On the characteristic polynomial of a supertropical adjoint matrix
aa r X i v : . [ m a t h . C O ] O c t ON THE CHARACTERISTIC POLYNOMIALOF A SUPERTROPICAL ADJOINT MATRIX
YAROSLAV SHITOV
Abstract.
Let χ ( A ) denote the characteristic polynomial of a matrix A overa field; a standard result of linear algebra states that χ ( A − ) is the reciprocalpolynomial of χ ( A ). More formally, the condition χ n ( X ) χ k ( X − ) = χ n − k ( X )holds for any invertible n × n matrix X over a field, where χ i ( X ) denotes thecoefficient of λ n − i in the characteristic polynomial det( λI − X ). We confirma recent conjecture of Niv by proving the tropical analogue of this result. The supertropical semifield is a relatively new concept arisen as a tool for study-ing problems of tropical mathematics [2]. The supertropical theory is now a devel-oped branch of algebra, and we refer the reader to [3] for a survey of basics andapplications. Our arguments make use of some other structures including fields andpolynomial rings over them, so it will be convenient for us to work with slightly un-usual equivalent description of supertropical semifield. In particular, we will denotethe operations by ⊕ and ⊙ to avoid confusion with standard operations + and · over a field. For the same reason, we will use the notation u ⊙ i in supertropical set-ting while u i will denote a power of element of a field. Similarly, we will denote thesupertropical determinant by det ◦ reserving the notation det for usual determinantover a field. Let us recall the definitions of concepts mentioned above.Let ( G , ∗ , , ≤ ) be an ordered Abelian group, and G (0) and G (1) be two copies of G .We consider the semiring S = G (0) ∪ G (1) ∪ { ε } with two commutative operations,denoted by ⊕ and ⊙ . Assume i, j ∈ { , } , s ∈ S , a, b ∈ G , and let a < b ; theoperations are defined by ε ⊕ s = s , ε ⊙ s = ε , a ( i ) ⊕ b ( j ) = b ( j ) , b ( i ) ⊕ b ( j ) = b (0) , a ( i ) ⊙ b ( j ) = ( a ∗ b ) ( ij ) . One can note that S is isomorphic to supertropical semifield ,and the elements from G (0) and G (1) correspond to ghost and tangible elements,respectively. We define the mapping ν sending a ( i ) to a ∈ G and ε to ε ; we saythat c, d ∈ S are ν -equivalent whenever ν ( c ) = ν ( d ), and we write c ≈ ν d in thiscase. Also, we write c | = d if either c = d or c = d + g , for some ghost element g ; this relation is known as ghost surpassing relation, which is one of fundamentalconcepts replacing equality in many theorems taken from classical algebra [3]. By u ⊙ i we denote the i th supertropical power of u , that is, the result of multiplying u by itself i times. Let A = ( a ij ) be a supertropical matrix; its determinant isdet ◦ A = M σ ∈ S n a σ (1) ⊙ . . . ⊙ a nσ ( n ) , where S n denotes a symmetric group on { , . . . , n } . The matrix A is said to be non-singular if det ◦ A is tangible; equivalently, A is non-singular if det ◦ A has a Mathematics Subject Classification.
Key words and phrases. tropical algebra, matrix theory. multiplicative inverse in S . The ( i, j )th cofactor of A is the supertropical determi-nant of the matrix obtained from A by removing i th row and j th column. By adj ◦ A we denote the adjoint of A , that is, the n × n matrix whose ( i, j )th entry equalsthe ( j, i )th cofactor. By χ k ◦ ( A ) we denote the supertropical sum of all principal k × k minors of A , that is, the coefficient of λ ⊙ ( n − k ) in the characteristic polyno-mial det ◦ ( A ⊕ λ ⊙ I ◦ ). Note that ε and 0 (1) are neutral elements with respect to ⊕ and ⊙ , respectively; therefore, the supertropical identity matrix I ◦ has elements0 (1) on diagonal and ε ’s everywhere else. The following has been an open problem. Conjecture 1. [1, Conjecture 6.2]
