ELA
GROUP INVERSE EXTENSIONS OF CERTAIN
M-MATRIX PROPERTIES
∗
K. APPI REDDY
†
, T. KURMAYYA
†
, AND K.C. SIVAKUMAR
‡
Abstract. In this article, generalizations of certain M -matrix properties are proved for the
group generalized inverse. The proofs use the notion of proper splittings of one type or the other.
In deriving certain results, a recently introduced notion of a B
#
-splitting is used. Applications in
obtaining comparison results for the spectral radii of matrices are presented.
Key words. M-Matrix, Group inverse, Proper splitting, Ps eudo regular splitting, Weak pseudo
regular splitting, B
#
-Splitting.
AMS subject classifications. 47B37, 15A09.
1. Introduction and motivation. Let R
n×n
denote the space of all real ma-
trices with n rows and n columns. A matrix A ∈ R
n×n
is called a Z-matrix if the
off-diagonal entries of A are nonpositive. A Z-matrix A can be written as A = s I −B,
where s ≥ 0 and B ≥ 0. A Z-matrix A is called an M -matrix if s ≥ ρ(B), where ρ(B)
denotes the spectral radius of B, viz., the maximum of the moduli of the eigenvalues
of B. It is well known that if s > ρ(B) in the representation described above, then M
is invertible and M
−1
≥ 0. We recall that if C is a matrix, by C ≥ 0 we mean that all
the entries of C are nonnegative. In fact, ther e are many interesting characterizations
of invertible M-matrices. The book by Berman and Plemmons [6] records more than
fifty equivalent conditions . For our purpose , we recall the following result:
Theorem 1.1. Let A ∈ R
n×n
be a Z-matrix with the representation A = sI − B.
Then the following statements are equivalent:
(a) A is invertible and A
−1
≥ 0.
(b) There exists x such that all the entries of x and Ax are positive.
(c) A is an M-matrix with s > ρ(B).
Let us consider square matrices satisfying condition (a) of the above theorem.
∗
Received by the editors on February 17, 2016. Accepted for publication on September 11, 2016.
Handling Editor: Oskar Mari a Baksalary.
†
Department of Mathematics, National Institute of Technology Warangal, Warangal - 506 004,
India (appireddy.kusuma@nitw.ac.in, kurmayya@nitw.ac.in).
‡
Department of Mathematics, Indian Institute of Technology Madras, Chennai - 600 036, India
(kcskumar@iitm.ac.in).
686
Electronic Journal of Linear Algebra ISSN 1081-3810
A publication of the International Linear Algebra Society
Volume 31, pp. 686-705, November 2016
http:/repository.uwyo.edu/ela
ELA
Group Inverse Extensions of Certain M -Matrix Properties 687
Such matrices are referr e d to as inverse positive matrices and are related to a notion
called monotonicity. A square real matrix A is called monotone if Ax ≥ 0 implies
x ≥ 0. Here , for y ∈ R
n
= R
n×1
, we use y ≥ 0 if all the entries of y are nonneg ative.
The concept o f monotonicity was first proposed by Collatz (see [7], for instance),
in connection with the application of finite difference methods for solving elliptic
partial differential equations. He showed that a matrix is monotone if and only
if it is invertible and the inverse is entrywise nonnegative. Hence, monotonicity is
equiva le nt to inverse po sitivity. Thus, another sta tement which is equivalent to the
three statements of Theorem 1.1 is: Ax ≥ 0 implies x ≥ 0.
The notion of mono tonicity has b een extended in a grea t variety of ways. Since
these generalizations are too many to be included, we only present a brief review,
here. Traditionally, splittings of matrice s have been used in studying these e xtensions.
For A ∈ R
n×n
, a decompositio n A = U − V , where U is nonsingular, is referred
to a s a splitting of A. With s uch a splitting, one associates an iterative sequence
x
k+1
= Hx
k
+ c, where H = U
−1
V is called the iteration matrix and c = U
−1
b,
for a nonnegative integer k and given an initial vector x
0
. It is well known that
this sequence converges to the unique solution of the system Ax = b (irrespective
of the choice of the initial vector x
0
) if and only if ρ(H) < 1. It is well known
that standard iterative methods a rise from different choices of U and V . For more
details one could refer to the books [6] and [16]. Next, we turn our attention to
two important types of splittings. A splitting A = U − V where (U is invertible)
U
−1
≥ 0 and V ≥ 0 is called a regular splitting. This was propos ed by Varga [16],
among others and it was shown that A is inverse p ositive if and only if for any regular
splitting A = U − V , one has ρ(U
−1
V ) < 1. A splitting A = U − V where (U
is invertible) U
−1
≥ 0 and U
−1
V ≥ 0 is calle d a weak regular splitting. This was
proposed by Ortega and Rheinboldt [13]. (Clearly, any regular splitting is a weak
regular splitting.) They showed that A is inverse positive if and only if for any weak
regular splitting A = U − V , one has ρ(U
−1
V ) < 1. This establishes a c onnection
between inverse positivity and convergence of an iterative scheme. It is pertinent to
point out the fact that if A is an invertible M -matrix with the usual representation
A = sI − B, then ρ
1
s
B
< 1. This observation immediately implies that the
two types of splittings disc ussed here are genuine generalizations of representations
of M -matrices. Let us also record the following: A splitting A = U − V where (U is
invertible) U ≥ 0, U
−1
≥ 0 and V ≥ 0 is called a completely regular splitting [1]. This
notion was used to prove the following result. This result gives a sufficient condition
for a matrix to b e inverse positive.
