1. Introduction
The set of all
complex matrices is denoted by
. The conjugate transpose of
A is
.
and
denotes the column and row spaces, respectively. The index of
A is denoted as ind(
A), which is the smallest integer (
k) such that
. The Moore–Penrose inverse of
A is denoted by
[
1,
2], and
X satisfies
and
. The Drazin inverse of
A is denoted by
[
3], which is a matrix that satisfies
and
, where
k is the index of
A. The set of all square matrices with
is denoted by
. The core-EP inverse of
A was introduced in [
4], and the symbol of this inverse is
;
X is called the core-EP inverse of
A if
, where
and
. The MPCEP inverse of an operator was introduced by Chen, Mosić and Xu [
5], and this concept was expanded on quaternion matrices by Kyrchei, Mosić and Stanimirović [
6,
7]. The MPCEP inverse of
A is denoted by
, which is a matrix (
) such that
. Let
X be the MPCEP inverse of
A; the authors of [
8] proved that
,
and
. A matrix (
X) is called the generalized Moore–Penrose inverse (or gMP) of
A [
9] (Theorem 1 and Definition 1) if
The symbol of the gMP inverse is
. More properties of the gMP inverse can be seen in [
10,
11].
The
inverse of
A is denoted by
[
12,
13,
14], which is a matrix (
Y) that satisfies
, where
. The
inverse of
A is an outer inverse of
A [
13,
15]. In [
16],
, where
, and the symbol
is the set of all inner inverses of
. The gMP inverse for bounded linear operators was introduced by Stojanović and Mosić [
9], which extends the Moore–Penrose inverse of the operator.
Let
. The core-nilpotent decomposition can be seen in [
17]; this matrix decomposition is unique. Let
be the core-nilpotent decomposition of
A; the core part of
A is
[
18] and
is the nilpotent part of
A. The core-EP decomposition of
with
was introduced by Wang [
19] (Theorem 2.1), which says that
A can be written as
, where
is group-invertible,
, and
. Let
. The EP-nilpotent decomposition of
A was introduced in [
20] by Wang and Liu.
2. New Characterizations of the gMP Inverse by Using the Core Part of the Core-EP Decomposition
In order to investigate the gMP relation, we need the following lemmas. The core-EP decomposition is unique, and there exists a matrix (
) such that
where
, the
matrix (
T) is invertible, the
matrix (
N) is nilpotent and
r is number of nonzero eigenvalues of
A, that is,
[
19].
For the convenience of readers, we provide a simple proof of the following lemma.
Lemma 1. Let and A have the matrix decomposition of (1); then, - (1)
is invertible;
- (2)
is invertible;
- (3)
is invertible.
Proof. “(1)” It is obvious that
is Hermitian. Then, for any
, we have
which suggests that
is a positive semi-definite matrix. Then, we have
, which is a positive definite matrix. Thus,
is invertible.
“(2)” It is easy to check that is a positive definite matrix; then, we can conclude that is invertible.
“(3)” Since , is invertible by , and T is invertible. □
The following lemma was proposed in [
19] (Theorem 2.3).
Lemma 2. Let and A have the matrix decomposition of (1); then, and . Lemma 3 ([
19] Corollary 3.3)
. Let . Then, . Lemma 4 ([
10] Lemma 2.5)
. Let and A have the matrix decomposition of (1). Then, Theorem 1. Let and A have the matrix decomposition of (1). Then, A is gMP-invertible with if and only if X satisfies Proof. “⇒” It is trivial according to [
9] (Theorem 1) and
.
“⇐” Let
According to
and
, we have
, where
. The conditions
and
yield
Thus, , which indicates that the proof is finished by Lemma 4. □
Theorem 2. Let and A have the matrix decomposition of (1). Then, A is gMP-invertible with if and only if X satisfies and . Proof. Suppose
X is the gMP inverse of
A; then,
and
according to Theorem 1. According to Lemma 3, we have
The condition holds if and only if according to Lemmas 2 and 3. Taking involution on yields , which implies . Post-multiplying by on yields . □
Theorem 3. Let and A have the matrix decomposition of (1). Then, A is gMP-invertible with if and only if X satisfies Proof. “⇒” It is trivial according to [
9] (Theorem 1) and
.
