Combinatorial Types of Tropical Eigenvectors

COMBINA TORIAL TYPES OF TR OPICAL EIGENVECTORS BERND STURMFELS AND NGOC MAI TRAN Abstract. The map whic h tak es a square matrix to its tropical eigenv alue-eigen v ector pair is piecewise linear. W e determine the cones of linearity of this map. They are simplicial but they do not form a fan. Motiv ated b y statistical ranking, we also study the restriction of that cone decomp osition to the subspace of sk ew-symmetric matrices. 1. Introduction Applications suc h as discrete even t systems [1] lead to the tropical eigen v alue equation A  x = λ  x. (1) Here arithmetic tak es place in the max-plus algebr a ( R , ⊕ ,  ), defined b y u ⊕ v = max { u, v } and u  v = u + v . The real n × n -matrix A = ( a ij ) is fixed. One seeks to compute all tr opic al eigenp airs ( λ, x ) ∈ R × R n , that is, solutions of (1). If ( λ, x ) is suc h a pair for A then so is ( λ, ν  x ) for any ν ∈ R . W e regard these eigenpairs as equiv alen t. The pairs ( λ, x ) are th us view ed as elements in R × TP n − 1 where TP n − 1 = R n / R (1 , 1 , . . . 1) is the tr opic al pr oje ctive torus [9]. Our p oin t of departure will b e the following result. Prop osition 1. Ther e exists a p artition of matrix sp ac e R n × n into finitely many c onvex p olyhe dr al c ones such that e ach matrix in the interior of a ful l-dimensional c one has a unique eigenp air ( λ, x ) in R × TP n − 1 . Mor e over, on e ach ful l-dimensional c one in that p artition, the eigenpair map A 7→ ( λ ( A ) , x ( A )) is r epr esente d by a unique line ar function R n × n → R × TP n − 1 . In tropical linear algebra [3] it is kno wn that the eigen v alue is unique, but the pro jectiv e tropical eigenspace can b e of dimension an ywhere b etw een 0 and n − 1. The prop osition implies that the set of matrices with more than one eigen v ector lies in the finite union of subspaces of co dimension one, and hence a generic n × n matrix has a unique eigenpair. The eigen v alue λ ( A ) is the maxim um cycle mean of the weigh ted directed graph with edge weigh t matrix A ; see [1, 3, 8]. As we shall see in (4) b elow, the map A 7→ λ ( A ) is the supp ort function of a con v ex polytop e, and hence it is piecewise linear. In this article w e study the refinemen t from eigen v alues to eigenv ectors. Our main result is as follows: Theorem 2. The op en c ones in R n × n on which the eigenp air map is r epr esente d by distinct and unique line ar functions ar e al l line arly isomorphic to R n × R n ( n − 1) > 0 . These c ones ar e indexe d by the c onne cte d functions φ : [ n ] → [ n ] , so their numb er is n X k =1 n ! ( n − k )! · n n − k − 1 . (2) F or n ≥ 3 , these c ones do not form a fan, that is, two c ones may interse ct in a non-fac e. 2000 Mathematics Subje ct Classific ation. Primary 05C99; Secondary 14T05, 91B12. Both authors were supp orted by the U.S. National Science F oundation (DMS-0757207 and DMS-0968882). 1 2 BERND STURMFELS AND NGOC MAI TRAN Here a function φ from [ n ] = { 1 , 2 , . . . , n } to itself is called c onne cte d if its graph is connected as an undirected graph. The coun t in (2) is the sequence A001865 in [10]. In Section 2, we explain this com binatorial representation and w e prov e both Prop osition 1 and Theorem 2. F or n = 3, the num b er (2) equals 17, and our cone decomp osition is represented b y a 5-dimensional simplicial complex with f-v ector (9 , 36 , 81 , 102 , 66 , 17). The lo cus in R n × n where the cone decomp osition fails to be a fan consists precisely of the matrices A whose eigenspace is positive-dimensional. W e explain the details in Section 3. In Section 4 w e restrict to matrices A that are skew-symmetric, in sym b ols: A = − A T . T ropical eigenv ectors of sk ew-symmetric matrices arise in p airwise c omp arison r anking , in the approach that was pioneered b y Elsner and v an den Driessche [5, 6]. In [11], the second author offered a comparison with tw o other metho ds for statistical ranking, and she noted that the eigenv alue map A 7→ λ ( A ) for sk ew-symmetric A is linear on (the cones o v er) the facets of the cographic zonotop e asso ciated with the complete graph on n vertices. The tropical eigenv ector causes a further sub division for man y of the facets, as seen for n = 4 in [11, Figure 1]. Our Theorem 8 c haracterizes these sub divisions in to cub es for all n . W e close with a brief discussion of the eigenspaces of non-generic matrices. 