Adi Niv
email: adi.niv@live.biu.ac.il.

ON PSEUDO-INVERSES OF MATRICES AND THEIR CHARACTERISTIC POLYNOMIALS IN SUPERTROPICAL ALGEBRA

Keywords: Tropical and supertropical linear algebra, characteristic polynomial, eigenvalues, Kleene star, permanent, definite matrices, pseudo-inverse 15A09 (Primary), 15A15, 15A18, 15A80, 15B33

The only invertible matrices in tropical algebra are diagonal matrices, permutation matrices and their products. However, the pseudo-inverse A ∇ , defined as 1 det(A) adj(A), with det(A) being the tropical permanent (also called the tropical determinant) of a matrix A, inherits some classical algebraic properties and has some surprising new ones. Defining B and B to be tropically similar if B = A ∇ BA, we examine the characteristic (max-)polynomials of tropically similar matrices as well as those of pseudo-inverses. Other miscellaneous results include a new proof of the identity for det(AB) and a connection to stabilization of the powers of definite matrices.

Introduction

The tropical max-plus semifield is an ordered group G (usually the set of real numbers R or the set of rational numbers Q), together with -∞, denoted as T = G {-∞}, equipped with the operations a b = max{a, b} and a b = a+b, denoted as a+b and ab respectively (see [START_REF] Akian | Handbook of Linear Algebra[END_REF] and [START_REF] Itenberg | Tropical Algebraic Geometry[END_REF]). The unit element 1 T is actually the element 0 ∈ Q. This arithmetic enables one to simplify non-linear questions by asking them in a (pseudo)linear setting (see [START_REF] Gondran | Path algebra and algorithms[END_REF]), which can be applied to discrete mathematics, optimization, algebraic geometry and more, as has been well reviewed in [START_REF] Fiedler | Linear optimization problems with inexact data[END_REF], [START_REF] Gaubert | Mathods and applications of (max, +) linear algebra[END_REF], [START_REF] Golan | Semirings and their applications[END_REF], [START_REF] Heidergott | Max Plus at Work: Modeling and Analysis of Synchronized Systems: A Course on Max-Plus Algebra and Its Applications[END_REF], [START_REF] Litvinov | Idempotent mathematics: correspondence principle and applications[END_REF] and [START_REF] Mikhalkin | Tropical geometry and its applications[END_REF]. In this max-plus language, we may also use notions of linear algebra to interpret combinatorial problems, such as eigenvectors being used to solve the Longest-Distance problem (see [START_REF] Butkovic | Max-algebra: the linear algebra of combinatorics?[END_REF]).

The intention of this paper is to use an analogous concept of the inverse matrix by passing to a wider structure called the supertropical semiring, equipped with the ghost ideal G = G ν , as established and studied by Izhakian and Rowen in [START_REF] Izhakian | Supertropical algebra[END_REF] and [START_REF] Izhakian | Supertropical linear algebra[END_REF]. The use of the term pseudoin this paper is the same as quasithroughout the work of Izhakian and Rowen.

We denote the standard supertropical semiring as

R = T G {-∞},
where T = G, which contains the so called tangible elements of the structure and where ∀a ∈ T we have a ν ∈ G are the ghost elements of the structure, as defined in [START_REF] Izhakian | Supertropical algebra[END_REF]. So G inherits the order of G. This enables us to distinguish between a maximal element a that is attained only once in a sum, i.e. a ∈ T which is invertible, and a maximum that is being attained at least twice, i.e. a + a = a ν ∈ G, which is not invertible. Note that ν projects the standard supertropical semiring onto G, which can be identified with the usual tropical structure.

In this new supertropical sense, we use the following order relation to describe two elements that are equal up to some ghost supplement: Definition 1.1. Let a, b be any two elements in R. We say that a ghost surpasses b, denoted a |= gs b, if a = b + ghost, i.e. a = b or a ∈ G with a ν ≥ b ν . We say a is ν-equivalent to b, denoted by a ∼ = ν b, if a ν = b ν .That is, when we use ν to project from R to G, identified with the tropical structure, ν-equivalence becomes equality.

For matrices A = (a ij ), B = (b ij ) ∈ M n×m (R) (and in particular for vectors) A |= gs B means a ij |= gs b ij ∀i = 1, ..., n and j = 1, ..., m.

For polynomials

f (x) = n i=1 a i x i , g(x) = n i=1 b i x i ∈ R[x],
we say that f (x) |= gs g(x), also denoted as f |= gs g, when a i |= gs b i ∀i.

Important properties of |= gs : 1. |= gs is a partial order relation.

See [START_REF] Izhakian | Supertropical matrix algebra II: solving tropical equations[END_REF]Lemma 1.5].

2. If a |= gs b then ac |= gs bc.

Considering this relation, we use the classical notion 1 det(A) adj(A) (where det(A) is the permanent and then adj(A) is defined as usual) to formulate results in the supertropical setting, which are inaccessible in the usual tropical setting. This notion was introduced and studied in tropical algebra by Yoeli [START_REF] Yoeli | A note on a generalization of boolean matrix theory[END_REF] and Cuninghame-Green [START_REF] Cunininghame-Green | Minimax algebra[END_REF], and was further investigated by Gaubert [START_REF] Gaubert | Théorie des systèmes linéaires dans les dioïdes[END_REF], Reutenauer and Straubing [START_REF] Reutenauer | Inversion of matrices over a commutative semiring[END_REF] and Sergeev [START_REF] Sergeev | Multiorder, Kleene stars and cyclic projectors in the geometry of max cones[END_REF]. We obtain tropical theorems by considering the tangible elements. By Izhakian's results in [START_REF] Izhakian | Tropical arithmetic and matrix algebra[END_REF], this notion satisfies that 1 det(A) adj(A)•A is equal to the Id matrix on the diagonal, and |= gs the Id matrix off the diagonal.

In section 3 we discuss type of matrices with 0 on the diagonal and 0 determinant, defined as definite matrices. In section 3.1 , we establish that for the set

N = {A ∇ : A ∈ M n (R)},
the operation ∇ is of order 2 (see Theorem 3.5), and in section 3.2 we will show that for a definite matrix A of order n

A ∇ ∼ = ν A ∇∇ ∼ = ν A * ∼ = ν A k , ∀k ≥ n -1
(see Theorem 3.6 and Proposition 3.7). These results are extended to the supertropical setting from the results obtained over the tropical structure in [START_REF] Yoeli | A note on a generalization of boolean matrix theory[END_REF] and [START_REF] Sergeev | Multiorder, Kleene stars and cyclic projectors in the geometry of max cones[END_REF], regarding these closure operations.

In section 4, we use the factorizability of matrices in N to give an alternative proof, analogous to the proof in classical linear algebra, of the property det(AB) |= gs det(A)det(B) stated in Theorem 2. [START_REF] Gaubert | Théorie des systèmes linéaires dans les dioïdes[END_REF].

In sections 5 we prove that f A ∇ BA |= gs f B , (see Theorem 5.4). In section 6, our considerations lead us to Conjecture 6.2:

|A|f A ∇ (x) |= gs x n f A (x -1 ),
where f M (x) = det(M +xI) denotes the characteristic polynomial of a square matrix M . We prove 6.2 for the determinant and trace coefficients and for every 2 × 2 or 3 × 3 and triangular matrix.

A consequence would be

f A ∇∇ = f A , whenever f A ∇∇ is tangible.
These properties provide a foundation for further research in representation theory and eigenspace decomposition.

Preliminaries

Matrices.

The work in [START_REF] Izhakian | Supertropical matrix algebra[END_REF], [START_REF] Izhakian | Supertropical matrix algebra II: solving tropical equations[END_REF] and [START_REF] Izhakian | Supertropical matrix algebra III: Powers of matrices and their supertropical eigenvalues[END_REF] shows that even though the semiring of matrices over the supertropical semiring lacks negation, it satisfies many of the classical matrix theory properties when using the ghost ideal G. Following [START_REF] Butkovic | Max-algebra: the linear algebra of combinatorics?[END_REF] and [START_REF] Izhakian | Supertropical matrix algebra[END_REF] for the tropical and supertropical notations, we give some basic definitions for this theory. One may also find in [START_REF] Butkovic | Max-algebra: the linear algebra of combinatorics?[END_REF] further combinatorial motivations for the objects discussed. Definition 2.1. A permutation track, of the permutation π ∈ S n , is the sequence

a 1,π(1) a 2,π(2) • • • a n,π(n) of n entries of the matrix A = (a i,j ) ∈ M n (R).
Definition 2.2. We define the tropical determinant of a matrix A = (a i,j ) to be

det(A) = σ∈Sn a 1,σ(1) • • • a n,σ(n) .
In the special case where a i,j ∈ R, ∀i, j, we refer to any permutation track yielding the highest value in this sum as a dominant permutation track.

The tropical determinant is actually the same as max permanent of [4]- [START_REF] Cunininghame-Green | Minimax algebra[END_REF], and the above definition is a reformulation of the optimal assignment problem. Definition 2.3. We define a matrix A ∈ M n (T) or A ∈ M n (R) to be tropically singular if there exist at least two different dominant permutation tracks. Otherwise the matrix is tropically non-singular.

Consequently a matrix

A ∈ M n (R) is supertropically singular if det(A) ∈ G {-∞} and supertropically non-singular if det(A) ∈ T . A matrix A is strictly singular if det(A) = -∞.
Notice that over the tropical semifield we cannot determine if the matrix is tropically non-singular from the value of its determinant, which is always invertible over T\{-∞}. Over the supertropical semiring however, a supertropically non-singular matrix has an invertible determinant, while a supertropically singular matrix has a non-invertible determinant.

As the definitions of singularity are identical over the tropical and supertropical structures, we will only indicate non-singular or singular and over T or over R(which will effect the value and invertibility of the determinant). Definition 2.4. Let T n be the free module (see [START_REF] Izhakian | Supertropical matrix algebra II: solving tropical equations[END_REF]) of rank n over the tropical semifield, and R n be the free module of rank n over the supertropical semiring. We define the standard base of T n , and therefore of R n , to be e 1 , ..., e n , where

e i = 1 T = 1 R , in the i th coordinate 0 T = 0 R , otherwise .
Definition 2.5. The tropical identity matrix in the tropical matrix semiring is the n × n matrix with the standard base for its columns. We denote this matrix as

I T = I R = I. Definition 2.6. A matrix A ∈ M n (R) is invertible if there exists a matrix B ∈ M n (R) such that AB = BA = I.
Definition 2.7. A square matrix P π = (a i,j ) is defined to be a permutation matrix if there exists π ∈ S n such that a i,j = 0 R , j = π(i) 1 R , j = π(i) .

Remark 2.8. A tropical matrix A is invertible if and only if it is a product of a permutation matrix P π and a diagonal matrix D with an invertible determinant. These type of products are defined as generalized permutation matrices.

Definition 2.9. Following the notation in [START_REF] Niv | Factorization of tropical matrices[END_REF], we define three types of tropical elementary matrices, corresponding to the three elementary matrix operations, obtained by applying one such operation to the identity matrix. A transposition matrix is obtained from the identity matrix by switching two rows (resp. columns). Multiplying a matrix A to the right of such a matrix (resp. to the left) will switch the corresponding rows (resp. columns) in A.

A diagonal multiplier is obtained from the identity matrix where one row (resp. column) has been multiplied by an invertible scalar. Multiplying a matrix A to the right of such a matrix (resp. to the left) will multiply the corresponding row (resp. columns) in A by the same scalar.

A Gaussian matrix is defined to differ from the identity matrix by having a nonzero entry r, in a non-diagonal position (i, j). Multiplying a matrix A to the right of such a matrix (resp. to the left) will add row j (resp. column), multiplied by r, to row (resp. column) i.

By Remark 2.8, a product of transposition matrices, which is a permutation matrix, is invertible. A product of diagonal multipliers, which is a diagonal matrix, is invertible. Thus a product of transposition matrices and diagonal multipliers, which is a generalized permutation matrix, is invertible. Gaussian matrices however, are not invertible, and therefore a product including a Gaussian matrix is not invertible. Proof. See Theorem 3.5 in [18, §3].

This theorem also follows from [START_REF] Gaubert | Théorie des systèmes linéaires dans les dioïdes[END_REF]Proposition 2.1.7]. Although S. Gaubert's proof is done in the symmetrized tropical semiring, it carries over to the supertropical case.

In this context, we note that the two transfer principles in [2, Theorem 3.3 and Theorem 3.4 ] allow one to obtain such results automatically in a wider class of semirings, including the supertropical semiring.

The fact that the determinant of a non-singular matrix A over R is tangible means that the matrix has one dominant permutation track. By using a permutation matrix we can relocate the corresponding permutation to the diagonal, and by using a diagonal matrix we can change the diagonal entries to 1 R , obtaining a non-singular matrix whose dominant Id-permutation track equals 1 R . That is, A = P Ā where P is an invertible matrix (See Remark 2.8) such that det(P ) = det(A) and Ā is a definite matrix. Definition 2.11. Ā is called the definite form of A, P the conductor of A and we say that P conducts the dominant permutation track in A.

The process of bringing a non-singular matrix A to its definite form, can be obtained by using transposition matrices and diagonal multipliers either on the columns or on the rows of A. Respectively, we denote these conducting matrices as the right conductor and the left conductor of A, which are invertible. The right (resp. left) definite form corresponds to the right (resp. left) conductor. Definite matrices are not the same as normal matrices, defined in [START_REF] Buchholz | Tropicalization of linear isomorphisms on plane elliptic curves[END_REF] and [START_REF] Butkovic | Max-algebra: the linear algebra of combinatorics?[END_REF] to have non-positive off diagonal entries. Over T the definite form is obtained for not strictly singular matrices, by conducting one of the dominant permutation tracks.

