1. Introduction
Let
denote the set of all
complex matrices. The symbols
and
will denote the range (column space) and the null space of a matrix
A, respectively. The rank of
is the dimension of
and the nullity of
is the dimension of
By
, we will denote the set of all matrices from
with a rank
denotes the identity matrix of order
n. We say that integers
k and
l are congruent modulo the positive integer
m, and we use the notation
if
For
, the group inverse [
1,
2,
3,
4] of
T is the unique (if it exists) matrix
such that:
The group inverse is a particular case of the generalized Drazin inverse. Recently, there has been interest in investigating the generalized Drazin inverse, for example [
5,
6].
In this paper, we will focus on the set of k-potent matrices, with k being a positive integer greater than one. This set of matrices is defined as In particular, if or then is called an idempotent (an oblique projector) or a tripotent matrix, respectively.
The research dealing with k-potent matrices, in particular idempotents and tripotents, is quite extensive. The fact that these matrices attract such attention is mostly due to their possible applications. Collections of results dealing with idempotent and tripotent matrices are available in several monographs emphasizing their usefulness in statistics, for instance [
7] (Section 12.4), [
8] (Chapter 7), and [
9] (Sections 8.6, 8.7, and 20.5.3). In addition to the papers [
10,
11,
12,
13,
14,
15,
16,
17,
18,
19,
20], each of which contains a systematic study over a selected topic concerning k-potent matrices, a collection of related isolated results was published in recent years in a number of independent articles.
Apart from the papers mentioned above, an inspiration for this paper was also the work of Baksalary et al. [
21], where the authors discussed the nonsingularity of a linear combination of two idempotent matrices, and the work of Koliha et al. [
22], where the authors considered the nullity and rank of linear combinations of two idempotent matrices. In this paper, we generalize the results given in [
22] to the cases of commuting
k-potent matrices. Furthermore, we give necessary and sufficient conditions for the nonsingularity of
and
, and we find some formulae for
and
under various conditions, where
are
k-potent matrices and
are nonzero complex numbers.
2. The Nullity and Rank of a Linear Combination of Two k-Potent Matrices
In the following theorem, we investigate the null space of a linear combination of two commuting k-potent matrices.
Theorem 1. Let be commuting k-potent matrices, and let such that Then, is isomorphic to
Proof. Let
First, we show that
Suppose that
Then,
Furthermore, suppose that
Now, we get:
Thus, i.e., and the inclusion is proven.
Next, we prove that is isomorphic to and for some
Let and Then, and i.e., Furthermore, the equality holds. The previous relations give and Therefore, we have This proves the item
Now, we show the inclusion
for
Suppose that
Applying
and
we obtain:
i.e.,
Hence,
is satisfied. From the previous inclusion and
it follows that
□
Following this, the results present that the nullity and the rank of a linear combination of two commuting k-potent matrices are constant.
Corollary 1. Let be commuting k-potent matrices, and let such that Then,
Proof. This follows from Theorem 1. □
Corollary 2. Let be commuting k-potent matrices, and let such that Then, .
Proof. This follows from the well-known general equality between the range of the column space and the null space of a matrix A, , and Corollary 1. □
Remark 1. Since is invertible if and only if by Corollary 1, we conclude that the invertibility of the linear combination of two commuting k-potent matrices does not depend on the choice of the constants where such that
3. Invertibility of A Linear Combination of Two k-Potent Matrices
In this section, we discuss the invertibility of a linear combination of two commuting
k-potent matrices. The next representation will be useful for further results. Let
A,
be two
k-potent matrices for some natural
Since
A is
k-potent, this matrix is diagonalizable, and its spectrum is contained in
(see [
23]). Therefore,
can be written as:
where
is nonsingular,
for
. Note that
is nonsingular and
Furthermore, we can write
as follows:
where
and
The following fact will be used very often: If
and
then for
:
Since where A is a k-potent matrix, we give the following results.
Theorem 2. Let and be k-potent matrices, and let
- (i)
If and then is nonsingular if and only if is nonsingular. Furthermore,where: - (ii)
If and then is nonsingular if and only if is nonsingular. Furthermore,
Proof. Let be of the form and .
