Abstract
We give the illustration of a new decomposition: the core-EP-nilpotent decomposition, which is on the basis of the core-EP decomposition and EP-nilpotent decomposition for some square matrices in this thesis. According to the new decomposition, we show the definitions and characteristics of two new orders: core-E-N partial order and core-E-S partial order. We also illustrate relations of the two orders under some restricted conditions.
1. Introduction
First of all, some mathematical notations are introduced as follows: denotes the matrices in the complex field. is the conjugate transpose, denotes the range space (or column space), and denotes the rank of . is the identity matrix of order . The index of , which is denoted by Ind , satisfies where is the smallest positive integer. The symbol stands for a set of matrices of index less than or equal to 1.
A unique matrix , which is called the Moore–Penrose inverse of , satisfies the equationsand then, it is usually denoted as . Furthermore, we denote . The unique matrix which is the group inverse of satisfies the equationsand then, it is usually denoted as .
The definition of core invertible matrix is defined as there can be at most one matrix such thatif equation (3) is satisfied. In this case, is the core inverse of and we denote . In [1], it had been proved that is core invertible if and only if .
We denote a set of EP matrices over by , where
As it is well known that contains many special types of matrices, such as Hermitian matrices, normal matrices, and nonsingular matrices.
Matrix decomposition and its research have been a hot research direction in recent years, among which some special matrices occupy the core position in matrix decomposition. For example, a decomposition named Toeplitz decomposition or Cartesian decomposition is introduced in [2, 3]. More details about other matrix decompositions can refer to [4–10]. Furthermore, there are some research issues about the systems of matrix equations by using generalized inverses of matrices which refer to [11–13].
In this paper, we will adopt Schur upper triangulation matrix decomposition, construct a new matrix decomposition based on the known core-EP decomposition and EP-nilpotent decomposition, and investigate related properties of this new matrix decomposition.
2. Preliminary Results
In this section, we give some preliminary results, (refer to [14], Theorem 5.4.1; [4], Theorem 2.1; [5], Theorem 2.1) which will be used in the next section.
Lemma 1 (Schur Decomposition). Let . Then, there exist a unitary matrix and an upper-triangular matrix such that
Lemma 2 (Core-EP decomposition). Let with Ind . Then, can be written as the sum of matrices and , i.e., , where(1);(2);(3).
Here, one or both of and can be null.
Lemma 3 (EP-nilpotent Decomposition). Let with Ind , , and . Then, can be written as the sum of matrices and , i.e., , where(1)(2)(3).
Here one or both of and can be null.
3. Main Results
First, we give a lemma as follows:
Lemma 4. Let . There exists a unitary matrix such thatand thus, can be written as , wherewhere and are nonsingular and upper-triangular, the main diagonal of and are the eigenvalues of , the last column of the except the main diagonal exists at least one nonzero element, is an upper-triangular matrix with zero elements on main diagonal, and , and are arbitrary matrices.
Proof. According to Lemma 1 and [4] (Theorem 2.2), for and can be written as follows:where is nonsingular and upper-triangular, the main diagonal of are the eigenvalues of , is an upper-triangular matrix with zero elements on main diagonal, and is arbitrary matrix. Moreover, using the block matrix method to decompose , we can derive that . Then, the forms of , and can easily be obtained.
Based on Lemma 4, we give a theorem of decomposition for some square matrices below.
Theorem 1. Let Ind , . A unique decomposition of , where the forms of , and are written as Lemma 4, where(i)(ii) or (iii)(iv), , Here , and can be null, and we denote , which represents the number of columns of and that have the nonzero element except the main diagonal.
Proof. First, by calculating, we can obviously find that the matrices , and satisfy all four conditions of the theorem. Next, we will illustrate the uniqueness of the decomposition. It follows from equation (8) thatWith Lemma 1, we can convert equation (9) as follows:where , , and . Therefore, with equation (10) and Lemma 4, we obtain the following equation:According to equation (11), we know that can be written as and uniquely. Moreover, with the restricted condition of and in Lemma 4, we can derive that when , the decomposition of can be written uniquely. When , the condition can always guarantee the decomposition of uniquely. In conclusion, can be uniquely written as , and .
According to Theorem 1, we give a definition of new decomposition as follows:
Definition 1. Let and Ind , . If the matrix decomposition satisfies Theorem 1, we say it is the core-EP-nilpotent decomposition.
