S ρ M C  Since (M/C) is con-s-k

      1

      

            T T

  We obtain,

       

         1 1 M C M C K ν K ν M C M C  

  Further using

  ρ

   

  ρ S ρ S

  . S

        1 2

  Thus Thus by [7] we get

      

  ρ S ρ S M C

      2 .

   

  0. S S M C

  N S M C N M C

    2

  Therefore,

      

     

     

    

  I M C M C

  I M C M C M D S M C

            

      and

       

  I M C M C

  ,”

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