# Core-EP-Nilpotent Decomposition and Its Applications

Authors: Yang Zhang; Zongyang Jiang
Journal: Journal of Mathematics
 (2023)

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## 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.


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## Body

## 1. Introduction

First of all, some mathematical notations are introduced as follows:ℂm,n denotes the m×n matrices in the complex field. A∗ is the conjugate transpose, ℜA denotes the range space (or column space), and rkA denotes the rank of A∈ℂm,n. In is the identity matrix of order n. The index of A∈ℂn,n, which is denoted by Ind A, satisfies rkAk+1=rkAk where k is the smallest positive integer. The symbol ℂnCM stands for a set of n×n matrices of index less than or equal to 1.A unique matrixX∈ℂn,m, which is called the Moore–Penrose inverse of A∈ℂm,n, satisfies the equations(1)1AXA=A,2XAX=X,3AX∗=AX,4XA∗=XA,and then, it is usually denoted as X=A†. Furthermore, we denote PA=AA†. The unique matrix X∈ℂn,n which is the group inverse of A∈ℂn,n satisfies the equations(2)1AXA=A,2XAX=X,5AX=XA,and then, it is usually denoted as X=A#.The definition of core invertible matrixA is defined as there can be at most one matrix X such that(3)AX=AA†,ℜX⊆ℜA,if equation (3) is satisfied. In this case, X is the core inverse of A and we denote . In [1], it had been proved that A∈ℂn,n is core invertible if and only if A∈ℂnCM.We denote a set of EP matrices overℂn,n by ℂnEP, where(4)ℂnEP=A|ℜA=ℜA∗,A∈ℂn,n.As it is well known thatℂnEP 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).
LetA∈ℂn,n. Then, there exist a unitary matrix U∈ℂn,n and an upper-triangular matrix B∈ℂn,n such that(5)A=UBU∗.Lemma 2 (Core-EP decomposition).
LetA∈ℂn,n with Ind A=k. Then, A can be written as the sum of matrices A1 and A2, i.e., A=A1+A2, where(1)
A1∈ℂnCM;(2)
A2k=0;(3)
A1∗A2=A2A1=0.Here, one or both ofA1 and A2 can be null.Lemma 3 (EP-nilpotent Decomposition).
LetA∈ℂn,n with Ind A=k, rkA=r, and rkAk=s. Then, A can be written as the sum of matrices A1 and A2, i.e., A=A1+A2, where(1)
A1∈ℂnEP(2)
A2k+1=0(3)
A2A1=0.Here one or both ofA1 and A2 can be null.