Looking at elementary matrices as the atoms of matrices, we present the following definitions: Definition 2.12. We define a matrix to be elementarily factorizable if it can be factored into a product of tropical elementary matrices. By Remark 2.8, an invertible matrix is always elementarily factorizable, while a non-invertible matrix is elementarily factorizable if and only if its definite form is factorizable.

Remark. Writing a permutation π as a product of disjoint cycles σ 1 , ..., σ t , the per-

mutation track a 1,π(1) • • • a n,π(n) can be decomposed into the cycle tracks C 1 , ..., C t ,
where C i is the cycle track corresponds to the cycle σ i .

We introduce a standard combinatorial property of definite matrices. Since this property will be in use in Section 5, we provide some details regarding the technique of its use. Lemma 2.13. For a definite matrix A = (a i,j ) ∈ M n (R), any sequence

(2.1) a i 1 ,i 2 a i 2 ,i 3 • • • a i k ,i 1 , where i j ∈ {1, ..., n} (not necessarily distinct)
either describes a permutation track, or is dominated by a subsequence which describes a permutation track.

Proof. As proved in [START_REF] Butkovic | Max-algebra: the linear algebra of combinatorics?[END_REF]Theorem 4.4], due to the determinant obtained by the 0-entries on the diagonal of a definite matrix, for every cycle track C we have that

C = C • C Id ≤ P Id ,
where P Id is the Id-permutation track and C Id is the Id-cycle track extending C to a permutation track. We use a standard fact from graph theory, that if a path (i

1 , i 2 , i 3 , • • • , i k , i 1
) has repeating nodes, then it can be recursively separated into (not necessarily disjoint) cycles, starting and ending at the points of repetition. Thus, (2.1) can be decomposed as the product of (not necessarily disjoint) cycle tracks satisfying For a matrix (and in particular for a vector) A = (a i,j ) we write

l j=1 C j ≤ C i = C i • C Id ∀i,
A ν = (a ν i,j ) and A = ( a i,j ).
For a polynomial f (x) = Σa i x i we write

f ν (x) = Σa ν i x i and f (x) = Σ âi x i . Definition 2.14.
A pseudo-zero matrix Z G is a matrix equal to 0 R on the diagonal, and whose off-diagonal entries are ghosts or 0 R .

A pseudo-identity matrix I G is a nonsingular, multiplicatively idempotent matrix equal to I + Z G , where Z G is a pseudo-zero matrix. Thus, for every matrix A and a pseudo-identity I G we have I G A |= gs A.

A ghost pseudo-identity matrix is a singular, multiplicatively idempotent matrix equal to

I ν + Z G .
Definition 2.15. The r, c-minor A r,c of a matrix A = (a i,j ) is obtained by deleting row r and column c of A. The adjoint matrix adj(A) of A is defined as the matrix (a i,j ), where a i,j = det(A j,i ).

The matrix A ∇ denotes 1 det(A) adj(A), when det(A) is invertible, and

1 det(A) ν adj(A),
when det(A) is not invertible. Thus over R, A ∇ is defined differently for singular and non-singular matrices. Over T, however, A ∇ is defined the same for every not strictly singular matrix. Remark. Notice that det(A j,i ) may be obtained as the sum of all permutation tracks in A passing through a j,i , with a j,i removed:

det(A j,i ) = π ∈ Sn : π(j) = i a 1,π(1) • • • a j-1,π(j-1) a j+1,π(j+1) • • • a n,π(n) .
When writing such a permutation as the product of disjoint cycles, det(A j,i ) can be presented as:

(2.2) det(A j,i ) = π ∈ Sn : π(j) = i (a i,π(i) • • • a π -1 (j),j )C π ,
where

(a i,π(i) • • • a π -1 (j),j
) is the cycle track missing a j,i , and C π is the product of the cycle tracks of π that do not include i and j.

Fact. The products AA ∇ and A ∇ A are pseudo-identities when det(A) is invertible, and ghost pseudo-identities otherwise. These identities can be deduced from [10, Proposition 2.1.2], by replacing Gaubert's a with a. Then a ν |= 0 R corresponds to a • ∇0 R . See [START_REF] Izhakian | Supertropical matrix algebra II: solving tropical equations[END_REF]Theorem 2.8] for the proof in the supertropical setting.

Definition 2.16. We say that A ∇ is the pseudo-inverse of A over R, denoting

I A = AA ∇ and I A = A ∇ A.
Theorem 2.17.

(i) det(A • adj(A)) = det(A) n . (ii) det(adj(A)) = det(A) n-1 .
Proof. [18, Theorem 4.9].

Remark 2.18. For a definite matrix A, we have A ∇ = 1 det(A) adj(A) = adj(A), which is also definite.

Proof. The diagonal entries in adj(A) are sums of cycle tracks of A, and thus the Id summand 1 R dominates every diagonal entry. Also, by Theorem 2.17(ii), we have

1 R = det(A ∇ ),
as required for definite matrices. Recall the classical Bruhat (LDU) decomposition, whose tropical analog in [START_REF] Litvinov | Universal algorithms for solving the matrix Bellmann equation over semirings[END_REF] is called the LDM decomposition. Lemma 2.20. a. If A is a not strictly singular triangular matrix over T (respectively non-singular over R), then A is elementarily factorizable. b. If A is a not strictly singular matrix over T (respectively non-singular over R), then A ∇ (respectively A ∇∇ ) is elementarily factorizable.

Proof. See the LDM decomposition in [START_REF] Litvinov | Universal algorithms for solving the matrix Bellmann equation over semirings[END_REF], or an alternative proof in [26, Lemma 6.5 and Corollary 6.6].

One can find a cruder factorization in [START_REF] Buchholz | Tropicalization of linear isomorphisms on plane elliptic curves[END_REF], where sufficient conditions are established for trop(AB) = trop(A)trop(B), when looking at the tropical structure as the image of a valuation over the field of Puiseux series, using the classical LDU decomposition over this field. In section 3 we will show the connection of definite matrices and factorization to the well known tropical closure operation * , studied in [START_REF] Sergeev | Multiorder, Kleene stars and cyclic projectors in the geometry of max cones[END_REF] and [START_REF] Sergeev | On visualization scaling, subeigenvectors and Kleene stars in max algebra[END_REF].

Polynomials.

As one can see in [START_REF] Izhakian | Supertropical algebra[END_REF], the polynomials over the tropical structure are rather straightforward to view geometrically. We notice that the graph of a monomial a i x i ∈ T[x] is a line, where the power i indicates the slope. Since T is ordered we may present its elements on an axis, directed rightward, where if a < b then a appears left to b on the T-axis, for every pair of distinct elements a, b ∈ T . It is now easy to understand that a tropical polynomial

n i=0 a i x i ∈ R[x]
takes the value of the dominant monomial among a i x i along the T-axis. That having been said, it is possible that some monomials in the polynomial would not dominate for any x ∈ T.

Definition 2.21. Let f (x) = n i=0 a i x i ∈ R[x]
be a supertropical polynomial. We call monomials in f (x) that dominate for some x ∈ R essential, and monomials in f (x) that do not dominate for any x ∈ R inessential. We write

f es (x) = i∈I a i x i ∈ R[x],
where a i x i is an essential monomial ∀i ∈ I, called the essential polynomial of f . Definition 2.24. We call an element r ∈ R a root of a polynomial f (x) if

Definition 2.22. We say that b is a k-root of a, for some k ∈ N, denoted as b = k √ a, if b k = a.
f (r) |= gs 0 R .
We distinguish between two kinds of roots of supertropical polynomials. Definition 2.25. We refer to roots of a polynomial being obtained as an intersection of two leading tangible monomials as corner roots, and to roots that are being obtained from one leading ghost monomial as non-corner roots.

Remark 2.26. Suppose f (x) = a i x i ∈ R[x]. We specialize to elements r ∈ R, starting with r small and then increasing. 1. The constant term a 0 and the leading monomial a n x n dominate first and last, respectively, due to their slopes. Furthermore, they are the only ones that are necessarily essential in every polynomial. 2. The intersection of an essential monomial a i x i and the next essential monomial a j x j where j > i, is the i th root of f (counting multiplicities), denoted as α i = k a i a j , and is of multiplicity k = j -i. Proof: The monomial a i x i dominates all monomials between a i x i and a j x j . Therefore, when r ∈ (α i-1 , α i ],

f (r) = a j r j + a i r i = a j r j + a i a j r i ∈ G ⇒ r j + a i a j r i = r i r j-i + a i a j ∈ G
which means (r + α i ) k ∈ G, and therefore α i is a root of f with multiplicity k.

Supertropical characteristic polynomials and eigenvalues.

We follow the description in [19, §5].

Definition 2.27. ∀v ∈ T n and A ∈ M n (R) such that ∃α ∈ T {0 R } where Av |= gs αv, we say that v is a supertropical eigenvector of A with a supertropical eigenvalue α.

The characteristic polynomial of A (also called the maxpolynomial) is defined to be f A (x) = det(xI + A). The tangible value of the roots of the characteristic polynomial f A are the eigenvalues of A, as shown in [START_REF] Izhakian | Supertropical matrix algebra[END_REF]Theorem 7.10]. The coefficient of x k in this polynomial is a sum of the tropical determinants of all n -k × n -k minors, obtained by deleting k chosen rows of the matrix, and their corresponding columns. These minors are defined as principal sub-matrices.

The combinatorial motivation for the tropical characteristic polynomial is the Best Principal Submatrix problem, and has been studied by Butkovic in [START_REF] Butkovic | On the coefficients of the max-algebraic characteristic polynomial and equation[END_REF] and [START_REF] Butkovic | Calculating essential terms of a characteristic maxpolynomial[END_REF].

Theorem 2.28. (Supertropical Hamilton-Cayley) Any matrix A satisfies its characteristic polynomial, in the sense that f A (A) is ghost.

Proof. See [START_REF] Straubing | A combinatorial proof of the Cayley-Hamilton Theorem[END_REF] or [18, Theorem 5.2] for a supertropical statement and proof.

Proposition 2.29. If α ∈ T 0 R is a supertropical eigenvalue of a matrix A ∈ M n (T ), then α i is a supertropical eigenvalue of A i . Proof. See [25, Proposition 3.2].
However, we notice that {α i : α is an eigenvalue of a matrix A} need not be the only supertropical eigenvalues of the matrix A i , as shown in the next example.

Example 2.30. Consider the 2 × 2 matrix

A = 0 0 1 2 . Then f A (x) = x 2 + 2x + 2 ⇒ f A (x) ∈ G when x = 0, 2.
However,

A 2 = 1 2 3 4 . Thus f A 2 (x) = x 2 + 4x + 5 ν ⇒ f A 2 (x) ∈ G when x ν ≤ 1 ν or x = 4. Theorem 2.31. Let A be in M n (R). If f A (x) = n i=0 α i x i and f A m (x) = n i=0 β i x i
are the characteristic polynomials of A and its m th power, respectively, then

f A m (x m ) |= gs f A (x) m .
Proof

The ∇ of a definite matrix

As shown in [START_REF] Niv | Factorization of tropical matrices[END_REF] and [START_REF] Sergeev | Multiorder, Kleene stars and cyclic projectors in the geometry of max cones[END_REF], reductions to definite matrices can simplify verifications of some complicated properties of matrices. On top of that, matrices of this form enjoy some interesting properties of their own. Remark 3.1. Since ghost equivalent implies equality in the tropical setting, we will formulate our results in terms of ghost equivalence (i.e. over R), with the understanding that the corresponding tropical equalities (i.e. over T) follow automatically.

Stabilization under the ∇ operation.

The goal in this section is to show that ∇ is a closure operation.

Lemma 3.2.

(i) P ∇ = P -1 whenever P is an invertible matrix.

(ii) det(PA)=det(P)det(A)=det(AP), for P an invertible matrix.

(iii) (P A) ∇ = A ∇ P ∇ where det(A) is invertible and P is an invertible matrix.

(iv) Let Ā be the left definite form of the matrix A, i.e. A = P Ā for some invertible matrix P . Then A ∇ = Ā∇ P -1 .

Proof. For (i), (iii) and (iv) see [START_REF] Niv | Factorization of tropical matrices[END_REF]Lemma 5.7].

(ii) We recall that for any permutation σ we have a corresponding permutation ρ such that σ(i) = ρ(j), σ(j) = ρ(i) and σ(k) = ρ(k) ∀k = i, j. As a result, if P = E i,j is a transposition matrix and A = (a i,j ), then

det(E i,j A) = σ a i,σ(j) a j,σ(i) k =j,i a k,σ(k) = ρ a i,ρ(i) a j,ρ(j) k =j,i a k,ρ(k) = det(A) = det(E i,j )det(A).
If P = E α•j is a diagonal multiplier and A = (a i,j ), then

det(E α•j A) = σ α•a j,σ(j) k =j a k,σ(k) = α σ a j,σ(j) k =j a k,σ(k) = αdet(A) = det(E α•j )det(A).
Inductively the claim holds for every invertible matrix P .

Claim 3.3. If A is a definite matrix, then A ∇ A ∼ = ν A ∇ ∼ = ν AA ∇ .
Proof. See [START_REF] Reutenauer | Inversion of matrices over a commutative semiring[END_REF] for general semirings, or [26, Claim 6.1] for the special case of the supertropical semiring. Theorem 3.5. We denote the application k times of ∇ to A as A ∇ (k) . Then

A ∇ (k) ∼ = ν A ∇ (k+2) ∀k ≥ 1.
That is, applying ∇ to A ∇ is an operator of order 2, which means that the ∇ operation on the set {A ∇ : A ∈ M n (R)} acts like the inverse operation.

