# The Tropical Matrix Groups with Symmetric Idempotents

Authors: Lin Yang
Journal: Discrete Dynamics in Nature and Society
 (2018)

---

## Abstract

In this paper we study the semigroupMn(T) of all n×n tropical matrices under multiplication. We give a description of the tropical matrix groups containing a diagonal block idempotent matrix in which the main diagonal blocks are real matrices and other blocks are zero matrices. We show that each nonsingular symmetric idempotent matrix is equivalent to this type of block diagonal matrix. Based upon this result, we give some decompositions of the maximal subgroups of Mn(T) which contain symmetric idempotents.


---

## Body

## 1. Introduction

Tropical algebra (also known as max-plus algebra or max-algebra) is the algebra of the real numbers extended by adding an infinite negative element-∞ when equipped with the binary operations of addition and maximum. It has applications in areas such as combinatorial optimization and scheduling, control theory, discrete event dynamic systems, and many other areas of science (see [1–9]). Many problems arising from these application areas are expressed using (tropical) linear equations, so many authors study tropical matrices, i.e., matrices over tropical algebra.For example, consider the multi-machine interactive production process (MMIPP) [4] where products P1,…,Pm are prepared using n machines, every machine contributing to the completion of each product by producing a partial product. It is assumed that every machine can work for all products simultaneously and that all these actions on a machine start as soon as the machine starts to work. Let aij be the duration of the work of the jth machine needed to complete the partial product for Pi(i=1,…,m,j=1,…,n). If this interaction is not required for some i and j, then aij is set to -∞. Denote the starting time of the jth machine by xj. Then all partial products for Pi(i=1,…,m) will be ready at time (1)max⁡x1+aij,…,xn+ain. Hence if bi(i=1,…,m) are given completion times then the starting times have to satisfy the system of equations: (2)∀i∈1,…,mmax⁡x1+aij,…,xn+ain=bi. The problem can be converted into a related problem in tropical matrices.From an algebraic perspective, a key object is the multiplicative semigroup of all square matrices of a given size over the tropical algebra. There are a series of papers in the literature considering this multiplicative semigroup (see [10–17]). Moreover, an important step in understanding tropical algebra is to understand the maximal subgroups of this semigroup. It is a basic fact of semigroup theory that every subgroup of a semigroup S lies in a unique maximal subgroup. Moreover, the maximal subgroups of S are precisely the H-classes (see Section 2 below for definitions) of S which contain idempotents element. Johnson and Kambites [16] give a classification of the maximal subgroups of the semigroup of all 2×2 tropical matrices under multiplication in 2011. Izhakian, Johnson, and Kambites [13] consider the case of matrices without -∞. They prove that every subgroup of the multiplicative semigroup of n×n finite tropical matrices is isomorphic to a direct product of the form R×Σ for some Σ≤Sn. In the same year, Shitov [17] gives a description of the subgroups of the multiplicative semigroup of n×n tropical matrices up to isomorphism; i.e., every subgroup of the semigroup admits a faithful representation with n×n tropical invertible matrices. In 2017, we showed that a maximal subgroup of the multiplicative semigroup of n×n tropical matrices containing a nonsingular idempotent matrix E is isomorphic to the group of all invertible matrices which commute with E as groups and proved that each maximal subgroup of the multiplicative semigroup of n×n tropical matrices with the identity of the rank r is isomorphic to some maximal subgroup of the multiplicative semigroup of r×r tropical matrices with nonsingular identity. Thus we shall turn our attention towards the invertible matrices that commute with the nonsingular idempotent. The main purpose of this paper is to study the invertible matrices that commute with a nonsingular symmetric idempotent and to give a decomposition of the maximal subgroups of n×n tropical matrices containing a nonsingular symmetric idempotent.This paper will be divided into five sections. In Section2 we introduce some preliminary notions and notation. The decompositions of the maximal subgroups of n×n tropical matrices containing an idempotent diagonal block matrix are established in Section 3. This result (see Theorem 11) develops the results obtained by Izhakian et al. in [13]. Finally, in the last section, we prove that each symmetric nonsingular idempotent matrix is equivalent to a block diagonal matrix and a decomposition of the maximal subgroup containing a symmetric idempotent matrix is given (see Theorem 17).