- (i)
By
,
, it follows that
B has the form:
where
. A simple induction argument shows there exists a sequence
such that:
Since
, we get that
. Thus,
is the
k-potent matrix. Now,
has the form:
Using
, we have:
and:
Therefore, the linear combination
is invertible for all constants
such that
Now, we conclude that
is nonsingular if and only if
G is nonsingular. Since:
and:
we conclude that
is nonsingular if and only if
G is nonsingular. Thus,
is nonsingular if and only if
is nonsigular and in case the inverse of
is defined by:
i.e.,
where
is given by (3). We observe that:
and:
Thus, the inverse has the form .
- (ii)
From
we can conclude that
B has the form:
where
A simple induction argument shows that there exists a sequence
such that:
Since
we get that
Hence,
is the
k-potent matrix. Therefore,
Since is nonsingular for all such that then is nonsingular if and only if G is nonsingular.
Analogously, as in Part we conclude that has the form and that is nonsingular if and only if G is nonsingular. Thus, the necessary and sufficient condition for the invertibility of is the invertibility of
Furthermore,
and
holds. □
Theorem 3. Let and be k-potent matrices, and let
- (i)
If and then is nonsingular if and only if is nonsingular. Furthermore, - (ii)
If and then is nonsingular if and only if is nonsingular. Furthermore,
Proof. This follows along the same lines as Theorem 2. □
In the next theorem, we investigate the invertibility of in the case when A and B are k-potent matrices such that or The following lemma is very important in the proof of this result.
Lemma 1. Let be a k-potent matrix, and let , , Then, is nonsingular and: Proof. Let
be of the form
and
. Then:
Obviously,
is nonsingular if and only if
is nonsingular. Now, by
it follows that:
Hence,
is nonsingular for all
such that
and
and:
Thus, the matrix is nonsingular for all such that and and holds. □
Theorem 4. Let and be k-potent matrices, and let If or then is nonsingular if and only if A is nonsingular. Furthermore, Proof. First suppose that the condition
holds. Furthermore, suppose that
A has the form
and
B has the form given by
. From
, we get:
where
, and
A simple induction argument shows that
has the form:
Since
we have that
Hence,
D is a
k-potent. Now,
has the form:
i.e.,
Since
D is a
k-potent matrix, then by Lemma 1, it follows that
is nonsingular for all constants
,
,
, and:
Now,
is nonsingular. Since
, then
is nonsingular if and only if
, i.e.,
is nonsingular if and only if
A is nonsingular. Then:
where
is given by
Thus, we can conclude that
holds. The proof under the condition
is similar to the proof given in the first part, so we omit it. □
The following theorem presents necessary and sufficient conditions for the nonsingularity of linear combinations of two commuting k-potent matrices and presents the form of its inverse.
Theorem 5. Let and be commuting k-potent matrices, and let , Then, is nonsingular if and only if is nonsingular. Furthermore, Proof. Let
be of the form
and
. We get that the condition
is equivalent to the fact that B has the form:
where
such that
and
A simple induction argument shows that
has the form:
Since
we get that
and
Hence,
D and
G are
k-potent matrices. Now,
Since
is a projector and:
we get that
is nonsingular for all constants
such that
. Based on
and the invertibility of
, we conclude that
is nonsingular for all constants
such that
and:
Now,
is nonsingular if and only if
G is nonsingular. We observe that
can be represented as (7) and that
is nonsingular if and only if
G is nonsingular. Hence,
is invertible if and only if
is invertible. In this case, the inverse
has the form:
i.e.,
where
is given by
Therefore,
holds. □
The following corollary is very useful in
Section 4.
Corollary 3. Let and be commuting k-potent matrices, and let , Then, is nonsingular if and only if is nonsingular.
Remark 2. The results in this section show us that the nonsingularity of the linear combination of two k-potent matrices does not depend on the choice of nonzero complex constants as long as the assumptions of the previous theorems are met.
4. Invertibility of a Linear Combination of Three k-Potent Matrices
In this section, we study the invertibility of a linear combination of three
k-potent matrices. We use the notions from previous sections. We also use the next representation for
where
and
We consider the invertibility of
where
are
k-potent matrices such that
Let
be of the form
and
. Using
and the nonsingularity of
K leads to
Hence,
where
A simple induction argument shows that
has the form:
Since C is a k-potent matrix, then T is also a k-potent matrix. Furthermore, the conditions from the theorems of the previous section for k-potent matrices apply.