Proof. In the same way as in Corollary 3.4 (b) we see that

(A ∇ ) ∇∇ = ( Ā∇ P -1 ) ∇∇ = (P Ā∇∇ ) ∇ ∼ = ν (P Ā∇ ) ∇ = Ā∇∇ P -1 ∼ = ν Ā∇ P -1 = A ∇ .
The general case then follows inductively.

In summary, ∇∇ is a sort of closure operation, which yields elementarily factorizable matrices.

Example 3.6. (i) For every non-singular 2 × 2 matrix

A = a b c d
we have that

A ∇ = det(A) -1 d b c a and A ∇∇ = a b c d = A,
imply that the formula in Theorem 3.5 holds in the 2×2 case for all k ≥ 0. We conclude that A is elementarily factorizable.

(ii) Let

A =   0 0 - -0 0 1 -0   ,
(where -denotes -∞). Thus

A ∇ =   -1 -1 -1 0 -1 -1 0 0 -1   , A ∇∇ =   0 0 -1 ν 0 ν 0 0 1 0 ν 0   , which is different than A ∇ , A ∇ (3) =   -1 -1 -1 0 -1 -1 0 0 -1   = A ∇ and therefore equal A ∇ (2k-1) ∀k ∈ N, and
A ∇ (4) =   0 0 -1 ν 0 ν 0 0 1 0 ν 0   = A ∇∇ and therefore equal A ∇ (2k) ∀k ∈ N.

The closure operation * and power stabilization.

Noticing that A ∇ arises from supertropical algebraic considerations (as its product with A gives a pseudo-identity), we would like to make a connection to the familiar tropical concept of the Kleene star, denoted as A * , which has been widely studied since the 60's. In the next theorem and proposition we give results relevant to [START_REF] Yoeli | A note on a generalization of boolean matrix theory[END_REF] and [START_REF] Sergeev | Multiorder, Kleene stars and cyclic projectors in the geometry of max cones[END_REF], obtained over the tropical setting, with the understanding that tropical equality implies supertropical ghost-equivalent. Since the definition for A * requires that det(A) is bounded by 1 R , we consider here the special case of definite matrices. According to Lemma 2.20(a), we can conclude from the LDM factorization of A * in [START_REF] Litvinov | Tropical and Idempotent Mathenatics[END_REF] that A * is elementarily factorizable. [START_REF] Yoeli | A note on a generalization of boolean matrix theory[END_REF]Theorem 4]. As described in [START_REF] Sergeev | Multiorder, Kleene stars and cyclic projectors in the geometry of max cones[END_REF], the equivalence to A n-1 is immediate from the definition of A * for a definite matrix A:

A k ∼ = ν A k+1 , ∀k ≥ n -1. Proposition 3.8. A ∇ ∼ = ν A * ∼ = ν A n-1 when A is definite. Proof. For A ∇ ∼ = ν A * see
(3.1) A * = i∈N 0 A i .
Indeed, according to [START_REF] Yoeli | A note on a generalization of boolean matrix theory[END_REF]Theorem 2] we have that

A ≤ A 2 ≤ ... ≤ A n-1 ∼ = ν A n ∼ = ν ...
(due to the diagonal, each position can only increase comparing to the corresponding position in the previous power and the value of each entry will stabilize at power n -1 at most).

Combining Corollary 3.4, Theorem 3.6 and Proposition 3.7 we get

A * ∼ = ν A n-1 ∼ = ν A ∇ ∼ = ν A ∇∇ ∼ = ν I A ,
for a definite matrix A of order n.

4. An alternative proof of Theorem 2.10:

det(AB) |= gs det(A)det(B)
The proof of Theorem 2.10 is given by means of a multilinear function. In the classical case there exists an alternative, somewhat easier proof, using the factorization of nonsingular matrices into the product of elementary matrices. Due to Lemma 2.20 we can now give a tropical analog of this proof.

If A or B are strictly singular (without lost of generality B), then according to [18, Theorem 6.5] the columns of B, denoted as C 1 , ..., C n , are tropically dependent:

α i C i ∈ G 0 R , α i ∈ T ∀i.
Denoting the rows of A as R 1 , ..., R n and the columns of AB as c 1 , ..., c n we get

α i c i =   α i R 1 C i . . . α i R n C i   ∈ G 0 R .
Therefore det(AB) ∈ G 0 R as required.

In order to obtain the ∇ operation for (not-strictly) singular matrices we work over the tropical semifield and will prove that det(AB) equals det(A)det(B) plus a term that is attained twice.

Using Theorem 2.17 we have that det(A ∇ ) = det(A) -1 , and it suffices to prove the theorem for A ∇ , B ∇ . By Lemma 3.2(ii), if A or B are transposition matrices and diagonal multipliers then the theorem holds with equality. Due to Lemma 2.20(b) we only have to show the impact of Gaussian matrices.

Let E = E row i+α•row j be a Gaussian matrix adding row j, multiplied by α, to row i. We denote the rows of A as R 1 , ..., R n . Then the rows of EA are the same as A's, except for the i th row which is R i + αR j , therefore

(4.1) det(EA) = σ∈Sn a 1,σ(1) • • • a i,σ(i) • • • a n,σ(n) + σ∈Sn a 1,σ(1) • • • αa j,σ(i) • • • a n,σ(n) ,
where the sum on the left is det(A). In order to analyze the sum on the right, we note that

(4.2) ∀i, j ∃!σ, ρ ∈ S n : σ(i) = ρ(j), σ(j) = ρ(i), and σ(k) = ρ(k) ∀k = i, j,
and let E fix i and j. Thus

(4.3) det(EA) = det(A)+ σ,ρ∈Sn (a 1,σ(1) • • • αa j,σ(i) • • • a j,σ(j) • • • a n,σ(n) + a 1,σ(1) • • • a j,ρ(j) • • • αa j,ρ(i) • • • a n,σ(n) ),
for σ, ρ as in (4.2). Therefore the sum is attained twice, as required. If E is multiplied on the right, then the proof is analogous using column operation instead of row operations.

As a result we have

|AB| = |(AB) ∇ | -1 |= gs |B ∇ A ∇ | -1 = |P -1 1 B∇ Ā∇ P -1 2 | -1 = |P 1 || B∇ Ā∇ | -1 |P 2 | |= gs |P 1 || B∇ | -1 | Ā∇ | -1 |P 2 | = |P -1 1 B∇ | -1 | Ā∇ P -1 2 | -1 = |B ∇ | -1 |A ∇ | -1
= |A||B|, where P 1 , P 2 are the right conductor of B and left conductor of A respectively, and as shown in Lemma 2.20, Ā and B are products of Gaussian matrices, for which the theorem holds according to (4.3).

Tropically conjugate and similar matrices

Although our factorization results for A ∇ follow from those for A * , the ∇ operation has additional applications in representation theory. These relations can be seen by direct computation, but are also a consequence of Theorem 5.4 below. Following the next motivating example, we have an analog to the theorem on similar matrices in classical algebra.

Example 5.3.

Let

A = 2 0 1 0 , B = 1 2 3 1 , B = 3 1 ν 5 3 .
We have A ∇ BA = B , and therefore

f B (x) = x 2 + 3 ν x + 6 ν |= gs f B (x) = x 2 + 1 ν x + 5.
As a result all supertropical eigenvalues of B are supertropical eigenvalues of A ∇ BA.

By taking

B = 0 0 1 2 , B = 1 3 1 2 , A = 0 1 ν -∞ 0 we get A ∇ BA = 2 ν 3 ν 1 2 ν |= gs B .
However, |B| = 2 and |B | = 4 do not ghost-surpass each other, which shows that there need not be a ghost surpassing relation between the characteristic polynomials of tropically conjugate matrices. Proof. Let B = A ∇ BA, where A is a non-singular matrix.

Let Ā be the right definite form of A and E be its right conductor which means

A = ĀE and A ∇ = E -1 Ā∇ .
Using Lemma 3.2, we notice that A ∇ BA and Ā∇ B Ā share the same characteristic polynomial, since

det(xI + A ∇ BA) = det(xI + E -1 Ā∇ B ĀE) = det(E) -1 det(xI + Ā∇ B Ā)det(E) = det(xI + Ā∇ B Ā).
Therefore it is sufficient to prove this theorem for the definite form of A, which according to Proposition 3.8 coincides with its Kleene star and n -1 power.

We consider A to be a definite matrix. We denote the characteristic polynomials of B and B as

f B (x) = x n + n-1 k=0 β k x n-k , f B (x) = x n + n-1 k=0 β k x n-k
respectively, and the entries of the matrices as

A = (a i,j ), A ∇ = A * = A n-1 = A n = (a i,j ) and B = (b i,j ).
We show that β k |= gs β k ∀k. That is, every summand in β k either has an identical summand in β k or appear twice, creating a ghost summand.

The entry of A ∇ BA appears in the (i, σ(i)) position is n t,s=1 a i,t b t,s a s,σ(i) . The summands in β k (the sum of determinants of k × k principal sub-matrices) are

(5.2) k j=1 a i j ,t j b t j ,s j a s j ,σ(i j ) ,
where σ ∈ S k , and t j , s j ∈ {1, ..., n}.

We can factor each permutation into disjoint cycles (i j σ(i j ) σ 2 (i j ) • • • σ -1 (i j )), and we denote i j as j. We abbreviate σ(j), σ 2 (j), ..., σ -1 (j) to σ, σ 2 , ..., σ -1 . Therefore, each permutation track can be factored into cycle tracks:

(5.3) cycles of σ (a j,t j b t j ,s j a s j ,σ )(a σ,tσ b tσ,sσ a sσ,σ 2 ) • • • (a σ -1 ,t σ -1 b t σ -1 ,s σ -1 a s σ -1 ,j ),
obtaining a product of brackets, where the left indices are distinct, describing a permutation. Changing the location of the brackets does not change the value of this term. We use this relocation of brackets to show that identical terms are being obtained on different permutation tracks.

The term a j,t j is a sum arising from the Kleene star:

a j,t j = n r 1 ,...,r n-2 =1 (a j,r 1 a r 1 ,r 2 • • • a r n-2 ,t j ).
Thus, we write (5.3) as cycles of σ (a j,r j,1 a r j,1 ,r j,2 • • • a r j,n-2 ,t j b t j ,s j a s j ,σ )(a σ,r σ,1 a r σ,1 ,r σ,2

• • • a r σ,n-2 ,tσ b tσ,sσ a sσ,σ 2 ) • • • (a σ -1 ,r σ -1 ,1 a r σ -1 ,1 ,r σ -1 ,2 • • • a r σ -1 ,n-2 ,t σ -1 b t σ -1 ,s σ -1 as σ -1 ,j
). Next, we divide our proof into disjoint cases as follows: a j,t j b t j ,s j a s j ,σ(j) Case 1

s j = σ(j) ∀j Case 2 ∃j : s j = σ(j) Case 1.1 j = t j ∀j ⇒ (5.2) is b j,σ(j) Case 1.2 ∃j : i j = t j ⇒ (5.2) is a j,t j b t j ,σ (j)
In case 1, s j = σ(j) ∀j. Thus a s j ,σ(j) is diagonal and therefore equals 0 for every j.

Next, if we have the special sub-case 1.1, j = t j ∀j, then a j,t j are also diagonal entries, equal to 0. Therefore (5.2) is b j,σ(j) , which is a summand in f B .

At this point we have established β k = β k + g k , where g k ∈ R ∀k. It remains to prove that g k ∈ G ∀k, that is every other summand is attained at least twice, creating a ghost summand. In sub-case 1.2, stating that ∃j : i j = t j , some a j,t j is not diagonal. Thus (5.3)

is cycles of σ (a j,r j,1 a r j,1 ,r j,2 • • • a r j,n-2 ,t j b t j ,σ )(a σ,r σ,1 a r σ,1 ,r σ,2 • • • a r σ,n-2 ,tσ b tσ,σ 2 ) • • • (a σ -1 ,r σ -1 ,1 a r σ -1 ,1 ,r σ -1 ,2 • • • a r σ -1 ,n-2 ,t σ -1 b t σ -1 ,j ).
We analyze the second index of each element: r j,1 , j = 1, ..., k.