Theorem 1.2. (Proposition 11, [1]) If A = U −V is a completely regular splitting,
and if U
−1
V or V U
−1
has an eigenvector x > 0 corresponding to an eigenvalue λ < 1 ,
then A
−1
≥ 0.
Electronic Journal of Linear Algebra ISSN 1081-3810
A publication of the International Linear Algebra Society
Volume 31, pp. 686-705, November 2016
http:/repository.uwyo.edu/ela
ELA
688 K. Appi Reddy, T. Kurmayya, and K.C. Sivakumar
Let us review some of the importa nt extensions of monotonicity. Mangasaria n
[10] called a rectangular matrix A to be monotone if Ax ≥ 0 ⇒ x ≥ 0. He showed,
using the duality theorem of linear programming, that A is monotone if and only if
A has a nonnegative left inverse. Berma n and Plemmons generalized the concept of
monotonicity in several ways, in a series of papers, where they studied their relation-
ships with no nnegativity of generalized inverses. The book by Berman and Plemmons
[6] documents these results. Several applications are also studied there. In order to
briefly rev ie w these extensions, we need the notion of ge ne ralized inverses.
The Moore-Penrose (generalized) inverse of a matrix A ∈ R
m×n
, is the unique
matrix X ∈ R
n×m
that satisfies the equations: A = AXA, X = XAX, (AX)
T
= AX
and (XA)
T
= XA. It is well known that the Moore- Penrose inverse exists for any
matrix; it is denoted by A
†
. The group (generalized) inverse of a matrix A ∈ R
n×n
(if it exists), denoted by A
#
is the unique matrix X satisfying A = AXA, X = XAX
and AX = XA. A necessary and sufficient condition for A
#
to exist is the condition
rank(A) = rank(A
2
). Of cours e, if A is nonsingular then A
#
= A
†
= A
−1
. Fo r more
details, we refer to [2].
Let us recall the following result that collects two characterizations for the non-
negativity of the two generalized inverses, viz., the Moor e-Penrose inverse and the
group inverse. These were proved in [3, Theorem 2] and [4, Thoerem 1], respectively.
R
n
+
denotes the nonnegative orthant of R
n
.
Theorem 1.3. Let A ∈ R
n×n
. Then the following hold:
(a) A
†
≥ 0 if and only if
Ax ∈ R
n
+
+ N (A
T
) and x ∈ R(A
T
) ⇒ x ≥ 0.
(b) A
#
exists and A
#
≥ 0 if and only if
Ax ∈ R
n
+
+ N (A) and x ∈ R(A) ⇒ x ≥ 0.
It is helpful to observe that A
†
≥ 0 and A
#
≥ 0 are extensions of A
−1
≥ 0
to singular matric es, whereas the second parts of statements (a) and (b) above are
generalizations o f the implication Ax ≥ 0 ⇒ x ≥ 0.
The notion of proper splitting of matrices has proved to be an important tool
in the study of nonnegativity of generalized inverses. Let us recall this br iefly. For
A ∈ R
m×n
, the set of all m × n matrices of reals, we denote the range spac e , the null
space and the transpose of A by R(A), N (A) and A
T
, respectively. A decomposition
A = U −V o f A ∈ R
m×n
is called a proper splitting if R(A) = R(U) and N(A) = N (U ).
This notion was introduced by Berman and Plemmons [5]. Analogous to the invertible
case, with such a splitting, one associates an iterative sequence x
k+1
= Hx
k
+ c,
where (this time) H = U
†
V is (again) called the iteration matrix and c = U
†
b, for
Electronic Journal of Linear Algebra ISSN 1081-3810
A publication of the International Linear Algebra Society
Volume 31, pp. 686-705, November 2016
http:/repository.uwyo.edu/ela
ELA
Group Inverse Extensions of Certain M -Matrix Properties 689
a nonnega tive integer k. Once again, it is well known that this sequence converges
to the unique solution of the system Ax = b (irrespective of the choice of the initial
vector x
0
) if and only if ρ(H) < 1. For details, refer to [6].
The authors in [5] showed that if A = U − V is a proper splitting with U
†
≥ 0
and U
†
V ≥ 0 then ρ(U
†
V ) < 1 if and only if A
†
≥ 0. Note that the type of splitting
given above is a verbatim extension of what we referred to earlier as a weak regular
splitting. We do not prefer to give a name to this type of a splitting. However, since
our concern is nonnegativity of the group inverse, we prop ose the following: A proper
splitting A = U − V will be referred to as a pseudo regular splitting if U
#
exists,
U
#
≥ 0 and V ≥ 0. A proper splitting A = U − V is called a weak pseudo regular
splitting if U
#
exists, U
#
≥ 0 and U
#
V ≥ 0. Let us observe that if A = U − V
is a proper splitting then A
#
exists if and only if U
#
exists. The following result,
characterizing the nonneg ativity of the group inverse of A if it has a weak pseudo
regular splitting can be considered the group inverse analogue of the result of Berman
and Plemmons, mentioned previously.