Theorem 6. Let and be k-potent matrices such that and , and let such that
- (i)
If and then is nonsingular if and only if is nonsingular. Furthermore,where: - (ii)
If then is nonsingular if and only if is nonsingular. Furthermore,
Proof. Suppose that have the form and respectively, and .
- (i)
Since
we can write
B as in
We note that
can be represented as:
where
are
k-potent matrices. Since
then
Analogous to the proof to Theorem 2, we conclude that
is invertible for all constants
such that
and that the inverse of
has the form
Therefore, the linear combination
is nonsingular if and only if
is nonsingular. Since
are commuting
k-potent matrices, we deduce that
is nonsingular if and only if
is nonsingular for all constants
such that
by Corollary 3. Furthermore,
and
is nonsingular if and only if
is nonsingular. Thus, the necessary and sufficient condition for the invertibility of and
is the invertibility of
.
By a direct computation, we get:
where
is given by (3). Furthermore,
and:
It is noteworthy that the group inverse of
is:
Hence, the formula holds.
- (ii)
The proof is similar to the one in Item
We use the form
for
Now,
where
are
k-potent matrices. Since
then
Note that
is nonsingular for all constants
such that
Analogous to the proof of Item
we conclude that
is nonsingular if and only if if
is nonsingular for all constants
such that
We observe that
can be represented as
and that
is nonsingular if and only if
is nonsingular. Therefore,
is nonsingular if and only if
is nonsingular for all constants
such that
The inverse of
is:
and can be represented as
The rest of the proof is similar to the one in Item (i). □
Theorem 7. Let and be k-potent matrices such that , and let If or then is nonsingular if and only if is nonsingular. Furthermore, Proof. We only consider the case when
The remaining case can be proven in the same way. Conditions
and
imply that
can be represented as (
1), (12), and (19), respectively. Then, we have:
Since
is nonsingular for all constants
,
,
(see the proof of Theorem 4, then
is nonsingular if and only if
G is nonsingular. We observe that:
and that
is invertible if and only if
G is invertible. Therefore,
is nonsingular if and only if
is nonsigular. In this case, the inverse of
has the form:
where
(see the proof of Theorem 4) and
is given by (13). Thus, (24) is satisfied. □
Theorem 8. Let and be commuting k-potent matrices such that , and let: , , and Then, is nonsingular if and only if is nonsingular. Furthermore, Proof. Let
be of the form
, and
respectively, and
Now,
where
and
are
k-potent matrices such that
and
As in the proof of Theorem 5, we derive that
is nonsingular for all constants
such that
and the inverse of
can be represented as in
Hence,
is nonsingular if and only if
is nonsingular. By Corollary 3, the nonsingularity of
is equivalent to the nonsingularity of
for all constants
such that
i.e.,
is nonsingular if and only if
is nonsingular. Using
and
, the form
follows immediately. □
Remark 3. The results in this section show us that the nonsingularity of the linear combination of two k-potent matrices does not depend on the choice of nonzero complex constants as long as the assumptions of the previous theorems are met.
5. Conclusions
In this paper, we used an elegant representation of
k-potent matrices and the matrix rank to investigate the invertibility of a linear combination of two or three
k-potent matrices under some conditions. Similar to [
21,
22] for idempotents, we proved that the invertibility of a linear combination of two commuting
k-potent matrices is independent of the choice of the nonzero complex constants. Furthermore, we proved that the invertibility of a linear combination of two or three
k-potent matrices is independent of the choice of the nonzero complex constants under various conditions. An open problem is if the given conclusion also holds for the invertibility of the linear combination of two or three arbitrary
k-potent matrices.
Author Contributions
Conceptualization, M.T. and E.L.; methodology, M.T. and N.K.; validation, V.S.; formal analysis, M.T. and E.L.; writing, original draft preparation, M.T.; writing, review and editing, N.K.; supervision, V.S. All authors read and agreed to the published version of the manuscript.
Funding
This research received no external funding.
Acknowledgments
The authors would like to thank the anonymous reviewers for their very useful comments, which helped to significantly improve the presentation of this paper.
Conflicts of Interest
The authors declare no conflict of interest.
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