If all these indices are distinct, then r σ l ,1 → r σ l+1 ,1 describes a permutation. Its track is obtained by moving the brackets one position forward:

cycles of σ a j,r j,1 )(a r j,1 ,r j,2 • • • a r j,n-2 ,t j b t j ,σ a σ,r σ,1 )(a r σ,1 ,r σ,2 • • • a r σ,n-2 ,tσ b tσ,σ 2 • • • a σ -1 ,r σ -1 ,1 )(a r σ -1 ,1 ,r σ -1 ,2 • • • a r σ -1 ,n-2 ,t σ -1 b t σ -1 ,j = cycles of σ (a r j,1 ,r j,1 a r j,1 ,r j,2 • • • a r j,n-2 ,t j b t j ,σ a σ,r σ,1 )(a r σ,1 ,r σ,1 a r σ,1 ,r σ,2 • • • a r σ,n-2 ,tσ b tσ,σ 2 a σ 2 ,r σ 2 ,1 ) • • • (a r σ -1 ,1 ,r σ -1 ,1 a r σ -1 ,1 ,r σ -1 ,2 • • • a r σ -1 ,n-2 ,t σ -1 b t σ -1 ,j a j,r j,1
). If r j,1 = r i,1 for some j = i, then:

If it occurs in the same cycle, we obtain a new permutation by factoring this cycle into two sub-cycles, using the points of repetition (we do not change any of the other cycles):

[(a j,r j,1 a r j,1 ,r j,2 • • • a r j,n-2 ,t j b t j ,σ ) • • • (a σ l ,r j,1 a r j,1 ,r σ l ,2 • • • a r σ l ,n-2 ,t σ l b t σ l ,σ l+1 ) • • • (a σ -1 ,r σ -1 ,1 a r σ -1 ,1 ,r σ -1 ,2 • • • a r σ -1 ,n-2 ,t σ -1 b t σ -1 ,j )] = [(a j,r j,1 a r j,1 ,r σ l ,2 • • • a r σ l ,n-2 ,t σ l b t σ l ,σ l+1 ) • • • (a σ -1 ,r σ -1 ,1 a r σ -1 ,1 ,r σ -1 ,2 • • • a r σ -1 ,n-2 ,t σ -1 b t σ -1 ,j )] [(a σ l ,r j,1 a r j,1 ,r j,2 • • • a r j,n-2 ,t j b t j ,σ ) • • • (a σ l-1 ,r σ l-1 ,1 a r σ l-1 ,1 ,r σ l-1 ,2 • • • a r σ l-1 ,n-2 ,t l-1 σ b t l-1 σ ,σ l )] (the squared brackets indicate cycles).
If an index r j,1 repeats in two different cycles, we obtain a new permutation by joining these cycles into a single cycle, using the points of repetition (we do not change any of the other cycles):

[(a j,r j,1 a r j,1 ,r j,2 • • • a r j,n-2 ,t j b t j ,σ ) • • • (a σ -1 ,t σ -1 b t σ -1 ,j )]• [(a i,r j,1 a r j,1 ,r i,2 • • • a r i,n-2 ,t i b t i ,σ(i) ) • • • (a σ -1 (i),t σ -1 (i) b t σ -1 (i) ,i )] = [(a j,r j,1 a r j,1 ,r i,2 • • • a r i,n-2 ,t i b t i ,σ(i) ) • • • (a σ -1 (i),t σ -1 (i) b t σ -1 (i) ,i ) (a i,r j,1 a r j,1 ,r j,2 • • • a r j,n-2 ,t j b t j ,σ ) • • • (a σ -1 ,t σ -1 b t σ -1 ,j )]
(the squared brackets indicate cycles).

In case 2, ∃j : s j = σ(j), and thus some a s j ,σ(j) is not diagonal.

We obtain the second permutation track using the algorithm of case 1.2, analyzing the indices s j , j = 1, ..., k. If all these indices are distinct, then s σ l → s σ l+1 describes a permutation. Even if this permutation coincides with σ, you still obtain a different summand of (5.2). Its track is obtained by moving the brackets one position backwards:

cycles of σ a j,t j b t j ,s j )(a s j ,σ a σ,tσ b tσ,sσ ) • • • (a s -2 σ ,σ -1 a σ -1 ,t σ -1 b t σ -1 ,s σ -1 )(a s σ -1 ,j = cycles of σ (a s σ -1 ,j a j,t j b t j ,s j )(a s j ,σ a σ,tσ b tσ,sσ ) • • • (a s -2 σ ,σ -1 a σ -1 ,t σ -1 b t σ -1 ,s σ -1 ).
(This case is dual to the previous argument in the sense that 1.2 starts where the entries of A are diagonal, and is being repeated by a permutation track with no such restriction.)

In all remaining summands s j = s i , for some j = i. Then like in case 1.2, we factor and disjoint cycles, using the points of repetition, without changing the non-relevant cycles:

[(a j,t j b t j ,s j a s j ,σ ) • • • (a σ l ,t σ l b t σ l ,s j a s j ,σ l+1 ) • • • (a σ -1 ,t σ -1 b t σ -1 ,s σ -1 a s σ -1 ,j )] = [(a j,t j b t j ,s j a s j ,σ l+1 ) • • • (a σ -1 ,t σ -1 b t σ -1 ,s σ -1 a s σ -1 ,j )]• [(a σ l ,t σ l b t σ l ,s j a s j ,σ ) • • • (a σ l-1 ,t σ l-1 b t σ l-1 ,s σ l-1 a s σ l-1 ,σ l )],
and

[(a j,t j b t j ,s j a s j ,σ ) • • • (a σ -1 ,t σ -1 b t σ -1 ,s σ -1 a s σ -1 ,j )]• [(a i,t i b t i ,s j a s j ,σ(i) ) • • • (a σ -1 (i),t σ -1 (i) b t σ -1 (i) ,s σ -1 (i) a s σ -1 (i) ,i )] = [(a j,t j b t j ,s j a s j ,σ(i) ) • • • (a σ -1 (i),t σ -1 (i) b t σ -1 (i) ,s σ -1 (i) a s σ -1 (i) ,i )• (a i,t i b t i ,s j a s j ,σ ) • • • (a σ -1 ,t σ -1 b t σ -1 ,s σ -1 a s σ -1 ,j )].
The assertion then follows.

Corollary 5.5. If B is tropically similar to B , then

1. Every supertropical eigenvalue of B is a supertropical eigenvalue of B . 2. If f B ∈ T [x] then f B = f B . 3. B satisfies f B in the sense that f B (B) is ghost.. Proof. 1. B is tropically similar to B . If λ is an eigenvalue of B, then f B (λ) |= gs f B (λ) ∈ G.
2. Follows immediately from the theorem. 3. By Theorem 2.28 we get f B (B) |= gs f B (B) ∈ G.

6.

The characteristic polynomial of the pseudo-inverse.

Current research regarding the eigenspaces of a supertropical matrix shows that these spaces may behave in an undesirable fashion, as shown in [START_REF] Izhakian | Supertropical matrix algebra II: solving tropical equations[END_REF]Example 5.7]. The following chapter provides properties regarding the characteristic polynomial, eigenvalues, determinant and trace of A ∇ , which may provide techniques for solving the dependency problem of supertropical eigenspaces.

We observe the following motivating example.

Example 6.1. Let A =   1 0 - 3 4 - --1   (where -denotes -∞). Then f A (x) = x 3 + 4x 2 + 5 ν x + 6 and A ∇ = 6 -1   5 1 - 4 2 - --5   =   -1 -5 - -2 -4 - ---1   .
As a result we get f A ∇ (x) = x 3 + (-1) ν x 2 + (-2)x -6. Multiplying by det(A) we get 6x 3 + 5 ν x 2 + 4x + 0, which has the same coefficients of f A but in opposite order. Consequently, the inverse of every supertropical eigenvalue of A is a supertropical eigenvalue of A ∇ , obtained in the opposite order.

In the following conjecture we formulate how the k th coefficient of the characteristic polynomial of A ∇ is related to the (n -k) th coefficient of the characteristic polynomial of A.

Conjecture 6.2. If A = (a i,j ) ∈ M n (R) is a non-singular matrix, then det(A)f A ∇ (x) |= gs x n f A (x -1 ). An equivalent formulation is det(A)b k |= gs a n-k ∀k = 0, ..., n.
We resolve this conjecture for some special cases described in Proposition 6. det(A) .

For a product of entries of A = (a i,j ), we abuse the graph theory notations of d out (r) and d in (r) to denote the number of appearances of r as a left index and as a right index, respectively. We use [START_REF] Izhakian | Supertropical matrix algebra[END_REF]Proposition 3.17], stating that a product of kn entries of A (6.2) kn j=1 a s j ,t j , where d out (s j ) = d in (t j ) = k, ∀s j , t j = 1, ..., n, is a product of k permutation tracks of A.

The numerator in 6.1 (6.3)

2 j=1 a i j ,σ(i j )
is the product of two entries of the adjoin of A, located in positions (i 1 , σ(i 1 )) and (i 2 , σ(i 2 )). That is, 6.3 is on the permutation track i 1 → σ(i 1 ), i 2 → σ(i 2 ) of the 2 × 2 principle sub-matrix of i 1 and i 2 in adj(A). Thus, 2 j=1 a i j ,σ(i j ) a σ(i j ),i j is the product of two permutation tracks of A: a σ(i j ),i j , which is a permutation track in a 2 × 2 principle sub-matrix of A, and

d out (σ(i j )) = d in (i j ) = 1, ∀j = 1, 2.
Analyzing 6.3, we would like to find that when multiplied by det(A) det(A) 2 , it either describes a summand in a 2 , dominated by one, or appears in another permutation track in a 2 × 2 principle sub-matrix of adj(A). We show that by applying a switch (a change of permutation on the same set of indices) on σ. That is, we multiply 6.3 by a σ(i 1 ),i 2 a σ(i 2 ),i 1 , instead of a σ(i 1 ),i 1 a σ(i 2 ),i 2 , as we did in 6.4: (6.6) a i 1 ,σ(i 1 ) a σ(i 1 ),i 2 a i 2 ,σ(i 2 ) a σ(i 2 ),i 1 .

Since the index counting in 6.6 do not chance (from 6.4), it also describes the product of two permutation tracks of A: We look for the location of a σ(i 2 ),i 1 and a σ(i 1 ),i 2 in 6.7.

The summands obtaining a 2 : If both appear in the same permutation track, without loss of generality π 1 , then the track of π 2 , which is equal to, or dominated by det(A), appears in 6.3. That is, we can consider π 2 to be the permutation attaining det(A), otherwise, it is dominated by the summand having the track attaining det(A) instead of π 2 .

Next, since a σ(i 1 ),i 2 a σ(i 2 ),i 1 is a track of a permutation of S 2 in A, included in the track of π 1 , we have that

a 1,π 1 (1)

Theorem 2 .

 2 10. (The rule of determinants) For n × n matrices A,B over the supertropical semiring R, we have det(AB) |= gs det(A)det(B).

  where C j are cycle tracks and C i • C Id is the extension of C i to a permutation track. Notation. For an element a ∈ R, we denote as a ν the element b ∈ G s.t. a ∼ = ν b, and as â the element b ∈ T s.t. a ∼ = ν b.

Proposition 2 .

 2 [START_REF] Izhakian | Supertropical matrix algebra II: solving tropical equations[END_REF]. adj(AB) |= gs adj(B)adj(A).

Proof. [ 18 ,

 18 Proposition 4.8].

Remark 2 . 23 .

 223 If a, b ∈ T and a k = b k then a k +b k = (a+b) k ∈ G, and therefore a+b ∈ G and a = b. That is, the k-root of a tangible element is unique.

  . See [25, Theorem 3.6]. Corollary 2.32. a. If f A m ∈ T [x] then equality holds in Theorem 2.31. b. Every corner root of f A m is an m th power of a corner root of f A . Proof. See [25, Corollary 3.7 and Corollary 3.10].

Corollary 3 . 4 .

 34 Let A be a matrix with left definite form Ā and right definite form Ã, i.e. A = P Ā = ÃQ for invertible matrices P and Q. Then a.Ā∇∇ ∼ = ν Ā∇ ∼ = ν I A , Ã∇∇ ∼ = ν Ã∇ ∼ = ν I A . b. A ∇∇ ∼ = ν P A ∇ P .Proof. a. According to [19, Corollary 4.4] we know that A ∇ ∼ = ν A ∇ A ∇∇ A ∇ . By applying Claim 3.3 to Ā∇ we can conclude Ā∇ ∼ = ν Ā∇ ( Ā∇∇ Ā∇ ) ∼ = ν Ā∇ Ā∇∇ ∼ = ν Ā∇∇ . According to Claim 3.3 Ā∇ ∼ = ν Ā∇ Ā = Ā∇ P -1 P Ā = A ∇ A = I A , and Ã∇ ∼ = ν Ã Ã∇ = ÃQQ -1 Ã∇ = AA ∇ = I A b. By Lemma 3.2 we have A ∇∇ = (P Ā) ∇∇ = P Ā∇∇ ∼ = ν P Ā∇ = P A ∇ P.

Theorem 3 . 7 .

 37 (See[START_REF] Yoeli | A note on a generalization of boolean matrix theory[END_REF] Theorem 2]) If A is an n×n definite matrix and k is a natural number, then

Definition 5 . 1 .

 51 A matrix B is tropically conjugate to B if there exists a nonsingular matrix A such that (5.1) A ∇ BA |= gs B . If equality holds in (5.1) then we say that B is tropically similar to B . Remark 5.2. If B is tropically similar to B , then a. det(B ) |= gs det(B), with equality when B is non-singular. b. tr(B ) |= gs tr(B).

Theorem 5 . 4 .

 54 If B is tropically similar to B , then f B |= gs f B .

3 and Example 6 . 4 .1≤i 1 <i 2 ≤n σ∈S 2 a i 1 ,σ(i 1 )

 64211 Proposition 6.3. Let A ∈ M n (R). We denote the characteristic polynomials of the matrix A = (a i,j ) andA ∇ = a i,j det(A) as f A (x) = n k=0 a k x k and f A ∇ (x) = n k=0 b k x k respectively, noting that x n f A (x -1 ) = n k=0 a n-k x k . Then, det(A)b k |= gs a n-k , where k ∈ {0, n -2, n -1, n}. Proof. (i) Since det(A) = det(A ∇ ) -1 , we have det(A)b 0 = det(A)det(A ∇ ) = 0 = a n and det(A)b n = det(A) = a 0 . (ii) We have det(A)b n-1 = a 1 since det(A)tr(A ∇ ) = det(A)b n-1 = det(A) a i,i det(A) = a i,i ,which is indeed the sum of determinants of all (n -1) × (n -1) principal sub-matrices in A. (iii) We have det(A)b n-2 |= gs a 2 , where b n-2 is the sum of determinants of 2×2 principal sub matrices in A ∇ , and (6.1) det(A)b n-2 = det(A) det(A) a i 2 ,σ(i 2 )

( 6 . 4 ) [a 1 ,π 1 ( 1 )

 64111 • • • a n,π 1 (n) ][a 1,π 2 (1) • • • a n,π 2 (n) π j (r) ,where a σ(i j ),i j appears in the track of π j ∈ S n , andd out (r) = d in (π j (r)) = 2, ∀r = 1, ..., n, j = 1

(6. 7 )

 7 [a 1,π 1 (1) • • • a n,π 1 (n) ][a 1,π 2 (1) • • • a n,π 2 (n) π j (r) ,where d out (r) = d in (π j (r)) = 2 and π j ∈ S n , ∀r = 1, ..., n, j = 1, 2.