## 3. Main Results

First, we give a lemma as follows:Lemma 4.
LetA∈ℂn,n. There exists a unitary matrix U∈ℂn,n such that(6)A=UT1S1S20T2S300NU∗,and thus, A can be written as A=A1+A2+A3, where(7)A1=UT1S10000000U∗,A2=U0000T20000U∗,A3=U00S200S300NU∗,where T1 and T2 are nonsingular and upper-triangular, the main diagonal of T1 and T2 are the eigenvalues of A, the last column of the T1 except the main diagonal exists at least one nonzero element, N is an upper-triangular matrix with zero elements on main diagonal, and S1,S2, and S3 are arbitrary matrices.Proof.
According to Lemma1 and [4] (Theorem 2.2), for A∈ℂn,n and A can be written as follows:(8)A=UTS0NU∗,where T is nonsingular and upper-triangular, the main diagonal of T are the eigenvalues of A, N is an upper-triangular matrix with zero elements on main diagonal, and S is arbitrary matrix. Moreover, using the block matrix method to decompose T, we can derive that A=UT1S1S20T2S300NU∗. Then, the forms of A1,A2, and A3 can easily be obtained.
Based on Lemma4, we give a theorem of decomposition for some square matrices below.Theorem 1.
LetA∈ℂn,n Ind A=k, rkAk=s. A unique decomposition of A=A1+A2+A3, where the forms of A1,A2, and A3 are written as Lemma 4, where(i)
A1∈ℂnCM(ii)
A2∈ℂnEP or A2∈ℂnEP,rkA2=l−1(iii)
A3k+1=0(iv)
A1∗A2=A2A1=0, A1A3∗=A3A1=0, A2A3∗=A3A2=0
HereA1,A2, and A3 can be null, and we denote l2≤l<s, which represents the number of columns of T1 and T2 that have the nonzero element except the main diagonal.Proof.
First, by calculating, we can obviously find that the matricesA1,A2, and A3 satisfy all four conditions of the theorem. Next, we will illustrate the uniqueness of the decomposition. It follows from equation (8) that(9)AkAk†=UIs000U∗.
With Lemma1, we can convert equation (9) as follows:(10)AkAk†=UIs1000Is20000U∗,where rkT1=s1, rkT2=s2, and s1+s2=s. Therefore, with equation (10) and Lemma 4, we obtain the following equation:(11)A1+A2=AAkAk†=A1+A2AkAk†,A3=A−AAkAk†.
According to equation (11), we know that A can be written as A1+A2 and A3 uniquely. Moreover, with the restricted condition of T1 and A=UT1S1S20T2S300NU∗ in Lemma 4, we can derive that when l=1, the decomposition of A1+A2 can be written uniquely. When l≥2, the condition rkA2=l−1 can always guarantee the decomposition of A1+A2 uniquely. In conclusion, A can be uniquely written as A1, A2, and A3.
According to Theorem1, we give a definition of new decomposition as follows:Definition 1.
LetA∈ℂn,n and Ind A=k, rkAk=s. 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)
A≤−B:A,B∈ℂm,n,rkB−rkA=rkB−A(b)
A≤#B:A,B∈ℂnCM,AA#=BA# and A#A=A#B(c)
A≤∗B:A,B∈ℂm,n,AA∗=BA∗ and A∗A=A∗B(d)
A≤㊮B:A,B∈ℂnCM,A㊮A=A㊮B and AA㊮=BA㊮(e)
A≤ENB:A,B∈ℂn,n,A1≤−B1 and A2≤−B2, in which A=A1+A2 and B=B1+B2 are the EP-nilpotent decompositions of A and B, respectively(f)
A≤ESB:A,B∈ℂn,n,A1≤#B1 and A2≤−B2, in which A=A1+A2 and B=B1+B2 are the EP-nilpotent decompositions of A and B, respectively
Next, 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). LetA,B∈ℂn,n, A=A1+A2+A3 and B=B1+B2+B3 be decomposed as Definition 1, where A1 and B1 are core invertible, A2 and B2 are EP, and A3 and B3 are nilpotent. We consider the binary operation:(12)A≤CENB:A1≤㊮B1,A2≤−B2andA3≤−B3.Theorem 2.
Operation (12) is called the core-E-N partial order.Proof.
Reflexivity of the relation is obvious. IfA≤CENB and B≤CENA, with equation (12), and the definitions of core partial order, and minus partial order, we can easily obtain that A1=B1, A2=B2, and A3=B3, i.e., A=B. The antisymmetry condition holds. Then, we suppose A≤CENB and B≤CENC, applying the decomposition of equation (12) and definition of core partial order, A1≤㊮B1 and B1≤㊮C1 can imply that A1≤㊮C1. Similarly, with the definition of minus partial order, we can derive that A2≤−C2 and A3≤−C3. By (12), we have A≤CENC. 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.
LetA,B∈ℂn,n. Then, A≤CENB if and only if there exists a unitary matrix U such that(13)A=UT1T2S1S20T40S3000S4000N1U∗,B=UT1T2S1S˜20T3+T4+D1T5D2D1T5+S˜1S˜30T5D2T5S˜4000N2U∗,where T1, T3, T4, and T5 are nonsingular and upper-triangular, N1 and N2 are nilpotent and upper-triangular, D1 and D2 are arbitrary matrices, and S2∗S3∗S4∗N1∗≤−S˜2∗S˜3∗S˜4∗N2∗.Proof.
