Min-plus eigenvalue of tridiagonal matrices in terms of the ultradiscrete Toda equation

The discrete Toda molecule equation can be used to compute eigenvalues of tridiagonal matrices over conventional linear algebra, and is the recursion formula of the well-known quotient difference algorithm for tridiagonal eigenvalues. An ultradiscretization of the discrete Toda equation leads to the ultradiscrete Toda (udToda) equation, which describes motions of balls in the box and ball system. In this paper, we associate the udToda equation with eigenvalues of tridiagonal matrices over min-plus algebra, which is a semiring with two operation types: and . We also clarify an interpretation of the udToda variables in weighted and directed graphs consisting of vertices and edges.


Introduction
The Toda lattice equation [20] is a mass-spring system with exponential decay. It is expressed as where x k (t) denotes displacement of the kth mass from the equilibrium position at continuous time t. The Toda lattice equation (1) is a completely integrable system with many good properties, such as soliton solutions, Lax pair, conserved quantities, and the Bäcklund transformation. The Toda lattice equation (1) Specifically, one step of the QR algorithm for the exponential of a symmetric matrix coincides with the time evolution from t to t + 1 in the Toda molecule equation (2) [17]. Furthermore, one step of the QR algorithm is identical to two steps of the LR algorithm [3], which is mathematically equivalent to Rutishauser's quotient difference (qd) algorithm for computing tridiagonal eigenvalues. Hirota et al [8] showed that the recursion formula of the qd algorithm is a time discretization of the Toda molecule equation (2), i.e. the discrete time Toda (dToda) equation: The dToda variables q (n) k and e (n) k denote values of q k and e k , respectively at discrete time n. Furthermore, it is known that certain extensions of the dToda equation are also related to the numerical algorithms. For example, the discrete relativistic Toda equation can generate the Laplace transformation [14] and the expansion into continued fractions [12]. The dToda equation (3) can be extended to the discrete hungry Toda (dhToda) equations [16,22], which are used to compute the eigenvalues of totally nonnegative matrices whose minors are all nonnegative [4,16].
Tokihiro et al [21] proposed an ultradiscretization technique, sometimes called the tropicalization, for transforming equations into piecewise linear equations with one of two operators min and max. The ultradiscretization is associated with min-plus, which we focus on in this paper, and max-plus algebras. Min-plus algebra has two binary operations ⊕ := min and ⊗ := + in the set R min := R ∪ {∞}. Min-plus algebra has a close relationship to weighted directed graphs constructed with sets of nodes and directed edges with weights. Weighted directed graphs naturally appear in various fields such as railway systems, automata, and petri nets used to model discrete event systems [6].
The so-called ultradiscrete Toda (udToda) equation [10,11] is, of course, an ultradiscretization of the dToda equation (3). The udToda equation describes motions of a finite number of balls in an array of infinite number of boxes [13]. For analysing the box and ball system (BBS) [18], conserved quantities of the udToda equation and the dToda equation (3) are useful. Conserved quantities of the dToda equation (3) are also useful in proving that the dToda equation (3) is applicable to computing tridiagonal eigenvalues. However, to the best of our knowledge, conserved quantities of the udToda equation have not yet been considered in matrix eigenvalue analysis. In this paper, we propose an application of the udToda equation to compute the matrix eigenvalue over min-plus algebra rather than linear algebra. We also show that the udToda equation can solve the minimum circuit problem in weighted and directed graphs. In [19], Tavakolipour and Shakeri analysed eigenvalues of tridiagonal Toeplitz matrices over max-plus algebra. They gives explicit formulas for the eigenvalues under certain conditions. The present paper also treats eigenvalues of tridiagonal matrices but is not restricted to Toeplitz matrices: diagonal and subdiagonal entries are not fixed to the same constants. Moreover the analytical approach is quite different from [19] in that the udToda equation can be applied to eigenvalue computation over min-plus algebra.
The remainder of this paper is organized as follows. In section 2, we give a brief explanation of the eigenvalues of matrices over min-plus algebra and describe the significance of the eigenvalues in weighted and directed graphs corresponding to these matrices. In section 3, we first introduce an ultradiscretization of the dToda equation (3), and then demonstrate how the udToda equation enables calculation of an eigenvalue of a tridiagonal matrix over minplus algebra. We also relate one of the udToda variables to the minimum circuit in weighted directed graphs. Finally, in section 4, we give concluding remarks.