Birmingham and an anonymous referee for their invaluable comments which were very helpful when preparing this paper.

This research was supported by the Israel Science Foundation (grant no. 1207/12).

The summands obtaining a ghost: If a σ(i 2 ),i 1 and a σ(i 1 ),i 2 appears in different permutation tracks, without loss of generality a σ(i 2 ),i 1 on the track of π 1 and a σ(i 1 ),i 2 on the track of π 2 , then (6.8) [

which is identical to 6.3 (we multiplied by a σ(i 2 ),i 1 a σ(i 1 ),i 2 instead of a σ(i 1 ),i 1 a σ(i 2 ),i 2 in the numerator, and then canceled it in the dominator), describes the product of two adjoint-entries, located in positions (i 1 , σ(i 2 )) and (i 2 , σ(i 1 )), and described by the permutations π 1 and π 2 respectively. As a result, products 6.3 and 6.8 describe two identical summands obtained on the permutation tracks of

) is a triangular, non-singular matrix, then A ∇ and all principal submatrices of A and A ∇ are triangular. Denoting A ∇ = (b i,j ), we have b i,i = a -1 i,i . Noting that the determinant of a triangular matrix comes from its diagonal, we get

2. By Proposition 6.3, if A ∈ M i (R), where i = 2, 3, then det(A)f A ∇ (x) |= x i f A (x -1 ) (with equality when i = 2), since the coefficient of x i-1 is the trace, the coefficient of x i-2 is b n-2 and the constant is the determinant. The 4 × 4 case has been verified by a computer calculation.

Consequences of Conjecture 6.2 would be:

1. The inverse of every supertropical eigenvalue of A is a supertropical eigenvalue of A ∇ , since if λ is an eigenvalue of A, then det(A)f A ∇ (λ -1 ) |= gs (λ -1 ) n f A (λ) ∈ G.

If f

Combining these two equations we get det(A)det(A) -1 f A ∇∇ (x) |= gs det(A)x n f A ∇ (x -1 ) |= gs x n x -n f A (x).

Then, if f A ∇∇ ∈ T [x], we have that f A ∇∇ = f A .


Adi Niv
email: adi.niv@inria.fr

Louis Rowen
email: rowen@macs.biu.ac.il

DEPENDENCE OF TROPICAL EIGENSPACES

We study the pathology that causes tropical eigenspaces of distinct supertropical eigenvalues of a non-singular matrix A, to be dependent. We show that in lower dimensions the eigenvectors of distinct eigenvalues are independent, as desired. The index set that differentiates between subsequent essential monomials of the characteristic polynomial, yields an eigenvalue λ, and corresponds to the columns of adj(A + λI) from which the eigenvectors are taken. We ascertain the cause for failure in higher dimensions, and prove that independence of the eigenvectors is recovered in case the "difference criterion" holds, defined in terms of disjoint differences between index sets of subsequent coefficients. We conclude by considering the eigenvectors of the matrix A ∇ := 1 det(A) adj(A) and the connection of the independence question to generalized eigenvectors.

Introduction

Although supertropical matrix algebra as developed in [START_REF] Izhakian | Supertropical matrix algebra[END_REF][START_REF] Izhakian | Supertropical matrix algebra II: solving tropical equations[END_REF] follows the general lines of classical linear algebra (i.e., a Cayley-Hamilton Theorem, correspondence between the roots of the characteristic polynomial and eigenvalues, Kramer's rule, etc.), one encounters the anomaly in [21, Remark 5.3 and Theorem 5.6] of a matrix whose supertropical eigenvalues are distinct but whose corresponding supertropical eigenspaces are dependent. In this paper we examine how this happens, and give a criterion for the supertropical eigenspaces to be dependent, which we call the difference criterion, cf. Definition 3.1 and Theorem 3.4. A pathological example (3.3) is studied in depth to show why the difference criterion is critical. We resolve the difficulty in general in Theorem 3.11 by passing to powers of A and considering generalized supertropical eigenspaces.

1.1. The tropical algebra and related structures. We start by discussing briefly the max-plus algebra, its refinements, and their relevance to applications.

The max-plus algebra was inspired by the function log, as the base of the logarithm approaches 0. In the literature, this structure is usually studied via valuations (see [START_REF] Izhakian | Supertropical semirings and supervaluations[END_REF] and [START_REF] Izhakian | A Glimpse at Supertropical Valuation Theory[END_REF]) over the field K = C{{t}} of Puiseux series with powers in Q, to the ordered group (Q, +, ≥). This valuation gives the lowest power of the series (indeed v(ab) = v(a) + v(b) and v(a + b) ≥ min(v(a), v(b))). Then, we look at the dual structure obtained by defining trop(a) = -val(a) and denoted as the tropicalization of a ∈ K. By setting trop(a + b) to be max{trop(a), trop(b)}, it is obvious that the tropical structure deals with the uncertainty of equality in the valuation, in the form of trop(a + a) = trop(a) (also equals to trop(-a)).

The first author is sported by the French Chateaubriand grant and INRIA postdoctoral fellowship.

The max-plus algebra.

The tropical max-plus semifield is an ordered group T (usually the additive group of real numbers R or the set of rational numbers Q), together with a formal element -∞ adjoined. The ordered group T is made into a semiring equipped with the operations a b = max{a, b} and a b = a + b, denoted here as a + b and ab respectively (see [START_REF] Akian | Handbook of Linear Algebra[END_REF], [START_REF] Itenberg | Tropical Algebraic Geometry. Oberwolfach Seminars[END_REF] and [START_REF] Izhakian | Tropical arithmetic and matrix algebra[END_REF]). The unit element 1 T is really the element 0 ∈ Q, and -∞ serves as the sero element. This arithmetic enables one to simplify non-linear questions by answering them in a linear setting (see [START_REF] Gondran | Path algebra and algorithms[END_REF]), which can be applied to discrete mathematics (see [START_REF] Akian | Discrete max-plus spectral theory[END_REF]), optimization (see [START_REF] Fiedler | Linear optimization problems with inexact data[END_REF]) and algebraic geometry (see [START_REF] Itenberg | Tropical Algebraic Geometry. Oberwolfach Seminars[END_REF]).

In [START_REF] Gaubert | Tropical Scaling of Polynomial Matrices[END_REF] Gaubert and Sharify introduce a general scaling technique, based on tropical algebra, which applies in particular to the companion form, determining the eigenvalues of a matrix polynomial. Akian, Gaubert and Guterman show in [START_REF] Akian | Tropical polyhedra are equivalent to mean payoff games[END_REF] that several decision problems originating from max-plus or tropical convexity are equivalent to zero-sum two player game problems. [START_REF] Litvinov | Tropical and Idempotent Mathenatics[END_REF] is a collection of papers put together by Litvinov and Sergeev. Here, the structure is introduced as a result of the Maslov dequantization applied to traditional mathematics over fields, built on the foundations of idempotent analysis, tropical algebra, and tropical geometry. Applications of idempotent mathematics were introduced by Litvinov and Maslov in [START_REF] Litvinov | Idempotent mathematics: correspondence principle and applications[END_REF].

On the pure mathematical side, contributions are made in [START_REF] Litvinov | Tropical and Idempotent Mathenatics[END_REF] on idempotent analysis, tropical algebras, tropical linear algebra and tropical convex geometry. Elaborate geometric background with applications to problems in classical (real and complex) geometry can be found in [START_REF] Mikhalkin | Tropical geometry and its applications[END_REF]. Here Mikhalkin viewed the tropical structure as a branch of geometry manipulating with certain piecewise-linear objects that take over the role of classical algebraic varieties and describes hypersurfaces, varieties, morphisms and moduli spaces in this setting.

Extensive mathematical applications have been made in combinatorics. In this maxplus language, we may use notions of linear-algebra to interpret combinatorial problems. In [START_REF] Jonczy | Algebraic Path Problems[END_REF] Jonczy presents some problems described by the Path algebra and solved by means of min and max operations. The main combinatorial surveys are [START_REF] Butkovic | On the coefficients of the max-algebraic characteristic polynomial and equation[END_REF], [START_REF] Butkovic | Max-algebra: the linear algebra of combinatorics?[END_REF] of Butkovic and [9] of Butkovic and Murfitt, which focus on presenting a number of links between basic max-algebraic problems on the one hand and combinatorial problems on the other hand. This indicates that the max-algebra may be regarded as a linearalgebraic encoding of a class of combinatorial problems. 1.3. Supertropical algebra. We pass to a cover of the max-plus semifield, called the supertropical semiring, equipped with the ghost ideal G := T ν , as established and studied by Izhakian and Rowen in [START_REF] Izhakian | Supertropical linear algebra[END_REF] and [START_REF] Izhakian | Supertropical algebra[END_REF].

We denote as R = T ∪ G ∪ {-∞} the "standard" supertropical semiring, which contains the so-called tangible elements of the structure and where we have a projection R → G given by a → a ν for a ∈ T . {a ν ∈ G, ∀a ∈ T } are the ghost elements of the structure, as defined in [START_REF] Izhakian | Supertropical algebra[END_REF]. So G inherits the order of T . We write 0 R for -∞, to stress its role as the zero element. This enables us to distinguish between a maximal element a that is attained only once in a sum, i.e., a ∈ T which is invertible, and a maximum that is being attained at least twice, i.e., a + a = a ν ∈ G, which is not invertible. We do not distinguish between a + a and a+a+a in this structure. Note that ν projects the standard supertropical semiring onto G, which can be identified with the usual tropical structure.

In this new supertropical sense, we use the following order relation to describe two elements that are equal up to a ghost supplement: Definition 1.1. Let a, b be any two elements in R. We say that a ghost surpasses b,

denoted a |= gs b, if a = b + ghost. That is, a = b or a ∈ G with a ν ≥ b ν .
We say a is ν-equivalent to b, denoted by a ∼ = ν b, if a ν = b ν . That is, in the tropical structure, ν-equivalence projects to equality.

We adjust these notations for matrices (and in particular for vectors) entry-wise, and for polynomials coefficient-wise.

Important properties of |= gs :

(1) |= gs is a partial order relation (see [START_REF] Izhakian | Supertropical matrix algebra II: solving tropical equations[END_REF]Lemma 1.5]).

( Considering this relation, we regain basic algebraic properties that were not accessible in the usual tropical setting, such as multiplicativity of the tropical determinant, the near multiplicativity of the tropical adjoint, the role of roots in the factorization of polynomials, the role of the determinant in matrix singularity, a matrix that acts like an inverse, common behavior of similar matrices, classical properties of adj(A), and the use of elementary matrices. Tropical eigenspaces and their dependences are of considerable interest, as one can see in [START_REF] Akian | Linear independence over tropical semirings and beyond[END_REF], [START_REF] Akian | Tropical bounds for eigenvalues of matrices[END_REF], [START_REF] Butkovic | On the coefficients of the max-algebraic characteristic polynomial and equation[END_REF], [START_REF] Izhakian | Supertropical linear algebra[END_REF], [START_REF] Izhakian | Supertropical matrix algebra II: solving tropical equations[END_REF] and [START_REF] Sergeev | Max-plus definite matrix closures and their eigenspaces[END_REF].

Many of these properties will be formulated in the Preliminaries section. We would also like to attain a supertropical analog to eigenspace decomposition, but we encounter the example of [START_REF] Izhakian | Supertropical matrix algebra II: solving tropical equations[END_REF]. Our objective in this paper is to understand how such an example arises, and how it can be circumvented, either by introduce the difference criterion of Definition 3.1 or by passing to generalized eigenspaces in § 3.3.3.

Preliminaries

In this section, we introduce the notations, definitions and fundamental results to be used throughout this paper. We present tropical polynomials, together with well-known and recent results. Then we introduce definitions and relevant properties of matrices and vectors in the tropical structure. We introduce extended definitions within the supertropical framework and demonstrate their use in the tropical framework.

(i.e., a k = a • • • a = a + • • • + a = ka), we may consider that a is a k-root of b, denoted as a = k √ b.
This operation is well-defined on T .

Clearly, any tropical polynomial takes the value of the dominant monomial along the T -axis. That having been said, it is possible that some monomials in the polynomial would not dominate for any x ∈ T .

Definition 2.3. Let f (x) = n i=0 α i x n-i ∈ R[x]
be a tropical polynomial. We call monomials in f (x) that dominate for some x ∈ R essential, and monomials in f (x) that do not dominate for any x ∈ R inessential. We write

f es (x) = k∈I α k x n-k ∈ R[x],
where α k x n-k is an essential monomial ∀k ∈ I, called the essential polynomial of f . In the classical sense, a root of a tropical polynomial can only be 0 R , which occurs if and only if the polynomial has constant term 0 R . We would like the roots to indicate the factorization of the polynomial, which leads to the following tropical definition of a root.

Definition 2.4. We define an element r ∈ R to be a root of a tropical polynomial

f (x) if f (r) |= gs 0 R .
We refer to roots of a polynomial being obtained as a simultaneous value of two leading tangible monomials as corner roots, and to roots that are being obtained from one leading ghost monomial as non-corner roots. We factor polynomials viewing them as functions. Then, for every corner root r of f , we may write f as (x + r) k g(x) for some g(x) ∈ R[x] and k ∈ N, where k is the difference between the exponents of the tangible essential monomials attaining r.