A binary relation on a nonempty set is called partial order if it satisfies reflexivity, transitivity, and antisymmetry. It is significant to establish the partial orders by using matrix decomposition. Here are some well-known partial orders such as minus, sharp, star, core, E-N, and E-S partial orders, which are defined as follows:(a)(b) and (c) and (d) and (e) and , in which and are the EP-nilpotent decompositions of A and B, respectively(f) and , in which and are the EP-nilpotent decompositions of A and B, respectivelyNext, based on the E-N and E-S partial orders, we will introduce two new partial order relations and describe some related properties of these two new partial orders.
Definition 2. (Core-E-N order). Let , and be decomposed as Definition 1, where and are core invertible, and are EP, and and are nilpotent. We consider the binary operation:
Theorem 2. Operation (12) is called the core-E-N partial order.
Proof. Reflexivity of the relation is obvious. If and , with equation (12), and the definitions of core partial order, and minus partial order, we can easily obtain that , , and , i.e., . The antisymmetry condition holds. Then, we suppose and , applying the decomposition of equation (12) and definition of core partial order, and can imply that . Similarly, with the definition of minus partial order, we can derive that and . By (12), we have . The transitivity condition holds. The proof is complete.
The constructional form in the following theorem is referred to [5] and by calculating with Lemma 4.
Theorem 3. Let . Then, if and only if there exists a unitary matrix such thatwhere , , , and are nonsingular and upper-triangular, and are nilpotent and upper-triangular, and are arbitrary matrices, and
Proof. Let , where and are decomposed as Definition 1. Then, , , and . Because of , , and , , we can derive the following equation:where , , , and are nonsingular and upper-triangular. Applying Definition 2, we can know thatwhere and are nilpotent and upper-triangular. Applying Definition 2 again, it follows from thatThe proof is complete.
It is worth noting that a few famous partial orders are not the minus type. Therefore, we need to study whether the core-E-N partial order described in equation (12) is a minus type or not; the following example will illustrate.
Example 1. Letin which and are decomposed as Definition 1, where and are core invertible, and are EP, and and are nilpotent. Then,We can easily check that , and , i.e., .
Since , by condition (a), we can get that is not below under the minus partial order and a corollary as follows:
Corollary 1. Core-E-N partial order is not the minus type.
Next, we will give another example to illustrate whether the minus partial order can lead to core-E-N partial order or not.
Example 2. Letin which the decompositions of and and the definitions of their components are the same as Example 1. Then,By calculating, we have , , , and . With condition (a), we have . However, as , that is, is not below under the minus partial order, we derive that is not below under the core-E-N partial order. Therefore, the minus partial order cannot lead to the core-E-N partial order.
Next, we characterize and introduce another new partial order.
Definition 3. (Core-E-S order). Let , and they are decomposed as Definition 1 where and are core invertible, and are EP, and are nilpotent. We consider the binary operation:
Theorem 4. Operation (21) is called the core-E-S partial order.
Proof. Similar to the proof of Theorem 2, according to Definition 3, we can easily conclude that the binary relation is partial order.
The following theorem works in a similar way to Theorem 3.
Theorem 5. Let . Then, if and only if there exists a unitary matrix satisfyingwhere , , , and are invertible and upper-triangular, and are nilpotent and upper-triangular, and .
Proof. Let , where and are decomposed as Definition 1. Then, , , and . Because of , and , , we can imply thatwhere , , , and are invertible and upper-triangular. Because of Definition 3, we can derive thatwhere and are nilpotent and upper-triangular. Since , we have the following results:The proof is complete.
Next, we will investigate whether core-E-S partial order is minus type or not.
Example 3. We assume which their forms as shown in the Example 1. We can check that , and by calculating, which implies . However, since , it is contradicted with condition (a) and we obtain the following corollary.
Corollary 2. The core-E-S partial order is not the minus type.
After the above discussion, we know that both the core-E-N and core-E-S partial orders are not minus type. Therefore, we will study the relationship between core-E-N partial order and core-E-S partial order under some conditions.
It is worth noting that when , we assume
By calculating, we can easily check that and . However,
With the definition of (b), we get that and do not hold sharp partial order relationship, i.e., . With the Definitions 2 and 3, we derive that .
According to ([15], Remark 4.2.2), it has been proved that , so we can draw a corollary as follows:
Corollary 3. Let , .
On the basis of [16] (Theorem 2.1), if , then . Moreover, we can get another way of defining core-E-S partial order as follows:where the decomposition of and are the core-EP-nilpotent decompositions.
Data Availability
No data were used to support this study.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
Acknowledgments
This paper was supported by the Fundamental Research Funds for the Central Universities (No. 2572021BC03).