## 2. Preliminaries

The following notation and definitions can be found in [3, 15, 18, 19]. We write T for the set R∪{-∞} equipped with the operations of maximum (denoted by ⊕) and addition (denoted by ⊗). Thus, we write (3)a⊕b=max⁡a,banda⊗b=a+b.As usual, the set of allm×n tropical matrices is denoted by Mm×n(T). In particular, we shall use Mn(T) instead of Mn×n(T). The operations ⊕ and ⊗ on T induce corresponding operations on tropical matrices in the obvious way. Indeed, if A, B∈Mm×n(T), C∈Mn×p(T), then we have (4)A⊕Bij=aij⊕bij,A⊗Cij=⨁k=1naik⊗ckj, where xij denotes the (i,j)th entry of the matrix X. For brevity, we shall write AC in place of A⊗C. It is easy to see that (Mn(T),⊗) is a semigroup. Other concepts such as transpose and block matrix are defined in the usual way. Unless otherwise stated, we refer to matrix as tropical matrix in the remainder of this paper. Recall that Green’s relations R and L [20] on the semigroup Mn(T) are, respectively, given by (5)ARB⇔∃X,Y∈MnTA=BX,B=AY;ALB⇔∃X,Y∈MnTA=XB,B=YA.Green’s relation H(D, resp.) is given by H=R∩L(D=R∘L, resp.). The H-class (D-class, resp.) containing the matrix A will be written as HA (DA, resp.).We shall be interested in the spaceTn ofaffine tropical vectors. We write xi for the ith component of a vector x∈Tn. We extend ⊕ to Tn componentwise so that (x⊕y)i=xi⊕yi for all i. And we define a scaling action of T on Tn by (6)λ⊗x1,x2,…,xn=λ⊗x1,λ⊗x2,…,λ⊗xn, for each λ∈T and each x∈Tn. These operations give Tn the structure of a T-semimodule.Atropical convex set in Tn is a subset closed under ⊕ and scaling by elements of T, that is, a T-subsemimodule of Tn. If S⊆Tn, then thetropical convex hull of S is the smallest tropical convex set containing S, that is, the set of all vectors in Tn which can be written as tropical linear combinations of finitely many vectors from S.LetX be a finitely generated tropical convex set in Tn. A set {x1,x2,…,xk}∈X is called aweak basis of X if it is a generating set for X minimal with respect to inclusion. It is known that every finitely generated tropical convex set admits a weak basis, which is unique up to permutation and scaling (see [[19], Theorem 5]). In particular, any two weak bases have the same cardinality, in view of which we may define thegenerator dimension of a finitely generated tropical convex set X to be the cardinality of a weak basis for X, or, equivalently, the minimum cardinality of a generating set for X.Given anm×n matrix A we define thecolumn space of A, denoted by Col(A), to be the tropical convex hull of the columns of A. Thus Col(A)∈Tm. Similarly, we define therow spaceRow(A)∈Tn to be the tropical convex hull of the rows of A. Thecolumn rank of A is the generator dimension for the column space of A. Therow rank of A is defined dually; it is well known that the row rank and column rank of a tropical matrix can differ (see [[15] Example 7.1]). The column rank (row rank, resp.) of A is denoted by c(A) (r(A), resp.). We denote the i-th row and the j-th column of A by ai∗ and a∗j, respectively. If c(A)=s and r(A)=r, then it is easy to see that there exist s columns a∗i1,…,a∗is of A such that {a∗i1,…,a∗is} is a weak basis of Col(A) and there exist r rows aj1∗,…,ajr∗ of A such that {aj1∗,…,ajr∗} is a weak basis of Row(A). The submatrix (7)a∗i1⋯a∗isaj1∗⋮ajr∗,aj1i1aj1i2⋯aj1isaj2i1aj2i2⋯aj2is⋮⋮⋮ajri1ajri2⋯ajris,resp.of A is said to be acolumn basis submatrix of A (arow basis submatrix of A, abasis submatrix of A, resp.). If c(A)=r(A)=r, then r is called therank of A. If c(A)=n(r(A)=m, resp.), then A is calledcolumn compressed (row compressed, resp.) [21]. The matrix A is callednonsingular if it is both column compressed and row compressed, andsingular otherwise.In the sequel, the following notions and notation are needed for us.(i)
Ann×n matrix A is called asymmetric matrix if AT=A.(ii)
diag(A1,A2,…,An) denotes the diagonal block matrix (8)A1-∞⋯-∞-∞A2⋯-∞⋮⋮⋮-∞-∞⋯Ak,where each diagonal blockAi is a square matrix, for all 1≤i≤n. Particularly, the matrix diag(a1,a2,…,an) will be calleddiagonal if all of a1,a2,…,an are real numbers.(iii)
In denotes theidentity matrix, i.e., the n×n matrix diag(0,0,…,0).(iv)
Ann×n matrix A is calledinvertible if there exists an n×n matrix B such that AB=BA=In. In this case, B is called an inverse of A and is denoted by A-1.(v)
Ann×n matrix is called amonomial matrix if it has exactly one entry in each row and column which is not equal to -∞.(vi)
Ann×n matrix is called apermutation matrix if it is formed from the identity matrix by reordering its columns and/or rows.(vii)
-∞ denotes thezero matrix, i.e., the matrix whose entries are all -∞.It is well known that ann×n matrix A is invertible if and only if A is monomial [22]. Also, the inverse of a permutation matrix is its transpose. Denote the set of all n×n monomial matrices (permutation matrices, resp.) by GLn(T)(Pn(T), resp.). Then GLn(T) and Pn(T) are group under the matrix multiplication.There are two types of elementary matrices corresponding to the two types of elementary operations.Type 1.
Anelementary matrix of Type1 is a matrix obtained by interchanging two rows (columns, resp.) of In. We write Ei,j as the matrix obtained by trading places of rows (or columns) i and j of In.Type 2.
Anelementary matrix of Type2 is a matrix obtained by multiplying a row (column, resp.) of In by a constant a≠-∞. We write Ei(a) as the matrix obtained by multiplying row (or column) i of the identity matrix by a≠-∞.Recall that ifA is an m×n matrix, and B is a matrix of the same size that is obtained from A by a single elementary row (column, resp.) operation, then there is an elementary matrix of size m (n, resp.) that will convert A to B via matrix multiplication on the left (right, resp.). Thus it is easy to see that a matrix is monomial if and only if it may be decomposed into the product of a finite number of elementary matrices. Also, it is worth mentioning that an elementary column (row, resp.) operation on a matrix does not change the linear relationship among the row (column, resp.) vectors. That is to say, if A,B∈Mm×n(T) and A=BM for some n×n monomial matrix M, then (9)ak∗=λ1⊗ai1∗⊕⋯⊕λsais∗⇔bk∗=λ1⊗bi1∗⊕⋯⊕λsbis∗,where ak∗,ai1∗,…,ais∗ are some rows of A, bk∗,bi1∗,…,bis∗ are the corresponding rows of B, and λ1,…,λs∈T.We say that matricesA and B areequivalent [23] (notation A≡B) if B=PNPT for some permutation matrix P, that is, B can be obtained by a simultaneous permutation of the rows and columns of A.