Matrices.

As defined over a ring, for matrices A = (a i,j

) ∈ M n×m (R), B = (b i,j ) ∈ M s×t (R)      A + B = (c i,j ) : c i,j = a i,j + b i,j , defined iff n = s, m = t , AB = (d i,j ) : d i,j = n k=1 a i,k b k,j , defined iff m = s . Definition 2.5. Let π ∈ S n and A = (a i,j ) ∈ M n (R). The permutation π of A is the word a 1,π(1) a 2,π(2) • • • a n,π(n) . The word a 1,1 a 2,2 • • • a n,
n is denoted as the identity or Id-permutation, corresponding to the diagonal of A. We write a permutation of A as a product of disjoint cycles C 1 , . . . , C t , where {C i } corresponds to the disjoint cycles composing π.

We define the tropical trace and determinant of A to be

tr(A) = n k=1 a k,k and det(A) = σ∈Sn a 1,σ(1) • • • a n,σ(n) ,
respectively.

In the special case where A ∈ M n (R), we refer to any entry attaining the trace as a dominant diagonal entry. We call a 1,σ(1) • • • a n,σ(n) the weight contributed by σ to the determinant, and any permutation whose weight has the same ν-value as the determinant is a dominant permutation of A.

If there is a single dominant permutation, its weight equals the determinant. Unlike over a field, the tropical concepts of singularity, invertability and factorizability do not coincide. We would like the determinant to indicate the singularity of a matrix. Hence, we define a matrix A ∈ M n (R) to be tropically singular if there exist at least two different dominant permutations. Otherwise the matrix is tropically nonsingular. Consequently, a matrix

A ∈ M n (R) is supertropically singular if det(A) |= gs 0 R and supertropically nonsingular if det(A) ∈ T . A matrix A is strictly singular if det(A) = 0 R .
A surprising result in this context is that the product of two nonsingular matrices might be singular, but we do have: Theorem 2.6. For n × n matrices A, B over the supertropical semiring R, we have

det(AB) |= gs det(A) det(B).
This theorem has been proved in [START_REF] Izhakian | Supertropical matrix algebra[END_REF]Theorem 3.5] due to considerations of graph theory, but also in [START_REF] Gaubert | Théorie des systèmes linéaires dans les dioïdes[END_REF]Proposition 2.1.7] by using the transfer principles (see [2, Theorem 3.3 and Theorem 3.4 ]). These theorems allow one to obtain such results automatically in a wider class of semirings, including the supertropical semiring.

Definition 2.7. Suppose R is a semiring. An R-module V is a semigroup (V, +, 0 V ) together with scalar multiplication R × V → V satisfying the following properties for all r i ∈ R and v, w ∈ V : (1) r(v + w) = rv + rw (2) (r 1 + r 2 )v = r 1 v + r 2 v (3) (r 1 r 2 )v = r 1 (r 2 v) (4) 1 R v = v (5) r • 0 V = 0 V (6) 0 R • v = 0 V .
For any semiring R, let R n be the free module of rank n over R. We define the standard base to be e 1 , . . . , e n , where

e i = 1 T = 1 R , in the i th coordinate 0 T = 0 R , otherwise .
The tropical identity matrix is the n × n matrix with the standard base for its columns. We denote this matrix as I T = I.

A matrix A ∈ M n (R) is invertible if there exists a matrix B ∈ M n (R) such that AB = BA = I.
From now on F := T ∪ G ∪ {0 F }, where its set T is presumed to be a group, and G is its ghost elements. We write V = F n , with the standard base {e 1 , . . . , e n }. Definition 2.8. We define vectors v 1 , . . . , v k in V to be tropically dependent if there exist a 1 , . . . , a k ∈ T such that k i=1 a i v i |= gs -→ 0 F . Otherwise, we define this set of tropical vectors to be independent.

We define two types of special matrices: Definition 2.9. An n × n matrix P = (p i,j ) is a permutation matrix if there exists π ∈ S n such that

p i,j = 0 F , j = π(i) 1 F , j = π(i) .
Since ∀π ∈ S n ∃!σ ∈ S n : σ = π -1 and 1 F is invertible, a permutation matrix is always invertible.

An

n × n matrix D = (d i,j ) is a diagonal matrix if ∃ a 1 , . . . , a n ∈ F : d i,j = 0 F , j = i a i , j = i ,
which is invertible if and only if det(D) is invertible (i.e., a i ∈ T , ∀i).

Remark 2.10. (See [20, Proposition 3.9]) A tropical matrix A is invertible if and only if it is a product of a permutation matrix and an invertible diagonal matrix. These types of products are called generalized permutation matrices, that is

(d i,j ) such that ∃ a 1 , . . . , a n ∈ T , π ∈ S n : d i,j = 0 F , j = π(i) a i , j = π(i) .
We define three types of tropical elementary matrices, corresponding to the three elementary matrix operations, obtained by applying one such operation to the identity matrix.

A transposition matrix is obtained from the identity matrix by switching two rows (resp. columns). This matrix is invertible: E -1 i,j = E i,j , and a product of transposition matrices yields a permutation matrix.

An elementary diagonal multiplier is obtained from the identity matrix where one row (resp. column) has been multiplied by an invertible scalar. This matrix is invertible:

E -1 α•i th row = E α -1 •i th row
, and a product of diagonal multipliers yields an invertible diagonal matrix.

A Gaussian matrix is defined to differ from the identity matrix by having a nonzero entry in a non-diagonal position. We denote as E i th row+α•j th row the elementary Gaussian matrix adding row j, multiplied by α, to row i. By Remark 2.10, this matrix is not invertible. 2.2.1. The supertropical approach. Having established that algebraically G ∪ {-∞} and |= gs effectively take the role of singularity and equality over F, we would like to extend additional definitions to the supertropical setting, using ghosts for zero. A quasi-zero matrix Z G is a matrix equal to 0 F on the diagonal, and whose off-diagonal entries are ghost or 0 F . A diagonally dominant matrix is a non-singular matrix with a dominant permutation along the diagonal. A quasi diagonally dominant matrix D G is a diagonally dominant matrix A whose off-diagonal entries are ghost or 0 F . A quasi-identity matrix I G is a nonsingular, multiplicatively idempotent matrix equal to I + Z G , where Z G is a quasi-zero matrix.

Thus, every quasi-identity matrix I G is quasi diagonally dominant. Using the tropical determinant, we attain the tropical analog for the well-known adjoint.

Definition 2.11. The r, c-minor A r,c of a matrix A = (a i,j ) is obtained by deleting row r and column c of A. The adjoint matrix adj(A) of A is defined as the matrix (a i,j ), where a i,j = det(A j,i ). When det(A) is invertible, the matrix A ∇ denotes 1 det(A) adj(A).

Notice that det(A j,i ) may be obtained as the sum of all permutations in A passing through a j,i , but with a j,i deleted:

det(A j,i ) = σ ∈ Sn : σ(j) = i a 1,σ(1) • • • a j-1,σ(j-1) a j+1,σ(j+1) • • • a n,σ(n) .
When writing each permutation as the product of disjoint cycles, det(A j,i ) can be presented as:

det(A j,i ) = σ ∈ Sn : σ(j) = i (a i,σ(i) • • • a σ -1 (j),j )C σ , where (a i,σ(i) • • • a σ -1 (j),j
) is the cycle with a j,i deleted, and C σ is the product of the cycles of σ that do not include i and j.

Definition 2.12. We say that A ∇ is the quasi-inverse of A over F, denoting

I A = AA ∇ and I A = A ∇ A,
where I A , I A are quasi-identities (see [START_REF] Izhakian | Supertropical matrix algebra II: solving tropical equations[END_REF]Theorem 2.8]).

As a result of these supertropical definitions, we gain a tropical version for two wellknown algebraic properties, proved in Proposition 4.8. and Theorem 4.9. of [START_REF] Izhakian | Supertropical matrix algebra[END_REF].

Proposition 2.13. adj(AB) |= gs adj(B) adj(A).

As a result, one concludes from the third property of |= gs (see Definition 1.1) that

(AB) ∇ |= gs B ∇ A ∇ . Theorem 2.14. (i) det(A • adj(A)) = det(A) n . (ii) det(adj(A)) = det(A) n-1 .
Remark 2.15. (see [START_REF] Niv | On Pseudo-inverses of matrices and their characteristic polynomials in supertropical algebra[END_REF]Remark 2.18]) For a definite matrix A we have

A ∇ = 1 det(A) adj(A) = adj(A),
which is also definite.

The following lemma has been proved in [START_REF] Niv | On Pseudo-inverses of matrices and their characteristic polynomials in supertropical algebra[END_REF]Lemma 3.2], and states the connection between multiplicity of the determinant and the quasi-inverse matrix: Lemma 2.16. Let P be an invertible matrix and A be non-singular.

(i) P ∇ = P -1 . (ii) det(P A) = det(P ) det(A). (iii) (P A) ∇ = A ∇ P ∇ (that is, equality holds in Proposition 2.13). (iv) If A = P Ā, where Ā is the definite form of A with left normalizer P , then A ∇ = Ā∇ P -1 where Ā∇ is definite, with right normalizer P -1 .

Matrix invariants

Let A ∈ M n (F). We continue the supertropical approach by defining v ∈ V , not all singular, such that ∃λ ∈ T ∪ {0 F } where Av |= gs λv, to be a supertropical eigenvector of A with a supertropical eigenvalue λ, having an eigenmatrix A + λI. The characteristic polynomial of A (also called the maxpolynomial, cf. [START_REF] Butkovic | Max-algebra: the linear algebra of combinatorics?[END_REF]) is defined to be

f A (x) = det(xI + A).
The tangible value of its roots are the eigenvalues of A, as shown in [START_REF] Izhakian | Supertropical matrix algebra[END_REF]Theorem 7.10]. Following to Definition 2.4, we may have corner eigenvalues and non-corner eigenvalues.

The coefficient of x n-k in this polynomial is the sum of determinants of all k × k principal sub-matrices, otherwise known as the trace of the k th compound matrix of A. Thus, this coefficient, which we denote as α k , takes the dominant value among the permutations on all subsets of indices of size k:

α k = I ⊆ [n] : |I| = k σ∈S k i∈I a i,σ(i) .
When α k ∈ T , we define the index set of α k , denoted by Ind k , a set I ⊆ [n] on which the dominant permutation defining α k is obtained.

Let f A (x) = n i=0 α i x n-i be the characteristic polynomial of A, with the essential polynomial

f es A (x) = k α i k x n-i k .
Let λ be the corner eigenvalue obtained between the essential monomial

α i k-1 x n-(i k-1 )
and the subsequent essential monomial α i k x n-i k . We denote This algorithm will be demonstrated in §3.2.

I λ = Ind i k \ Ind i k-1 .
The Supertropical Cayley-Hamilton Theorem has been proved in [START_REF] Izhakian | Supertropical matrix algebra[END_REF]Theorem 5.2], and is as follows:

Theorem 2.18. Any matrix A satisfies its tangible characteristic polynomial f A , in the sense that f A (A) is ghost.

One can find a combinatorial proof in [START_REF] Straubing | A combinatorial proof of the Cayley-Hamilton Theorem[END_REF] and a proof using the transfer principle in [START_REF] Akian | Linear independence over tropical semirings and beyond[END_REF].

In analogy to the classical theory, we have Remark 2.20. Recall that a supertropical polynomial is r-primary if it has the unique supertropical root r. It is well-known that any tropical r-primary polynomial has the form (x + r) m for some m ∈ N, and any tropical essential polynomial f A can be factored as a function to a product of primary polynomials, and thus of the form i g i where Another classical property attained in this extended structure is:

g i = (x + r i ) m i .
Proposition 2.21. If λ ∈ T ∪{0 F } is a supertropical eigenvalue of a matrix A ∈ M n (F)
with eigenvector v, then λ i is a supertropical eigenvalue of A i , for every i ∈ N, with respect to the same eigenvector.

Theorem 2.22. Let A be a non-singular matrix.

(1) ( [27, Theorem 3.6]) For any m ∈ N we have

f A m (x m ) |= gs (f A (x)) m ,
implying that the m th -root of every corner eigenvalue of A m is a corner eigenvalue of A.

(2) ( [6, Theorem 5.1]) For A ∇ , the quasi-inverse of A, we have

det(A)f A ∇ (x) |= gs x n f A (x -1 ),
implying that the inverse of every corner eigenvalue of A ∇ is a corner eigenvalue of A.

Dependence of eigenvectors

A well-known decomposition of F n , where F is a field, is the decomposition to eigenspaces of a matrix A ∈ M n (F ). In particular, this decomposition is obtained when the eigenvalues are distinct since, in the classical case, eigenspaces of distinct eigenvalues are linearly independent, which compose a basis for F n . In the tropical case, considering that dependence occurs when a tropical linear combination ghostsurpasses -→ 0 F , such a property need not necessarily hold. In the upcoming section we analyze the dependence between eigenvectors, using their definition according to the algorithm described in Theorem 2.17. We present special cases in which this undesired dependence is resolved. Definition 3.1. The matrix A satisfies the difference criterion if the sets I λ , such that λ is a corner root of f A , are disjoint.

Eigenspaces in lower dimensions.

In the following proposition, we verify that independence of eigenvectors, of distinct eigenvalues, holds in dimension 2 and 3. Proposition 3.2. Let A = (a i,j ) be an n × n nonsingular matrix, where n ∈ {2, 3}, with a tangible characteristic polynomial (coefficient-wise) and n distinct eigenvalues. Then the eigenvectors of A are tropically independent.