Min-plus algebra and associated graph
In this section, we briefly review scalar and matrix arithmetics over min-plus algebra, graphs corresponding to matrices over min-plus algebra, and some known results about relationships of min-plus matrices to associated graphs.
We begin by explaining scalar arithmetic and its elementary properties over min-plus algebra. For two R min numbers a and b, min-plus algebra has the following two binary operations: We can easily check that ⊕ and ⊗ are both associative and commutative, ⊗ is distributive with respect to ⊕, and ε := +∞ and e := 0 are identities with respect to ⊕ and ⊗, respectively. If the R min number b satisfies then b is the inverse of a with respect to ⊗. For convenience, we hereafter employ an auxiliary operator as the inverse of ⊗ such that Matrices whose entries are all R min numbers are called min-plus matrices, and the set of all m-by-n min-plus matrices is denoted by R m×n min . Since the min-plus matrices appearing in the next section are all m-by-m square matrices, we hereinafter focus on the case of R m×m min matrices. For two R m×m , and the multiplication of A by the R min number α, , are respectively given as These operations are intuitively similar to those over conventional linear algebra. With respect to eigenvalues and eigenvectors, the following definition gives reasonable min-plus analogues.
then λ and x are an eigenvalue of A and its corresponding eigenvector, respectively.
Determinants of R m×m min matrices, however, are not directly defined as min-plus analogues of linear determinants. This is because min-plus algebra has no operation corresponding to subtraction over linear algebra. The following definition gives the min-plus analogue of determinants over linear algebra.
where S m is the symmetric group of permutations of {1, 2, . . . , m}. Moreover, for an R m×m min matrix A, the characteristic polynomial g A (x) is given by where I denotes the m-by-m identity matrix whose (i, j) entries are 0 if i = j, or ε otherwise.
We now consider the so-called min-plus polynomial of degree n with respect to x, = kx and c 1 , c 2 , . . . , c n ∈ R min are the coefficients. Regarding p(x) as the min-plus function with respect to x, we see that p(x) is piecewise linear because It is emphasized here that, in contrast to linear algebra, two distinct min-plus polynomials p(x) and p(x) are sometimes factorized using common linear factors [2]. If the linear factorizations of p(x) are the same as those of p(x), then p(x) is considered 'equivalent' to p(x). To distinguish this from equality, namely, Figure 1 shows the graphs of the min-plus polynomial func- Those min-plus polinomials p(x) and p(x) are not equal but equivalent since p(x) and p(x) define the same function. So we represent this relation as p(x) ≡p(x).
The following proposition describes the factorization and the minimum root of the characteristic polynomial g A (t).

of g A (t) coincides with the eigenvalue of A.
Min-plus algebra differs from linear algebra in that all the roots of characteristic polynomials of R m×m min matrices do not always coincide with eigenvalues of the R m×m min matrices. Over min-plus algebra, only the minimum roots are certainly eigenvalues.
A directed graph (digraph) G = (V, E) consists of a vertex set V and a directed edge set Vertices v 0 and v s are respectively called the initial and the terminal vertices of the path P. If the initial vertex of a path P coincides with the terminal vertex, then P is called a circuit. A digraph G is called strongly connected if, for any vertices v i and v j , there exists at least one path from v i to v j . Furthermore, if G is a weighted digraph, then a real number w(e) is assigned to each edge e, and is called the weight. The following definition gives the so-called weighted adjacency matrices associated with weighted digraphs.
It is clear that the weighted adjacency matrix A(G) is a min-plus matrix. Conversely, for any R m×m min matrix A, there exists a weighted digraph whose weighted adjacency matrix is A. We hereinafter denote such a weighted digraph by G (A).
Moreover, in a circuit C on a weighted digraph G(A), we refer to the number of edges and the sum of edge weights as the length (C) and the weight sum w(C), respectively. These values are used to define the average weight of the circuit C as follows.
Definition 2.5. For a circuit C, the average weight w ave (C) is the ratio of the weight sum w(C) to the length (C), namely: Then the following theorem gives interesting relationships between an eigenvalue of the

The ultradiscrete Toda equation and min-plus eigenvalue
In this section, after referring to the basic properties of the dToda equation (3) related to computing eigenvalues of tridiagonal matrices, we show that an ultradiscrete analogue of the dToda equation (3) has an application to computing eigenvalues of tridiagonal min-plus matrices. Now, we prepare the m-by-m lower and upper bidiagonal matrices L (n) and R (n) involving the dToda variables q By observing the entries in L (n+1) R (n+1) and R (n) L (n) , we can easily check that Considering that R (n) L (n) = R (n) (L (n) R (n) )(R (n) ) −1 , we see that the dToda equation (3) generates the similarity transformation from A (n) := L (n) R (n) to A (n+1) as With respect to the asymptotic behaviour of the dToda equation (3) as n → ∞, Henrici [7] and Rutishauser [15] showed that where λ 1 , λ 2 , …, λ m denote eigenvalues of A: = A (0) satisfying λ 1 > λ 2 > · · · > λ m > 0. This suggests that the dToda equation (3) with initial settings q  (3), taking the logarithm, multiplying ε on both sides, and taking the limit → +0, we obtain the ultradiscrete Toda (udToda) equation: According to Nagai et al [13], the udToda equation (4) describes dynamics of the BBS introduced in Takahashi and Satsuma [18]. The BBS is considered as a dynamics of solitons in which Q (n) k and E (n) k respectively correspond to the number of balls appearing in the kth sequence of balls from the left and the number of empty boxes between the kth and (k + 1) th sequential balls at discrete time n. As ultradiscrete analogues of L (n) and R (n) , we introduce two R m×m min matrices: Similarly to the discrete case, we define tridiagonal R m×m min matrices A (n) as the product of L (n) and R (n) , namely: Figure 2 shows the weighted digraph G(A (n) ) in the case where A (n) is the weighted adjacency matrix. From figure 2, we observe that G(A (n) ) is strongly connected. Furthermore, C 1 , C 2 , …, C m are loops, C 12 , C 23 , …, C m−1,m are circuits, and their weights are expressed by Thus, from theorem 2.6, the tridiagonal R m×m min matrix A (n) has an eigenvalue that coincides with the minimum value of the average weights of the loops C 1 , C 2 , . . . , C m , and the circuits C 12 , C 23 , . . . , C m−1,m .
To grasp an eigenvalue of matrix A (n) without the corresponding weighted digraph, we consider the characteristic polynomials of min-plus tridiagonal matrices A (n) .