## 3. Tropical Matrix Groups Containing a Diagonal Block Idempotent

In this section, we study the tropical matrix groups containing a diagonal block idempotent. First, we will need the following notation and results in [13]. Let E be an n×n nonsingular idempotent matrix. We denote the set of all monomial matrices commuting with E by GE. That is to say, (10)GE=M∣M∈GLnT,ME=EM. The H-classes containing an n×n idempotent matrix are the maximal subgroups of the semigroup Mn(T). By Theorems 4.3 and 5.3 in [13], we have the following.Lemma 1.
LetE be an idempotent of rank r. Then HE is isomorphic to GE¯ as groups, where E¯ is a basis submatrix of E.Since each basis submatrix of an idempotent is a nonsingular idempotent matrix, we need only to study the groupGE, in which E is a nonsingular idempotent matrix. Indeed it is easy to see the following.Lemma 2.
E = diag ⁡ E 1 , E 2 , … , E k is a nonsingular idempotent matrix if and only if E1,E2,…,Ek are nonsingular idempotent matrices.We can say immediately thatGIn=GLn(T), which is isomorphic to R≀Sn as groups. More generally, we have the following.Lemma 3.
If(11)E=F-∞⋯-∞-∞F⋯-∞⋮⋮⋮-∞-∞⋯Fis an n×n nonsingular idempotent matrix, where the diagonal blocks are k real square matrices, then GE=(12)
M
=
M
11
M
12
⋯
M
1
k
M
21
M
22
⋯
M
2
k
⋮
⋮
⋮
M
k
1
M
k
2
⋯
M
k
k
∈
G
L
n
T
∣
∃
σ
∈
S
k
∀
i
,
j
∈
k
M
i
j
∈
G
F
j
=
σ
i
M
i
j
=
-
∞
j
≠
σ
i
.Proof.
Suppose thatE=diag⁡F,F,…,F is an n×n nonsingular idempotent matrix and that F is a real matrix. Then by Lemma 2 we can find that F is an (n/k)×(n/k) real nonsingular idempotent matrix. If M∈GE, then partition M in the same manner of E, i.e., (13)M=M11M12⋯M1kM21M22⋯M2k⋮⋮⋮Mk1Mk2⋯Mkk, where Mij are all (n/k)×(n/k) matrices, and we have (14)EM=ME. Thus we can see that(15)FMij=MijF,for any i,j∈[k].
Now we claim that(16)ifMij∉GLn/kT,thenMij=-∞.If Mij∉GLn/k(T), then Mij has some row where entries are all -∞ or Mij has some column where entries are all -∞, since Mij is a submatrix of the monomial matrix M. Without loss of generality, we assume that Mij has one row where entries are all -∞; thus MijF=FMij has one row where entries are all -∞. Since F is real matrix, it follows that Mij=-∞, for otherwise FMij does not have one row where entries are all -∞.
If, on the other hand,Mij∈GLn/k(T) such that (15), then Mij∈GF. This completes our proof.For any matrixF, we denote the matrix diag⁡F,F,…,F by F~.As a consequence, we have the following.Corollary 4.
G F ~ is isomorphic to GF≀Sn/k as groups, in which the matrix F~ has the form given in Lemma 3.Next, we shall want to consider the type of matrices in Lemma9. And we need some lemmas at first. By [21, Theorem 102], we immediately have the following.Lemma 5.
LetE be an n×n nonsingular idempotent matrix. Then (17)DE=MEN∣M,N∈GLnT.Lemma 6 (see [24] Proposition 4.5).
LetE be a nonsingular idempotent matrix. If there exists a monomial matrix M, such that EME=E, then M=In.Lemma 7.
LetE,F be nonsingular idempotent matrices. Then EDF if and only if there exists a monomial matrix M such that E=MFM-1, i.e., such that EM=MF.Proof.
Suppose thatE,F are nonsingular idempotent matrices.
IfEDF, then by Lemma 5 we can see that E=MFN, for some monomial matrices M and N. It follows that MFN=E=E2=MFNMFN. This implies that F=FNMF. Now by Lemma 6 we have that NM=In. Hence E=MFM-1, and so EM=MF.
To prove the converse half, if there exists a monomial matrixM such that EM=MF, then we let C=EM=MF, and we can see that ERCandCRF. Hence EDF as required.IfM=(mij)n×n is a monomial matrix, then there exists a unique σ∈Sn, such that miσ(i)∈R and mij=-∞ for all j≠σ(i). Thus from the definition of matrix multiplication it is easy to show that the map (18)φ:GLnT→Sn,M→σ is a homomorphism of groups. Now we can show thatProposition 8.