“⇐” Let
According to
and
, we have
which yields
According to
and
, we have
which yields
According to Lemma 1 and equalities (
6) and (
7), we have
Thus,
according to (
8), which indicates that the proof is finished by Lemma 4. □
Theorem 4. Let , and A have the matrix decomposition of (1). Then, is the gMP inverse of A if and only if and . Proof. Suppose
X is the right gMP inverse of
A; then,
and
according to Theorem 3. According to Lemmas 2 and 3, we have
Then,
by
is idempotent according to [
9]. Thus,
holds according to
. □
3. New Characterizations of the gMP Inverse by Using Equations and Subspaces
Motivated by Theorem 2 in [
9], in the following theorem, we show that the condition
in [
10] (Theorem 3.1(b)–(e)) can be relaxed as the condition
.
Theorem 5. Let and . Then, the following statements are equivalent:
- (1)
X is the gMP inverse of A;
- (2)
, ;
- (3)
, ;
- (4)
, ;
- (5)
, .
Proof. “
(2)–(4)” It is obvious according to [
10] (Theorem 3.1).
“
” We have
for some
according to
; then,
According to Lemma 3, we have
According to
, equalities (
9) and (
10) yield
Assuming
and equality (
11) yield
Equality (
12) implies
for some
. Then,
By assuming
, equality (
13) and Theorem 2 in [
9], we have
X, which is the gMP inverse of
A.
“
” From the proof of
, we have
, so
X is the gMP inverse of
A based on the assumption of
and Theorem 2 in [
9].
“
” From the proof of
, we have
; then by assuming
, we have
thus,
X is the gMP inverse of
A according to [
9] (Theorem 1).
“” is trivial according to . □
Motivated by Theorem 2 in [
9], in the following theorem, we show that the condition of
in Theorem 3.2(b)–(e) of [
10] can be relaxed as the condition
.
Theorem 6. Let and . Then, the following are equivalent:
- (1)
X is the gMP inverse of A;
- (2)
, ;
- (3)
, ;
- (4)
, ;
- (5)
, .
Proof. “
” It is obvious according to [
10] (Theorem 3.2(b)).
“” Post-multiplying by on yields , and is an outer inverse of A.
“
” According to Lemma 3, we have
and
, which implies that
, so
according to
. Since
,
according to
, which implies that
. Thus,
“
” According to the proof of “
”
,
The proof is finished by Theorem 1 of [
9]. □
The condition
in [
10] (Theorem 3.3(b)) can be relaxed as the condition
.
Theorem 7. Let and . Then, X is the gMP inverse of A if and only if , and .
Proof. “⇒” is obvious according to [
10] (Theorem 3.3(b)).
“⇐” By assuming
, we have
for some
. Then,
; thus,
X is the gMP inverse of
A according to
and [
10] (Theorem 3.2(f)). □
The condition
in [
10] (Theorem 3.3(b)) can be relaxed as the condition
.
Theorem 8. Let and . Then, X is the gMP inverse of A if and only if , and .
Proof. “⇒” is obvious according to [
10] (Theorem 3.3(b)).
“⇐” By assuming
, we have
. By assuming
and the assumption condition
, we have
, that is,
; thus,
X is the gMP inverse of
A according to
and [
10] (Theorem 3.1(f)). □
The relationship between the core-EP inverse and the gMP inverse is investigated in the following theorem. Note that Stojanović and Mosić proposed a condition such the core-EP inverse coincides with the gMP inverse, that is, if and only if . According to Theorem 5, we have . One can also prove that ; thus, we have the following:
Proposition 1. Let and . Then, the core-EP inverse coincides with the gMP inverse if and only if . Moreover, the condition can be replaced by , or .
According to the definition of the core-EP inverse and [
10] (Theorem 3.3(b)), we have the following table. Note that the relationship between the core-EP inverse and the gMP inverse can be determined by the following
Table 1. We can also determine the the relationship between the MPCEP inverse and the gMP inverse according to [
8].
For the convenience of readers, in the following proposition, some properties of the gMP inverse are collected.
Proposition 2. Let , A have the matrix decomposition of (1) and be the gMP inverse of A. Then, - (1)
and ;
- (2)
and ;
- (3)
;
- (4)
;
- (5)
.