Let A ∈ S n × n be a non-singular matrix. Then, χ k ◦ (adj ◦ A ) | = (det ◦ A ) ⊙ ( k − ⊙ χ n − k ◦ ( A ) holds for all k ∈ { , . . . , n } . Actually, Niv formulates this conjecture in a slightly different but equiva-lent way. If A is tropically non-singular, then its pseudoinverse is defined as A ∇ = (det A ) ⊙ ( − ⊙ (adj ◦ A ). Multiplying both sides of equality in Conjecture 1by (det A ) ⊙ (1 − k ) , one gets (det ◦ A ) ⊙ χ k ◦ ( A ∇ ) | = χ n − k ◦ ( A ), which is exactly theformulation given by Niv. To prove Conjecture 1, we consider the n × n matrix V consisting of variables ( v ij ), and we define polynomials α, β ∈ S [ V ] as α = χ k ◦ (adj ◦ V ) , β = (det ◦ V ) ⊙ ( k − ⊙ χ n − k ◦ ( V ) . Note that any coefficient of α and β is either 0 (0) or 0 (1) ; define γ ∈ S [ V ] as thesupertropical sum of those monomials that appear in β with tangible coefficients. Claim 2. If A ∈ S n × n is a non-singular matrix, then β ( A ) = γ ( A ) .Proof. By definition, β is supertropical sum of monomials m µ = m µ ⊙ m µ , where m µ = n K i =1 v iσ ( i ) ! ⊙ . . . ⊙ n K i =1 v iσ k − ( i ) ! , m µ = K j ∈ J v jτ ( j ) , over all tuples µ = ( σ , . . . , σ k − , J, τ ) such that σ , . . . , σ k − ∈ S n , a subset J ⊂{ , . . . , n } has n − k elements, and τ is a permutation of J .Assume β ( A ) = ε . Suppose that there exist distinct tuples µ and µ ′ =( σ ′ , . . . , σ ′ k − , J ′ , τ ′ ) satisfying m µ = m µ ′ and β ( A ) ≈ ν m µ ( A ). Then, since thereis a unique permutation σ such that J ni =1 a iσ ( i ) ≈ ν det ◦ A , we have that all σ t andall σ ′ t are equal to σ . In particular, we have m µ = m µ ′ , which implies m µ = m µ ′ .Note that the latter condition in turn implies J = J ′ and τ = τ ′ , a contradiction.Therefore, either β ( A ) = ε or ν ( m ( A )) = ν ( β ( A )) holds for any monomial m that appears in β with ghost coefficient. This means that we can remove all thesemonomials from β without changing the value of β ( A ). (cid:3) Claim 3.
If a monomial v ⊙ k ⊙ . . . ⊙ v ⊙ k nn nn appears with a tangible coefficient ineither α or β , then it appears in both α and β with coefficients different from ε .Proof. Let X = ( x ij ) be a matrix whose entries are variables of the polynomial ring C [ x , . . . , x nn ], and define ϕ = χ k (adj X ), ψ = (det X ) k − χ n − k ( X ). Let us getrid of brackets by distributivity in the standard expressions of ϕ and ψ and denotethe expressions we obtain before canceling terms by ϕ and ψ , respectively. Now,if we replace any monomial ± x k . . . x k nn nn by 0 (1) ⊙ v ⊙ k ⊙ . . . ⊙ v ⊙ k nn nn in ϕ and ψ , we get α ( V ) and β ( V ). Since the equality ϕ = ( − n ψ is true for matrices overa field, the total number of appearances in ϕ and ψ is even for any monomial. (cid:3) N THE CHARACTERISTIC POLYNOMIAL OF A SUPERTROPICAL ADJOINT MATRIX 3
Claim 4.
Let A ∈ S n × n be a non-singular matrix. If α ( A ) is a tangible element,then there is s ∈ S such that β ( A ) = α ( A ) ⊕ s .Proof. By assumption, there is a monomial m = v ⊙ k ⊙ . . . ⊙ v ⊙ k nn nn appearing in α with tangible coefficient such that m ( A ) = α ( A ). But m appears in β as well byClaim 3, so the result follows. (cid:3) Claim 5.
There is a polynomial ρ ∈ S [ V ] such that α = γ ⊕ ρ .Proof. By definition of γ , it is the sum of monomials that appear in β with tangiblecoefficients. All these monomials appear in α as well by Claim 3. (cid:3) Proof of Conjecture 1.
By Claims 2 and 5, there is u ∈ S such that α ( A ) = β ( A ) ⊕ u .Claim 4 rules out the case when u is tangible and greater than β ( A ). It remainsto note that α ( A ) and β ( A ) are, respectively, the left-hand and right-hand sides ofthe assertion of Conjecture 1. (cid:3) References [1] A. Niv, On pseudo-inverses of matrices and their characteristic polynomials in supertropicalalgebra, Linear Algebra Appl. 471 (2015) 264–290.[2] Z. Izhakian, L. Rowen, Supertropical algebra, Adv. Math. 225 (2010) 2222–2286.[3] Z. Izhakian, L. Rowen, A guide to supertropical algebra, Trends in Mathematics, Advancesin Ring Theory (2010) 283–302.
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