LetE=(eij)n×n and F=(fij)n×n be real nonsingular idempotent matrices. Then EDF if and only if there exists σ∈Sn, such that, for all i,j∈[n],(19)ei1-ej1-eij=fσiσ1-fσjσ1-fσiσj,Proof.
Suppose thatE=(eij)n×n and F=(fij)n×n are real nonsingular idempotent matrices.
IfEDF, then by Lemma 7 we have that there exists a matrix M=(mij)n×n∈GE such that EM=MF. It follows that (20)EM=eijmij=MF=mijfij.This implies that, for any i,j∈[n], (21)eij⊗mjσj=miσi⊗fσiσj. Since for all i,j∈[n], eij,mjσ(j),miσ(i) and fσ(i)σ(j) are real numbers, then we have (22)eij+mjσj=miσi+fσiσj. Thus we can see that, for any i,j∈[n], (23)miσi-mjσj=eij-fσiσj=miσi-m1σ1-mjσj-m1σ1=ei1-fσiσ1-ej1-fσjσ1. Hence for any i,j∈[n] we have(24)ei1-ej1-eij=fσiσ1-fσjσ1-fσiσj.
Conversely, if there existsσ∈Sn such that, for any i,j∈[n], (25)ei1-ej1-eij=fσiσ1-fσjσ1-fσiσj, then the system(26)m2σ2-m1σ1=e21-fσ2σ1m3σ3-m1σ1=e31-fσ3σ1⋯mnσn-m1σ1=en1-fσnσ1m1σ1-m2σ2=e12-fσ1σ2m3σ3-m2σ2=e32-fσ3σ2⋯mnσn-m2σ2=en2-fσnσ2⋯mn-1,σn-1-mnσn=en-1,n-fσn-1σnhas the solutions (27)miσi=λ⊗ei1-fσiσ1, where λ∈R. This means that if σ satisfies (24), then there exists a monomial matrix M, whose (i,σ(i))th entry is the real number miσ(i) and the other entries are -∞, such that EM=MF, and so EDF.Lemma 9.
Let(28)E=E1-∞⋯-∞-∞E2⋯-∞⋮⋮⋮-∞-∞⋯Ekbe an n×n nonsingular idempotent matrix, where the matrix Ei is a real matrix of order ni, i∈[k], and for any i,j∈[k], (Ei,Ej)∉D(i≠j). Then (29)GE=M=diag⁡M11,M22,…,Mkk∈GLnT∣∀i∈kMii∈GEi.Proof.
LetE=diag⁡E1,E2,…,Ek be an n×n nonsingular idempotent matrix. Then by Lemma 2 we can see that Ei is an (ni)×(ni) real nonsingular idempotent matrix. Suppose that M∈GE. Then partition M into k2 blocks (30)M11M12⋯M1kM21M22⋯M2k⋮⋮⋮Mk1Mk2⋯Mkk, where Mij is an ni×nj matrix. It follows EM=ME that(31)EiMij=MijEj,for any i,j∈[k]. Since Mij is a submatrix of the monomial matrix M, it has at most one entry in each row and column which is not equal to -∞. We now distinguish two cases: (32)ii≠j,iii=j.
In case (i), suppose thatMij is a monomial matrix such that (31). Then by Lemma 7 we have that EiDEj. This contradiction implies that Mij is not a monomial matrix. It follows by a closely similar proof of the claim (16) that Mij=-∞.
In case (ii),Mii is a monomial matrix such that (31), since Mij=-∞(i≠j) and M is a monomial matrix. This implies that Mii∈GLni(T) such that EiMii=MiiEi, and so Mii∈GEi. This completes our proof.We now immediately deduce the following.Corollary 10.
If the matrixE has the form in Lemma 9, then GE is isomorphic to GE1×GE2×⋯×GEk as groups.By the connection between the elementary operations and elementary matrices, it follows by Lemma7 that if E=diag⁡(E1,E2,…,Ek) is a nonsingular idempotent matrix, then there exists a monomial matrix N, such that (33)NEN-1=diag⁡E~i1,E~i2,…,E~is, where Ei1,Ei2,…,Eis are diagonal blocks of E and for any h,g∈{1,…,s}, (Eih,Eig)∉D(h≠g). It is easy to see that the mapping φ:GE→GNEN-1 defined by (34)φM=NMN-1M∈GE is a group isomorphism. Thus we obtain that GE is isomorphic to GNEN-1 as groups. Hence we have the following theorem.Theorem 11.
Let(35)E=E1-∞⋯-∞-∞E2⋯-∞⋮⋮⋮-∞-∞⋯Ekbe an n×n nonsingular idempotent matrix, where E1,E2,…,Ek are real square matrices. Then there exists a monomial matrix N, such that (36)NEN-1=E~i1-∞⋯-∞-∞E~i2⋯-∞⋮⋮⋮-∞-∞⋯E~is, where Ei1,Ei2,…,Eis are diagonal blocks of E and for any h,g∈{1,…,s}, (Eih,Eig)∉D(h≠g). Furthermore, GE is isomorphic to(37)GEi1≀Sk1×GEi2≀Sk2×…×GEis≀Sks as groups, where nh is the order of the matrix E~ih and kh is the number of the diagonal blocks of E~ih, h∈{1,…,s}.It follows by Lemma1 and Theorem 11 that each tropical matrix group containing an idempotent of the form in Theorem 11 is isomorphic to some direct products of some wreath products. This result develops the decomposition of maximal subgroups of the semigroup of n×n real matrices under multiplication as direct products of R with finite groups in [13].