Proof. “(1)” is obvious according to [
9] (Theorem 2(iv),(vii)).
“(2)” Pre-multiplying by
A on
, we have
, which yields
according to Lemma 2. Post-multiplying by
A on
, we have
, which yields
according to [
9] (Theorem 1).
“(3)” We have
according to [
9] (Theorem 1); then,
“(4)” We have
according to (2), so we have
. Thus, the proof is finished by Theorem 3.3 of [
10] and the proof of
in Theorem 5.
“(5)” The equality
is trivial according to the condition
in (2). The equality
holds according to [
10] (Theorem 3.3). According to Lemma 3, we have
and
, which yields
. □
Remark 1. The condition in Proposition 2
is very significant for investigating the gMP inverse of a complex matrix. Since the condition yields and yields , one can see that is an idempotent matrix [
9].
Let and be the gMP inverse of A. According to Proposition 2, is a inverse of A. It is easy to check that and . In the following example, we show that is not a inverse of A.
Example 1. Let in . Then, and . Therefore, and Thus, and are not Hermitian matrices, so is not a inverse of A.
For maximal classes, the gMP inverse was investigated by Stojanović and Mosić in [
9] (Theorem 6). In the following theorem, we show that
is the inverse along
and
U.
Theorem 9. Let . Then, is the inverse along and U.
Proof. Since
,
where
and
. □
Remark 2. Let . Then, is the inverse along and N. The proof of a such fact is similar to the proof Theorem 9.
4. The gMP Relation
Several necessary and sufficient conditions for the binary relation based on the gMP inverse are obtained.
Definition 1. Let . Then, A is below B under the gMP relation ifwhere is the gMP inverse of A. If A is below B under the gMP relation, then the symbol denotes this relation. Example 2. The gMP relation is not a partial ordering. Let and . Then, and . It is not difficult to check that . Moreover, the binary relation is not anti-symmetric according to in view of and .
The core-EP inverse can be expressed by the core-EP decomposition as the following lemma.
Lemma 5 ([
19] Theorem 3.2)
. Let if is the core-EP decomposition of A and , as in (
1)
. Then, . Lemma 6 ([
21] Lemma 2)
. If is partitioned as , then if and only if , where It is well known that for a complex matrix (), we have Thus, .
Lemma 7. Let be partitioned as . Then, if and only if , where
Theorem 10. Let and A have the matrix decomposition of (1). Then, if and only if B can be written as Proof. According to [
9] (Theorem 1), we have
if and only if
We have
according to Lemma 5, where
. According to Lemmas 1, 7 and (
17), we have
where
. According to (
17) and (
18), we have
Let
. According to Lemma 5 and (
18), we have
according to (
16), (
19) and (
20), we have
which is equivalent to
Note that
, so (
22) is equivalent to
As
V and
T are invertible, (
23) is equivalent to
that is,
According to (
25), matrix
B can be written as
According Lemma 5, (
18) and (
26), we have
According to (
16), (
27) and (
28), we have
As
V and
T are invertible, (
29) is equivalent to
Equality (
30) yields
. □
For the convenience of the beginner, we provide a simple proof of the following theorem; the remaining proof is similar to the proof of Theorem 10.
Theorem 11. Let , and A have the matrix decomposition of (1). Then, if and only if B can be written as Proof. According to [
9] (Theorem 1), we have
if and only if
which is equivalent to
According to the proof of Theorem 10, we have
where
. Let
. According to Lemma 5, (
33) and (
34), we have
As
V and
T are invertible, equality (
37) implies
and
. Equality (
38) implies
. □
5. Conclusions
A one-sided gMP inverse for matrices was introduced in this paper. Some conditions are proposed such that a left (or right) gMP-invertible matrix is gMP-invertible. The necessary and sufficient conditions of or are proposed such that X is gMP-invertible. The relationship between the core-EP inverse and the gMP inverse is also proposed. The following future perspectives for research are proposed:
Part 1. The reverse order law of the gMP inverse;
Part 2. The rank properties of the gMP inverse, such as ;
Part 3. The weighted gMP inverse of matrices;
Part 4. Centralizer applications of the gMP inverse in rings.