Theorem 3.1. The characteristic polynomials g
Proof. If m = 1, then, because E 1 . If m = 2, the characteristic polynomials g A (n) (t) can be factorized as It is worth noting that the factors Q According to definition 2.2, cofactor expansions of tropical determinants can be considered as analogues of linear cofactor expansions. Applying the min-plus cofactor expansions along the mth column to tropdet(A (n) ⊕ t ⊗ I), we obtain k are the principal R k×k min submatrices of A (n) , namely: Under the assumption that tropdet(A which implies that (6) holds for any m. □ m−1 are the roots of the characteristic polynomials g A (n) (t). Combining this with (5), we realize that the roots of the characteristic polynomials g A (n) (t) coincide with the loop weights w(C 1 ), w(C 2 ), …, w(C m ). Using proposition 2.3, we see that the minimum root of g A (n) (t) is an eigenvalue λ (n) of A (n) . Thus, we can express the eigenvalues λ (n) using the udToda variables Q A study on the BBS also gives the significance of (7) as follows. [22]). For any discrete time n, conserved quantities of the udToda equation (4) are given by

Proposition 3.2 (Tokihiro et al
In the BBS, the conserved quantities C are equal to the minimum number of sequential balls corresponding to the amplitude of the shortest soliton. Equations (7) and (8) imply that λ (n) are constants independent of discrete time n. In other words, the tridiagonal R m×m min matrices A (n) have the same eigenvalue λ = λ (0) as A = A (0) .

Theorem 3.3. For any discrete time n, an eigenvalue
Theorem 3.3 also suggests that the udToda equation (4) gives the so-called min-plus similarity transformations of the R m×m min matrices A (n) . Reconsidering the BBS properties is the key to analyzing the asymptotic behaviour as n → ∞ of Q  [22]). In the udToda equation (4), there exists a discrete time N such that, for any n N   Here, we present a numerical example to verify that the udToda equation (4) enables us to find an eigenvalue of A = A (0) , which coincides with the minimum values of average weights of circuits in a weighted digraph corresponding to A, without considering tropical determinants and weighted digraphs. We set the initial values of the udToda equation (4) as Q (0)

Proposition 3.4 implies that Q
min tridiagonal matrix A is given by  (4). Figure 3 illustrates the position of the balls in the BBS and weighted digraphs corresponding to A (n) at discrete times n = 0, 6, 12. From figure 3, we observe that all the minimum values of the average weights of the circuits at n = 0, 6, 12 are 1 which is exactly the same as the eigenvalue of A (n) . Table 1 shows the values of Q (n) 3 at n = 0, 1, . . . , 12, and suggests that the eigenvalue 1 corresponds to one of the udToda variables at any n and appears in the values of Q (n) 1 at n 10.

Concluding remarks
In this paper, we showed that the ultradiscrete Toda (udToda) equation can be used to compute an eigenvalue of a tridiagonal R m×m min matrix where the eigenvalue is equal to the minimum value of the average weights of all circuits in the corresponding directed and weighted graph. Considering cofactor expansions of tropical determinants, we presented the factorization of the min-plus characteristic polynomial of the tridiagonal R m×m min matrix associated with the udToda equation. By relating the roots of the characteristic polynomial to the minimum value of the average weights of all circuits in the directed and weighted graph corresponding to the tridiagonal R m×m min matrix, we expressed an eigenvalue using the udToda variables at an arbitrary discrete time. Next, using known properties of the BBS, we proved that all tridiagonal R m×m min matrices generated using the udToda equation have at least a common eigenvalue, and that one of the udToda variables coincides with the eigenvalue after a sufficient discrete time lapse.
The discrete hungry Toda (dhToda) equations are extensions of the discrete Toda equation. The dhToda equations differ from the discrete Toda equation in that they are applicable to computing the eigenvalues of band matrices including tridiagonal matrices [4,16]. Our future Table 1. Discrete time evolution from n = 0 to 12 in the udToda equation (4). The bold 1s mean the minimum value (eigenvalue) at each discrete time.
n Q (n) 1 work aims to relate the ultradiscrete hungry Toda equation [4] to eigenvalues of R m×m min band matrices. This will be discussed in a separate paper.