## 4. Tropical Matrix Groups Containing a Symmetric Nonsingular Idempotent Matrix

In this section we shall prove that each symmetric nonsingular idempotent matrix is similar to a diagonal block matrix. On this basis, we give a decomposition of the maximal subgroups containing an idempotent of this kind. For this aim, the following lemmas are needed.Lemma 12 (see [24] Corollary 4.4).
All main diagonal entries of a nonsingular idempotent matrix are0.It is easy to verify the following lemma.Lemma 13.
LetE be an n×n matrix. Then the following are true. (i)
IfE=(eij)n×n is a nonsingular idempotent matrix, then (38)eik⊗ekj≤eij,for alli,j,k∈[n];(ii)
IfE is a nonsingular symmetric idempotent matrix, then so is PEPT for any P∈Pn(T).We can now prove the following proposition.Proposition 14.
LetE be a nonsingular symmetric idempotent matrix. Then there exists a permutation matrix P such that (39)PEPT=E1-∞⋯-∞-∞E2⋯-∞⋮⋮⋮-∞-∞⋯Ek, where E1,E2,…,Ek are real nonsingular symmetric idempotent matrices.Proof.
Suppose thatE=E(1)=(eij(1)) is an n×n nonsingular symmetric idempotent matrix. Then we shall show that E can be reduced to a diagonal block form using some simultaneous elementary rows and columns operations.
Step 1. Since E is a nonsingular idempotent matrix, it follows by Lemma 12 that all main diagonal entries of E are 0. If the i-th row of E has the most -∞ entries, then we can interchange 1-row and i-row of E and interchange 1-column and i-column of E. By Lemma 13 (ii), a new nonsingular symmetric idempotent matrix obtained will be (40)E2=P1E1P1T=eij2n×n,where P1=E1,i is an elementary matrix.
Step 2. By some synchronous permutations of the rows and columns of E(2), we can move the all -∞ entries of the first row to the end of this row. This means that we can take a suitable permutation matrix P2 and obtain another new matrix (41)E3=P2E2P2T=0e123⋯e1k3-∞⋯-∞e2130⋯e2k3e2,k+13⋯e2n3⋮⋮⋮⋮⋮ek13ek23⋯0ek,k+13⋯ekn3ek+1,13ek+1,23⋯ek+1,k30⋯ek+1,n3⋮⋮⋮⋮⋮en13en23⋯enk3en,k+13⋯0, where the first row has the most -∞ entries and e1j(3)=-∞ iff j>k. By Lemma 13 (ii) we have that E(3) is a nonsingular symmetric idempotent matrix. It follows by Lemma 13 (i) that e1t(3)⊗etj(3)≤e1j(3), for all t,j∈[n]. When t≤k,j>k, we can see that e1t(3)∈R and e1j(3)=-∞, and so etj(3)=-∞. Thus we have (42)
E
3
=
0
e
12
3
⋯
e
1
k
3
-
∞
⋯
-
∞
e
21
3
0
⋯
e
2
k
3
-
∞
⋯
-
∞
⋮
⋮
⋮
⋮
⋮
e
k
1
3
e
k
2
3
⋯
0
-
∞
⋯
-
∞
e
k
+
1,1
3
e
k
+
1,2
3
⋯
e
k
+
1
,
k
3
0
⋯
e
k
+
1
,
n
3
⋮
⋮
⋮
⋮
⋮
e
n
1
3
e
n
2
3
⋯
e
n
k
3
e
n
,
k
+
1
3
⋯
0
=
E
11
3
-
∞
E
21
3
E
22
3
. On the other hand, since E(3) is symmetric, it now follows that E21(3)=-∞. Hence (43)E3=E113-∞-∞E223. We can find that E11(3) is a real matrix, since the first row of E(3) has the most -∞ entries. Now the matrices E11(3) and E22(3) are nonsingular symmetric idempotent matrices. It follows that we can use the same method to reduce E22(3).