Theorem 1.4. (Theor e m 3.5, [17], paraphrased) Let A ∈ R
n×n
with index 1.
Let A = U − V be a weak pseudo regular splitting. Then the following statements are
equivalent:
(i) A
#
≥ 0.
(ii) A
#
V ≥ 0.
(iii) ρ(U
#
V ) =
ρ(A
#
V )
1 + ρ(A
#
V )
< 1.
Let us revert back to the case of the inverse positive matrices to provide a mo-
tivation to the results of this article. Peris [14] studied a certain extension of the
notion of Z- matrices by proposing what are c alled B-splittings. We will not get into
the specific details here, but only mention that inverse positivity was characterized
in terms of the existence of B-splittings. In this regard, mention must be made of
the work of Barker [1] who, considered regular splittings and completely regular split-
tings of a matrix and considered several extensions of the prope rties of M -matrices.
One notable contribution in this work is the use of cones in place of the nonnegative
orthant of the real Euclidean space. Irreducibility and imprimitivity of matrices were
also studied in that work. Also, r ather recently, the authors of [8] studied comparison
results for certain nonnegative splittings and studied their relations hips with inverse
positive matrices. The work of Peris mentioned above, has been extended to the case
of the Moore-Penrose inverse by Mishra and Sivakumar [1 2]. In this paper, certain
extensions of some of the results of [1] and [8] on group inverses ar e proved. Also,
a generalization of a nice re sult of Fan [9] which c oncerns the M-matrix property of
an invertible matrix A of the type I − A
−1
, is proposed. In the nex t section, some
Electronic Journal of Linear Algebra ISSN 1081-3810
A publication of the International Linear Algebra Society
Volume 31, pp. 686-705, November 2016
http:/repository.uwyo.edu/ela
ELA
690 K. Appi Reddy, T. Kurmayya, and K.C. Sivakumar
notations, definitions and results are introduced. In Section 3, the main results are
proved.
2. Notation, definitions and preliminaries. Let L, M be complementary
subspaces of R
n
, i.e., L + M = R
n
and L ∩ M = {0}. Then P
L,M
denotes the
(not necessarily orthogonal) projection of R
n
onto L along M . So, we have P
2
L,M
=
P
L,M
, R(P
L,M
) = L and N (P
L,M
) = M . If in addition L⊥M , then P
L,M
will be
denoted by P
L
. In such a case, we also have P
T
L
= P
L
.
The notions of the Moore-Penrose inverse and the group inverse were discussed
in the introduction, where one necessary and sufficient condition for the existence of
the group inverse was stated. Another characterization fo r the group inverse of A
to exist is that the subspaces R(A) and N(A) are complementary. (It is now easy
to deduce that if A = U − V is a prope r splitting, then A
#
exists if and only if
U
#
exists). Next, we collect some well known properties of A
#
which will be used:
R(A) = R(A
#
); N(A) = N (A
#
); AA
#
= P
R(A),N (A)
= A
#
A . In particular, if
x ∈ R(A) then x = A
#
Ax. For the proofs of these and other results, we refer to [2].
The following is a fundamental res ult concerning systems of linear equations. This
will be rather frequently used in deriving some of our results. We refer the reader to
[2] for its proof.
Lemma 2.1. Let A ∈ R
n×n
with index 1 and b ∈ R
n
. Then the system of linear
equations Ax = b has a solution if and only if AA
#
b = b. In such a case, the general
solution is given by the formula x = A
#
b + z for some z ∈ N (A).
We frequently use the following result in proving the main results of this paper.
Theorem 2.2. (Theorem 3.4.1, [12]) Let A = U − V be a proper splitting of
A ∈ R
n×n
. Suppose that A
#
exists. Then the following hold:
(a) U
#
exists.
(b) AA
#
= U U
#
and A
#
A = U
#
U.
(c) A = U (I − U
#
V ).
(d) I − U
#
V is invertible.
(e) A
#
= (I − U
#
V )
−1
U
#
.
As mentioned in the intro duction, a matrix A is called nonnegative if all the
entries of A are nonnegative; this is denoted by A ≥ 0. A is called positive if all the
entries of A are p ositive; this is denoted by A > 0. For A, B ∈ R
m×n
, the notation
A ≤ B means tha t B − A ≥ 0. A vector x ∈ R
n
is called nonnegative and is denoted
by x ≥ 0 if all its coordinates are no nnegative; x is called positive if all its coordinates
Electronic Journal of Linear Algebra ISSN 1081-3810
A publication of the International Linear Algebra Society
Volume 31, pp. 686-705, November 2016
http:/repository.uwyo.edu/ela