LetA≤CENB, where A and B are decomposed as Definition 1. Then, A1≤㊮B1, A2≤−B2, and A3≤−B3. Because of A1,B1∈ℂnCM, A1≤㊮B1, and A2,B2∈ℂnEP, A2≤−B2, we can derive the following equation:(14)A1=UT1T2S10000000000000U∗,A2=U00000T40000000000U∗,B1=UT1T2S100T3S˜1000000000U∗,B2=U00000T4+D1T5D2D1T500T5D2T500000U∗,where T1, T3, T4, and T5 are nonsingular and upper-triangular. Applying Definition 2, we can know that(15)A3=U000S2000S3000S4000N1U∗,B3=U000S˜2000S˜3000S˜4000N2U∗,where N1 and N2 are nilpotent and upper-triangular. Applying Definition 2 again, it follows from A3≤−B3 that(16)S2∗S3∗S4∗N1∗≤−S˜2∗S˜3∗S˜4∗N2∗.
The 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.
Let(17)A=1100001011001000000000000,B=1100101010001000000000000,in which A and B are decomposed as Definition 1, where A1 and B1 are core invertible, A2 and B2 are EP, and A3 and B3 are nilpotent. Then,(18)A1=1100001000000000000000000,A2=0000000000001000000000000,A3=0000000011000000000000000,B1=1100001000000000000000000,B2=0000000000001000000000000,B3=0000100010000000000000000.
We can easily check thatA1≤㊮B1, A2≤−B2 and A3≤−B3, i.e., A≤CENB.
SincerkB−rkA=0≠rkB−A=1, by condition (a), we can get that A is not below B 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.
Let(19)A=1101011100000000,B=1101011100100000,in which the decompositions of A and B and the definitions of their components are the same as Example 1. Then,(20)A1=1100010000000000,A2=0000000000000000,A3=0001001100000000,B1=1100011000100000,B2=0000000000000000,B3=0001000100000000.
By calculating, we haverkB−A=1, rkB=3, rkA=2, and rkB−rkA=1. With condition (a), we have A≤−B. However, as rkB3−A3=1≠rkB3−rkA3=−1, that is, A3 is not below B3 under the minus partial order, we derive that A is not below B 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). LetA,B∈ℂn,n, and they are decomposed as Definition 1 where A1 and B1 are core invertible, A2 and B2 are EP, A3 and B3 are nilpotent. We consider the binary operation:(21)A≤CESB:A1≤㊮B1,A2<#=B2andA3≤−B3.Theorem 4.
Operation (21) is called the core-E-S partial order.Proof.
Similar to the proof of Theorem2, according to Definition 3, we can easily conclude that the binary relation is partial order.
The following theorem works in a similar way to Theorem3.Theorem 5.
LetA,B∈ℂn,n. Then, A≤CESB if and only if there exists a unitary matrix U satisfying(22)A=UT1T2S1S20T40S3000S4000N1U∗,B=UT1T2S1S˜20T3+T4S˜1S˜300T5S˜4000N2U∗,where T1, T3, T4, and T5 are invertible and upper-triangular, N1 and N2 are nilpotent and upper-triangular, and S2∗S3∗S4∗N1∗≤−S˜2∗S˜3∗S˜4∗N2∗.Proof.
LetA≤CESB, where A and B are decomposed as Definition 1. Then, A1≤㊮B1, A2≤#B2, and A3≤−B3. Because of A1,B1∈ℂnCM, A1≤㊮B1 and A2,B2∈ℂnEP, A2≤#B2, we can imply that(23)A1=UT1T2S10000000000000U∗,A2=U00000T40000000000U∗,B1=UT1T2S100T3S˜1000000000U∗,B2=U00000T40000T500000U∗,where T1, T3, T4, and T5 are invertible and upper-triangular. Because of Definition 3, we can derive that(24)A3=U000S2000S3000S4000N1U∗,B3=U000S˜2000S˜3000S˜4000N2U∗,where N1 and N2 are nilpotent and upper-triangular. Since A3≤−B3, we have the following results:(25)S2∗S3∗S4∗N1∗≤−S˜2∗S˜3∗S˜4∗N2∗.
The proof is complete.
Next, we will investigate whether core-E-S partial order is minus type or not.Example 3.
We assumeA,B which their forms as shown in the Example 1. We can check that A1≤㊮B1, A2≤#B2 and A3≤−B3 by calculating, which implies A≤CESB. However, since rkB−A=1≠rkB−rkA=0, 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 whenA,B∈ℂnEP, we assume(26)A=100000000,B=211111111∈ℂ3EP.By calculating, we can easily check thatA≤−B and A≤CENB. However,(27)AB=211000000≠BA=200100100.With the definition of (b), we get thatA and B do not hold sharp partial order relationship, i.e., A≤−B⇏A≤#B. With the Definitions 2 and 3, we derive that A≤CENB⇏A≤CESB.According to ([15], Remark 4.2.2), it has been proved that A≤#B⇒A≤−B, so we can draw a corollary as follows:Corollary 3.
LetA,B∈ℂnEP, A≤CESB⇒A≤CENB.
On the basis of [16] (Theorem 2.1), if A,B∈ℂnEP, then A≤#B⇔A≤∗B. Moreover, we can get another way of defining core-E-S partial order as follows:(28)A≤CESB:A1≤㊮B1,A2≤∗B2andA3≤−B3,where the decomposition of A and B are the core-EP-nilpotent decompositions.



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