After finite steps we will end up with a diagonal block matrix(44)Em=E1-∞⋯-∞-∞E2⋯-∞⋮⋮⋮-∞-∞⋯Ek=Pm-1Em-1Pm-1T=Pm-1Pm-2Em-2Pm-2TPm-1T=Pm-1Pm-2⋯P1E1P1T⋯Pm-2TPm-1T=PE1PT=PEPT, where P=Pm-1Pm-2⋯P1 is a permutation matrix and E1,E2,…,Ek are real nonsingular symmetric idempotent matrices. This completes our proof.This proposition shows that each nonsingular symmetric idempotent matrixE is equivalent to a diagonal block matrix diag⁡(E1,E2,…,Ek), which is a Frobenius normal form [23] of E, where E1,E2,…,Ek are real matrices. In the following, we will study GE, where E is a diagonal block idempotent whose diagonal blocks are all real matrices.By a similar argument in Proposition8, we have the following.Lemma 15.
LetE be an n×n real nonsingular idempotent matrix. Then (45)φGE=σ∈Sn∣∀i,j∈1,2,…,nei1-ej1-eij=eσiσ1-eσjσ1-eσiσjand GE is isomorphic to the group R×φ(GE).In [13], Izhakian, Johnson, and Kambites give a result that GE≅R×Σ for some Σ∈Sn. We use a different method to prove this result in the above lemma and give a necessary and sufficient condition for some permutation σ in Σ. And we can easily verify that (46)σ∈Σ⇔∀i,j∈1,2,…,nei1-ej1-eij=eσiσ1-eσjσ1-eσiσj⇔∀i,j,k∈1,2,…,neik-ejk-eij=eσiσk-eσjσk-eσiσj.Especially ifE is an n×n symmetric real nonsingular idempotent matrix, then we have the following.Proposition 16.
LetE be an n×n real symmetric nonsingular idempotent matrix. Then (47)φGE=σ∈Sn∣∀i,j∈1,2,…,neij=eσiσjand (48)GE=λP∣λ∈R,P∈GE∩PnT,which is isomorphic to the group R×φ(GE).Proof.
Following the proof of Proposition8, we have that for all i,j∈{1,2,…,n}(49)eij=ejiandeσiσj=eσjσi Thus (26) reduce to(50)miσi-mjσj=eij-eσiσj=eji-eσjσi=mjσj-miσi.Then we know that the set of solutions to (50) is not empty if and only if (51)eij=eσiσj∀i,j∈1,2,…,n and the solutions are (52)miσi=λ∀i∈1,2,…,n, in which λ∈R. Hence there exist a real number λ and a permutation matrix P∈GE, such that (53)M=λP. Thus GE=λP∣λ∈R,P∈GE∩Pn(T), and so GE is isomorphic to R×φ(GE).Proposition16 enables us to compile the following algorithm.If the idempotent matrixE is real nonsingular, we have discussed GE. In the following, we will study the symmetric nonsingular idempotent matrix, which is ont only a real matrix. In summation, from Theorem 11 and Proposition 16, we have the following.Theorem 17.
LetE be an n×n symmetric nonsingular idempotent matrix. Then there exists a monomial matrix N, such that (54)NEN-1=E~1-∞⋯-∞-∞E~2⋯-∞⋮⋮⋮-∞-∞⋯E~s, where E1,E2,…,Es are symmetric real nonsingular idempotent matrices and for any i,j∈{1,…,s}, (Ei,Ej)∉D(i≠j). Moreover, GE is isomorphic to (55)R×Σ1≀Sk1×R×Σ2≀Sk2×⋯×R×Σs≀Sks as groups, where Σi≤Sni/ki, ni is the order of the matrix E~i, and ki is the number of the diagonal blocks of E~i, i∈{1,2,…,s}.Since each basis submatrix of a symmetric idempotent matrix is a symmetric nonsingular idempotent matrix, it follows by Lemma1 and Theorem 17 that each tropical matrix group containing a symmetric idempotent matrix is isomorphic to some direct products of some wreath products.Our next aim is to provide an algorithm forGE of any nonsingular idempotent E∈Mn(T).



---

*Source: 4797638-2018-12-02.xml*