Proof. The 2 × 2 case: Let f A (x) = x 2 + tr(A)x + det(A) be the characteristic polynomial of A. If A has two distinct eigenvalues, then these must be λ 1 = tr(A) and λ 2 = det(A) tr(A) . We must have λ 1 > λ 2 , for otherwise either

f A (λ 2 ) = det(A) tr(A) det(A) tr(A) + tr(A) ν = det(A) tr(A) 2 ∈ T ,
or λ 1 = λ 2 , which means the polynomial has one root with multiplicity 2. Without loss of generality, we may assume that tr(A) = a 1,1 . According to the algorithm, since I λ 1 = {1}, λ 1 has the eigenvector obtained by the tangible value of the first column of its eigenmatrix. Since I λ 2 = {2}, λ 2 has the eigenvector obtained by the tangible value of the second column of its eigenmatrix. The determinant is either:

det(A) = a 1,1 a 2,2
, where a 1,1 > a 2,2 and a 1,1 a 2,2 > a 1,2 a 2,1 , (and then the eigenvalues are a 1,1 and a 2,2 ,) or

det(A) = a 1,2 a 2,1 , where a 1,1 a 2,2 < a 1,2 a 2,1 ,
(and then the eigenvalues are a 1,1 and a 1,2 a 2,1

a 1,1 , satisfying a 1,1 > a 1,2 a 2,1 a 1,1
> a 2,2 ). In both cases, the first column of adj(A + λ 1 I) is (a 1,1 , a 2,1 ) and the second column of adj(A + λ 2 I) is (a 1,2 , a 1,1 ), which are tropically independent since a 2 1,1 > a 1,2 a 2,1 . The 3 × 3 case:

This case indicates key techniques for understanding and motivating the general proof on matrices satisfying the difference criterion in §3.3.1. Let

f A (x) = 3 i=0 a i x 3-i = x 3 + tr(A)x 2 + αx + det(A)
be the characteristic polynomial of A. We assign tr(A) to be a 1,1 , i.e., (3.1) a 1,1 > a t,t ∀t = 1.

For the determinant we have six permutations of S 3 . In order to obtain three distinct eigenvalues, we must have

(3.2) λ 1 = tr(A) > λ 2 = α tr(A) > λ 3 = det(A) α ,
for otherwise ∃t, s : f A (λ t ) ∈ T or λ t = λ s . Thus

(3.3) λ 1 λ 2 = α and λ 1 λ 2 λ 3 = det(A).
As a result, Ind 1 ⊆ Ind 2 ; otherwise, a 1,1 together with α yields a permutation whose weight is dominated by det(A), and we get

λ 1 = a 1,1 < det(A)•a 1,1 α•a 1,1 = λ 3 , contrary to (3.2). Therefore,      I λ 1 = {1} \ ∅ = {1} I λ 2 = {1, j} \ {1} = {j}, I λ 3 = {1, j, k} \ {1, j} = {k},
where 1, j, k are distinct. Without loss of generality, we may take j = 2 and k = 3, and obtain the eigenmatrices:

A + λ 1 I =   λ 1 a 1,2 a 1,3 a 2,1 λ 1 a 2,3 a 3,1 a 3,2 λ 1   (since a 1,1 > a t,t ∀t = 1), A + λ 2 I =   λ 1 a 1,2 a 1,3 a 2,1 λ 2 a 2,3 a 3,1 a 3,2 λ 2   (since α • a t,t tr(A) • a t,t ≥ a t,t ∀t = 1)
and

A + λ 3 I =   λ 1 a 1,2 a 1,3 a 2,1 β a 2,3 a 3,1 a 3,2 λ 3   (since det(A) • a 3,3 α • a 3,3 ≥ a 3,3 ),
where β = max{a 2,2 , λ 3 }.

Recalling the algorithm in Theorem 2.17, we let W be the matrix with the (tangible value of the) eigenvectors for its columns

W =       λ 2 1 a 1,2 λ 2 + a 1,3 a 3,2 a 1,3 α + a 1,2 a 2,3 a 2,1 λ 1 + a 2,3 a 3,1 λ 1 λ 2 a 2,3 λ 1 + a 2,1 a 1,3 a 3,1 λ 1 + a 3,2 a 2,1 a 3,2 λ 1 + a 3,1 a 1,2 λ 1 β + a 1,2 a 2,1      
, Furthermore, the (3, 3) position is α = λ 1 λ 2 , since λ 2 > λ 3 :

If α = a 1,2 a 2,1 , then both λ 1 a 2,2 < α and λ 1 λ 3 < α, implying λ 1 β + a 1,2 a 2,1 = α. If α = a 1,1 a 2,2 , then λ 3 < a 2,2 a 1,1 a 1,1 = λ 2 , so β = a 2,2 and λ 1 a 2,2 + a 1,2 a 2,1 = α.
Due to relations (3.1), (3.2) and (3.3), det(W ) = λ 2 1 (λ 1 λ 2 )(λ 1 λ 2 ) is obtained solely from the diagonal and therefore is tangible. (We further elaborate this statement in the generalization proved in Theorem 3.4.)

The pathology appears.

We follow Example 3.3, introduced in [START_REF] Izhakian | Supertropical matrix algebra II: solving tropical equations[END_REF], to show how independence of eigenspaces might fail for dimensions higher then 3, due to the increased variety of indices. While applying the eigenvectors-algorithm, we demonstrate how the classical algorithm for finding the eigenvectors by reduction of the eigenmatrix, still holds in the tropical case, when treating the ghosts as "zero-elements". This illustrative example will provide the motivation for Theorem 3.4, Conjecture 3.5 and Conjecture 3.6, generalizing the connection of the index sets to the dependence of the eigenvectors.

Example 3.3. Let A =     10 10 9 - 9 1 -- ---9 9 ---     . The characteristic polynomial of A is f A (x) = x 4 + 10x 3 + 19x 2 + 27x + 28,
obtained from the permutations (1), (1 2), (1 3 4), (1 3 4)(2), respectively. Therefore,

(3.4)          I λ 1 = {1} \ ∅ = {1}, I λ 2 = {1, 2} \ {1} = {2}, I λ 3 = {1, 3, 4} \ {1, 2} = {3, 4}, I λ 4 = {1, 2, 3, 4} \ {1, 3, 4} = {2}
where λ 1 = 10, λ 2 = 9, λ 3 = 8 and λ 4 = 1, are the eigenvalues of A. As we saw in §3.1, the overlap of the second and fourth sets cannot occur in lower dimensions. The eigenmatrices and eigenvectors are as follows:

For λ 1 : For λ 2 :

A + 10I =    
A + 9I =     10 10 9 - 9 9 -- --9 9 9 --9     ,
and the tangible value of the second column of its adjoint is v 2 = (28, 28, 28, 28) = 28 (0, 0, 0, 0) . This can also be obtained when multiplying the eigenmatrix by

E 4 th row+2•3 rd row E 4 th row+1•2 nd row E 2 nd row+1 st row E 1,4
on the left:

    9 --9 9 ν 9 -9 --10 9 10 ν 10 ν 9 ν 9 ν     ,
and solving the tropically linear system

     9x + 9w ∈ G, 9y + 9w ∈ G, 9z + 9w ∈ G,
which yields (0, 0, 0, 0) , a multiple of v 2 .

For λ 3 :

A + 8I =     10 10 9 - 9 8 -- --8 9 9 --8     ,
and the tangible value of the third column of its adjoint is

v 3 = ( 25, 26, 27, 26) = 25 (0, 1, 2, 1) .
This can also be obtained when multiplying the eigenmatrix by

E 4 th row+1•3 rd row E 4 th row+2•2 nd row E 2 nd row+1 st row E 1,4
on the left:

    9 --8 9 ν 8 -8 --8 9 11 ν 10 ν 9 ν 10 ν     ,
and solving the tropically linear system

     9x + 8w ∈ G, 8y + 8w ∈ G, 8z + 9w ∈ G,
which yields (7, 8, 9, 8) = 7 (0, 1, 2, 1) , a multiple of v 3 .

For

λ 4 A + 1I =     10 10 9 - 9 1 ν -- --1 9 9 --1     ,
and the tangible value of the second column of its adjoint is v 4 = (12, 27, 28, 20) = 12 (0, [START_REF] Izhakian | Tropical arithmetic and matrix algebra[END_REF][START_REF] Izhakian | Supertropical semirings and supervaluations[END_REF][START_REF] Butkovic | Max-algebra: the linear algebra of combinatorics?[END_REF] .

This can also be obtained when multiplying the eigenmatrix by

E 4 th row+(-1)•1 st row E 4 th row+•2 nd row E 2 nd +(-1)•1 st row on the left:     10 10 9 - 9 ν 9 8 - --1 9 9 ν 9 ν 8 ν 1 ν     ,
and solving the tropically linear system

     10x + 10y + 9z ∈ G, 9y + 8z ∈ G, 1z + 9w ∈ G,
which yields (x, 8, 9, 1) , where x ≤ 8.

From the fourth position of Av |= gs λv, we get 9x |= gs 2 which implies x = -7. Thus the eigenvector is (-7, 8, 9, 1) = -7(0, 15, 16, 8), a multiple of v 4 . Next, we examine the dependence of the eigenvectors, using the matrix W having these vectors for its columns: The determinant of W is 112 ν and is obtained by the permutations (1)(2)(3 4) and (1)(2 4)(3). One can see that the ghost part of the product is attained in the principal sub-matrix {2, 3, 4} × {2, 3, 4}, where the pathology of the index sets occurs. We rewrite W using the eigenvalues and the entries of A = (a i,j ), in order to understand this dependence:

W =    
W =          λ 3 1 a 1,2 λ 2 2 λ 2 3 a 1,3 λ 2 4 a 1,2 λ 2 1 a 2,1 λ 1 λ 2 2 λ 3 a 2,1 a 1,3 a 1,3 a 3,4 a 4,1 λ 1 a 3,4 a 4,1 a 3,4 a 4,1 a 1,2 λ 3 a 1,2 a 2,1 a 3,4 a 4,1 a 1,2 λ 2 1 a 4,1 λ 2 a 4,1 a 1,2 λ 3 a 4,1 a 1,3 λ 4 a 4,1 a 1,2          .
The determinant is attained by

λ 3 1 (λ 1 λ 2 2 )(a 3,4 a 4,1 a 1,2 )(λ 3 a 4,1 a 1,3 ) and λ 3 1 (a 1,3 a 3,4 a 4,1 )(λ 3 a 1,2 a 2,1 )(λ 2 a 4,1 a 1,2
). All elements are identical except for λ 1 λ 2 , which appears in the first term, and a 1,2 a 2,1 , which appears in the second term, but we know that

λ 1 λ 2 = a = a 1,2 a 2,1 .
That is, the determinant is not attained twice because of arbitrarily equal values (for example 1 • 6 = 3 • 4), or because of repeating values in the entries of A (such as 9 and 10), but rather is attained twice by precisely the same entries of A after a permution:

λ 3 1 (λ 1 λ 2 2 )(a 4,1 a 1,2 a 3,4 )(a 4,1 a 1,3 λ 3 ) = λ 3 1 [a][λ 2 a 4,1 a 1,2 ][a 3,4 a 4,1 a 1,3 ][λ 3 ] = λ 3 1 [λ 2 a 4,1 a 1,2 ][a 3,4 a 4,1 a 1,3 ][a 1,2 a 2,1 λ 3 ].
3.3. Resolving the pathology. In this section we offer sufficient conditions for independence, and provide two conjectures on the eigenvectors of the quasi-inverse of a matrix.

3.3.1.

The resolution by means of disjoint index sets.

We show that the pathology is a Zariski-closed condition.

Theorem 3.4. Let A = (a i,j ) be an n×n nonsingular matrix, with tangible characteristic polynomial (coefficient-wise) and n distinct eigenvalues. If A satisfies the difference criterion, then the eigenvectors of A are tropically independent.

Proof. Let f A (x) = n i=0 α i x n-i be the characteristic polynomial of A, which means α 0 = 0, α 1 = tr(A) and α n = det(A).

Without loss of generality, we may assume that tr(A) is a 1,1 , i.e., (3.5) a 1,1 > a t,t ∀t = 1.

In order to get n distinct eigenvalues, we must have f A (x) = f es A (x), or equivalently:

(3.6) λ 1 = tr(A) > λ 2 = α 2 tr(A) > λ 3 = α 3 α 2 > • • • > λ n-1 = α n-1 α n-2 > λ n = det(A) α n-1 ,
where {λ l } n l=1 are the corner-roots of f A . Otherwise ∃t, s such that f A (λ t ) ∈ T or λ t = λ s , contrary to hypothesis. Thus

λ 1 = a 1,1 ≤ α k α k-1 • a 1,1 • a 1,1 = λ k , contradicting (3.6).
The indices Ind k \ {1} = {j 2 , . . . , j k } are where the pathology lies. Straightforward, since Ind 0 = ∅, we have that ∀s ∈ {2, ..., k} ∃i ≤ k : j s ∈ I λ i . Assume that ∀l ≤ k Ind l-1 ⊆ Ind l and ∃s ∈ {2, ..., k} : j s / ∈ Ind k+1 . The index j s must appear again in the index set of some subsequent coefficient (since Ind n of the determinant includes all of the indices). Denoting the first such coefficient as α t , that means j s ∈ Ind t \ Ind t-1 = I λt , t > k, contradicting the difference criterion.

Thus we require that the Ind k ⊆ Ind k+1 , ∀k, and without loss of generality we may assume that j k = k, ∀k = 1, . . . , n. As a result, I λ k = {k} and (3.5)-(3.6) become:

(3.8) a 1,1 = λ 1 > a t,t , ∀t = 1,
and ∀k > 1 :

(3.9)

a 1,1 > λ k = α k • a t,t α k-1 • a t,t ≥ a t,t , ∀t ≥ k,
with equality only when k = t, and

(3.10) β k,t = max{λ k , a t,t }, ∀t < k.

Thus, the eigenmatrices are (A written off the diagonal means that the off-diagonal entries are identical to those of A):

A+λ 1 I=     λ 1 A . . . A λ 1     , A+λ 2 I=      λ 1 A λ 2 . . . A λ 2      , A+λ 3 I=         λ 1 A β 3,2 λ 3 . . . A λ 3         , . . .

Or in general

A + λ k I = (b (k)
i,j ), where (3.11) b

(k) i,i =      λ 1 , i = 1 β k,i , 1 < i < k λ k , i ≥ k and b (k) i,j = a i,j , ∀i = j.
Let adj(A) = (a i,j ) and W = (w i,j ) be the matrix with the (tangible value of the) eigenvectors for its columns.

On the one hand the (k, k) position of A + λ k I is λ k , and the (k, k) position of

(A + λ k I)(adj(A + λ k I)) is det(A + λ k I) = f A (λ k ) ∈ G 0 F , attained by α k λ n-k k + α k-1 λ n-k+1 k = (λ 1 • • • λ k-1 λ n-k+1 k ) ν .
Therefore, the (k, k) position of adj(A + λ k I) (and therefore of W ) is at most

λ 1 • • • λ k-1 λ n-k k .
On the other hand, one of the summands in the (k, k) position of adj(A + λ k I) is

λ n-k k det(M k,...,n ),
where M k,...,n is the (k -1) × (k -1)-principal sub-matrix of A + λ k I, obtained by rows and columns 1, . . . , k-1. Since it differs from the corresponding (k-1)×(k-1)-principle sub-matrix of A by having a greater diagonal, we get:

det(M k,...,n ) ≥ ν α k-1 = λ 1 • • • λ k-1 .
Thus, the (k, k) position in adj(A + λ k I) is at least

λ 1 • • • λ k-1 λ n-k k .
As a result, the diagonal positions of W are:

(3.12) w k,k = λ 1 • • • λ k-1 λ n-k k .
We recall that when A is nonsingular, det(adj(A)) = det(A) n-1 is attained solely by the permutation σ -1 , where σ attains det(A).

Since det

(A) = λ 1 • • • λ n , we have det(adj(A)) = λ n-1 1 • • • λ n-1 n < n i=1 w i,i .
Next, using the relations (3.5)-(3.12), we show that each permutation of W weights at least as much as the corresponding permutation in adj(A), with the unique maximal weight coming from the Id-permutation.

Examining the entries of W , we notice that the diagonal entries of A are replaced according to (3.11), and that λ n does not appear. Every non-Id-permutation π of W includes cycles of A, and therefore satisfies at least one of the following properties:

• π is a single k-cycle. Then the permutation π of W is smaller than or equal to λ k = λ 1 • • • λ k ∈ T (with Ind k = {1, . . . , k}). • π contains the product of two different k-cycles. Then the permutation π of W is strictly smaller than λ 2 k , because every coefficient is tangible. • π is a k-cycle that does not include one of the indices 1, . . . , k. Then the permutation π of W is strictly smaller than λ k . • The permutation π of W includes, other than entries of A, only the eigenvalue λ 1 .

Then the cycles of π do not include the index 1, and since λ 1 = a 1,1 it will expend a k-cycle into a k + 1-cycle which is strictly smaller than (λ

1 λ 2 • • • λ k-1 λ k )λ k .
• The permutation π of W does not include λ t for any t. Then π is a product of n-1 permutations of A, whose product is smaller than or equal to det(adj(A)). As a result, every permutation other than Id is strictly smaller than the Id. Thus, the determinant of W is attained solely by the diagonal and therefore is tangible.

The resolution by means of the quasi-inverse.

We examine the implications of this pathology on adj(A), which according to Theorem 2.14 and Lemma 2.16 has determinant det(A) n-1 attained solely by the permutation σ -1 , where det(A) is attained solely by σ. This will lead us to Conjecture 3.5 and Conjecture 3.6.

Calculating the adjoint of the matrix A from Example 3. (It is easy to see that the determinant is attained by the permutation invert to the one attaining det(A), and taking on its value three times. f A ∇ is obtained by dividing the k th coefficient by det(A) k for every k: f A ∇ (x) = x 4 +(-1)x 3 +(-9)x 2 +(-10 ν )x+(-28), which satisfies Theorem 5.1 in [START_REF] Akian | Identities on exterior power of tropical matrices[END_REF].) Therefore,

         {2} \ ∅ = {2}, {3, 4} \ {2} = {3, 4}, {2, 3, 4} \ {3, 4} = {2}, {1, 2, 3, 4} \ {2, 3, 4} = {1},
respectively. However, calculating the eigenvalues of adj(A) we discover that f es adj(A) (x) = x 4 + 27x 3 + 74 ν x + 84. That is, the dependence in the principal sub-matrix {2, 3, 4} × {2, 3, 4} (identical to the location of dependence in W ) increased the coefficient of x, which caused the monomial of x 2 to be inessential.

Indeed, the dependence in W caused by

λ 2 (λ 1 λ 2 )(a 4,1 a 1,2 a 3,4 )(a 4,1 a 1,3 )λ 3 = λ 2 (a 4,1 a 1,2 )(a 3,4 a 4,1 a 1,3 )(a 1,2 a 2,1 )λ 3 ,
would yield the equality

(λ 1 λ 2 )(a 4,1 a 1,2 a 3,4 )(a 4,1 a 1,3 ) a 4,1 a 1,3 a 3,4 a 1,1 = (a 4,1 a 1,2 )(a 3,4 a 4,1 a 1,3 )(a 1,2 a 2,1 ) a 4,1 a 1,3 a 3,4 a 1,1 ,
using entries of A to express λ 2 λ 3 and later λ 1 λ 2 . We recall Theorem 2.22 for the following conjecture.

Replacing

Conjecture 3.5. Let A be a non-singular matrix with n distinct eigenvalues. If the eigenvectors of A are dependent, then • det(A)f A ∇ (x) strictly ghost-surpasses x n f A (x -1 ).

• f A ∇ = f es A ∇ , which means A ∇ has fewer distinct eigenvalues than A. Conjecture 3.6. Let A be a non-singular matrix. If A ∇ has n distinct eigenvalues then their corresponding eigenvectors are independent.

The resolution by means of generalized eigenspaces.

Eigenspaces are studied in [START_REF] Izhakian | Supertropical matrix algebra II: solving tropical equations[END_REF] and are defined in [START_REF] Izhakian | Supertropical matrix algebra III: Powers of matrices and their supertropical eigenvalues[END_REF] to be spanned by supertropical eigenvectors. Let V = F n . Definition 3.7. A tangible vector v ∈ V is a generalized supertropical eigenvector of A, with generalized supertropical eigenvalue λ ∈ T , if (A + λI) m v is ghost for some m ∈ N. If A m v is itself ghost for some m, we call the eigenvector v degenerate.

The minimal such m is called the multiplicity of the eigenvalue (and also of the eigenvector).

The generalized supertropical eigenspace V λ with generalized supertropical eigenvalue λ ∈ T is the set of generalized supertropical eigenvectors with generalized supertropical eigenvalue λ.

Note that if v is a degenerate eigenvector, then it belongs to V λ for all sufficiently small λ. Lemma 3.8. V λ is indeed a supertropical subspace of V .

Proof. Let v, u ∈ V λ . Thus ∃m, t : (A + λI) m v |= gs 0 F and (A + λI) t u |= gs 0 F , and therefore for any a ∈ F (A + λI) (m+t) (v + au) = (A + λI) t (A + λI) m v + a(A + λI) m (A + λI) t u |= gs 0 F . Remark 3.9. We have the following hierarchy:

Av |= gs λv, implies A m v |= gs λ m v, implies A m v+λ m v |= gs 0 F , implies (A+λI) m v |= gs 0 F . This approach gives some insight into the difference criterion. Lemma 3.10. For m = n!, the diagonal is a dominant permutation of A m . The difference criterion is satisfied for A iff the diagonal entries of A m are distinct.

Proof. The first assertion is clear.

Suppose that in A m some index i appears in both I k and I k for k < k , where k is taken minimal such. Then all the previous I j are disjoint, so, rearranging the diagonal entries, we may assume that i appears in the |I 1 |+• • •+|I k-1 |+α position in the diagonal for some 1 ≤ α ≤ |I k |. But i must also appear in the analogous position arising from I k , which is further down, so A m has a double entry, and f A has a double root. Lemma 3.11. If A is nonsingular and diagonally dominant, then the diagonal of A is tangible.

Proof. The determinant is the product of the diagonal entries, so each is tangible.

In view of Remark 2.20, we can refine V λ . Write f A = i g i where g i = (x + λ i ) t i , with the λ i distinct, and let fi = j =i g j . (Thus, f A = g i fi .) Suppose v ∈ fi (A)V λ . Then g i (A)v ∈ f A (A)V is ghost, implying v ∈ V λ . Thus, we can define the subspace

V λ i = j =i g j (A) V.
Definition 3.12. A matrix A is strongly nonsingular if A m is nonsingular for all m.

  ) If a |= gs b then ac |= gs bc. (3) If a |= gs b and c |= gs d then a + c |= gs b + d and ac |= gs bd.

2. 1 .

 1 Tropical Polynomials. Notation 2.1. Throughout, for an element a ∈ R, we consider a ν to be the element b ∈ G s.t. a ∼ = ν b, and â to be the element b ∈ T s.t. a ∼ = ν b. (We define 0ν R = 0 R = 0 R .)We adjust these notations for matrices (and in particular for vectors) entry-wise, and for polynomials coefficient-wise.

Definition 2 . 2 .

 22 Let k ∈ N. Defining b = a k to be the tropical product of a by itself k times

Theorem 2 .

 2 17. (The eigenvectors algorithm, see [21, Remark 5.3 and Theorem 5.6].) Let t ∈ I λ . The tangible value of the t th -column of adj(λI + A) is a tropical eigenvector of A with respect to the eigenvalue λ.

Proposition 2 .

 2 19. ([20, Proposition 7.7]) The roots of the polynomial f A (x) are precisely the supertropical eigenvalues of A.

10 ν 10

 10 value of the first column of its adjoint is v 1 = (30, 29, 28, 29) = 28 (2, 1, 0, 1) . This can also be obtained when multiplying the eigenmatrix byE 2 4 th row+1•3 rd row E 4 th row+1•2 nd row E 2 nd row+1 st row E 1,4 on the left: ∈ G, 10y + 10w ∈ G, 10z + 9w ∈ G,which yields (11, 10, 9, 10) = 9 (2, 1, 0, 1) , a multiple of v 1 .

(3. 7 )

 7 λ 1 • • • λ k = α k , ∀k, and in particular λ 1 • • • λ n = det(A), where α k is attained solely by this product since f A ∈ T [x]. Moreover, {1} = Ind 1 ⊆ Ind k , for every k ≥ 1. Otherwise, a 1,1 together with α k-1 yields a permutation on k indices, dominated by α k , and we get

  f adj(A) (x) = x 4 + 27x 3 + 47x 2 + 74 ν x + 84,obtained by the permutations (2), (3 4), (2 4 3) and (2)(3 4), (3 1 4)(2), respectively.

a 4 ,1 a 1,2 a 1 , 1 by a 2 , 2 I λ 1 =

 411221 on both sides, we obtain the equality (a 1,2 a 2,1 a 3,4 )(a 4,1 a 1,3 a 3,4 )(a 4,1 a 1,3 a 2,2 ) = (a 3,4 a 4,1 a 1,2 )(a 2,1 a 1,3 a 3,4 )(a 4,1 a 1,3 )a 2,2 which causes the coefficient of x to be a ghost.As a result, {2} \ ∅ = {2},I λ 2,3 = {2, 3, 4} \ {2} = {3, 4}, I λ 4 = {1,2, 3, 4} \ {2, 3, 4} = {1} , where λ 1 = 27, λ 2,3 = 23.5 (with multiplicity 2), λ 4 = 10, and independent eigenvectors: v 1 = (66, 81, 82, 74) = 66(0, 15, 16, 8), v 2,3 = 65 -1 (65, 69.5, 74, 69.5) from the third column = (0, 4.5, 9, 4.5) = 69.5 -1 (69.5, 74, 78.5, 74) from the fourth column , and v 4 = (74, 65, 55, 65) = 55(19, 10, 0, 10).

  The supertropical version of this is given in[START_REF] Izhakian | Supertropical algebra[END_REF] Theorem 8.25 and Theorem 8.35].

In this case, A has no nonzero degenerate eigenvectors, for if

Theorem 3.13. If A is strongly nonsingular, then the V λ are independent.

Proof. We can replace A by A n! and assume that A is diagonally dominant and that V λ are eigenspaces of A. We use the notation following Lemma 3.11. Given a ghost dependence, i.e., u i=1 γ i fi (A)v i ghost for tangible γ i , the fact that A is strongly nonsingular implies that λ tu u dominates all λ tu-j u β j , for all β < λ u . Hence, when x is to be specialized to these β, λ tu u dominates j λ tu-j u

x j u = g u , and thus some component of γ u λ tu u fu (A)v u is dominant in γ u g u (A) fu (A)v u , a ghost. Therefore f u (A) = fu (A)g u (A)v u is ghost, implying some power of A ghost annihilates v u , in contradiction to A being strongly nonsingular.