2. Tropical Eigenv alues and Eigenvectors W e first review the basics concerning tropical eigen v alues and eigen v ectors, and w e then pro v e our t w o results. Let A b e a real n × n -matrix and G ( A ) the corresponding w eigh ted directed graph on n vertices. It is known that A has a unique tropical eigenv alue λ ( A ). This eigenv alue can b e computed as the optimal v alue of the following linear program: Minimize λ sub ject to a ij + x j ≤ λ + x i for all 1 ≤ i, j ≤ n. (3) Cuninghame-Green [8] used the form ulation (3) to sho w that the eigen v alue λ ( A ) of a matrix A can b e computed in p olynomial time. F or an alternative approac h to the same problem we refer to Karp’s article [7]. The linear program dual to (3) takes the form Maximize P n i,j =1 a ij p ij sub ject to p ij ≥ 0 for 1 ≤ i, j ≤ n, P n i,j =1 p ij = 1 and P n j =1 p ij = P n k =1 p ki for all 1 ≤ i ≤ n. (4) The p ij are the v ariables, and the constraints require ( p ij ) to be a probabilit y distribution on the edges of G ( A ) that represen ts a flow in the directed graph. Let C n denote the n ( n − 1)-dimensional con v ex polytop e of all feasible solutions to (4). By strong duality , the primal (3) and the dual (4) ha v e the same optimal v alue. This implies that the eigen v alue function A 7→ λ ( A ) is the supp ort function of the conv ex p olytop e C n . Hence the function A 7→ λ ( A ) is con tinuous, con v ex and piecewise-linear. By the eigenvalue typ e of a matrix A ∈ R n × n w e shall mean the cone in the normal fan of the p olytop e C n that con tains A . Since each vertex of C n is the uniform probabilit y distribution on a directed cycle in G ( A ), the eigen v alue λ ( A ) is the maxim um cycle mean of G ( A ). Th us, the op en cones in the normal fan of C n are naturally indexed b y cycles in the graph on n vertices. The cycles corresp onding to the normal cone containing the matrix A are the critic al cycles of A . The union of their v ertices is called the set of critic al vertic es in [3, 5] Example 3. Let n = 3. There are eight cycles, t w o of length 3, three of length 2 and three of length 1, and hence eigh t eigenv alue types. The p olytop e C 3 is six-dimensional: it is the threefold pyramid o v er the bipyramid formed b y the 3-cycles and 2-cycles. 2 COMBINA TORIAL TYPES OF TROPICAL EIGENVECTORS 3 W e ha v e seen that the normal fan of C n partitions R n × n in to p olyhedral cones on whic h of the eigen v alue map A 7→ λ ( A ) is linear. Our goal is to refine the normal fan of C n in to cones of linearit y for the eigenv ector A 7→ x ( A ) map. T o pro ve our first result, we in tro duce some notation and recall some prop erties of the tropical eigenv ector. F or a matrix A ∈ R n × n , let B := A  ( − λ ( A )). F or a path P ii 0 from i to i 0 , let B ( P ii 0 ) denote its length (= sum of all edge weigh ts along the path) in the graph of B . W e write { Γ ii 0 } := argmax P ii 0 B ( P ii 0 ) for the set of paths of maximum length from i to i 0 , and write Γ ii 0 if the path is unique. Note that B (Γ ii 0 ) is well-defined even if there is more than one maximal path, and it is finite since all cycles of B are non-p ositiv e. If j, j 0 are intermediate v ertices on a path P ii 0 , then P ii 0 ( j → j 0 ) is the path from j to j 0 within P ii 0 . It is kno wn from tropical linear algebra [2, 3] that the tropical eigen v ector x ( A ) of a matrix A is unique if and only if the union of its critical cycles is connected. In such cases, the eigen v ector x ( A ) can b e calculated by first fixing a critical v ertex ` , and then setting x ( A ) i = B (Γ i` ) , (5) that is, the en try x ( A ) i is the maximal length among paths from i to ` in the graph of B . Pr o of of Pr op osition 1. F ollo wing the preceding discussion, it is sufficien t to construct the refinemen t of each eigen v alue t yp e in the normal fan of C n . Let A lie in the in terior of suc h a cone. Fix a critical v ertex ` . Since the eigenv alue map is linear, for any path P i` the quantit y B ( P i` ) is giv en b y a unique linear form in the entries of A . A path Q i` is maximal if and only if B ( Q i` ) − B ( P i` ) ≥ 0 for all paths P i` 6 = Q i` . Hence, by (5), the co ordinate x ( A ) i of the eigenv ector is given by a unique linear function in the en tries of A (up to c hoices of ` ) if and only if { Γ i` } has cardinality one, or, equiv alen tly , if and only if B ( Q i` ) − B ( P i` ) > 0 for all paths P i` 6 = Q i` . (6) W e no w claim that, as linear functions in the en tries of A , the linear forms in (6) are indep enden t of the choice of ` . Fix another critical v ertex k . It is sufficient to prov e the claim when ( ` → k ) is a critical edge. In this case, for an y path P i` , the path R ik := P i` + ( ` → k ) is a path from i to k with B ( R ik ) = B ( P i` ) + a `k − λ ( A ). Con versely , for an y path R ik , tra v ersing the rest of the cycle from k back to ` giv es a path P i` := R ik + ( k → . . . → ` ) from i to ` , with B ( P i` ) = B ( R ik ) − ( a `k − λ ( A )), since the critical cycle has length 0 in the graph of B . Hence, the map B ( P i` ) 7→ B ( P i` ) + a `k − λ ( A ) is a bijection taking the lengths of paths from i to ` to the lengths of paths from i to k . Since this map is a tropical scaling, the linear forms in (6) are unchanged, and hence they are indep endent of the choice of ` . W e conclude that (6) defines the cones promised in Prop osition 1. 2 Tw o p oints should b e noted in the pro of of Prop osition 1. Firstly , in the interior of eac h eigenpair cone (6), for any fixed critical vertex ` and any other vertex i ∈ [ n ], the maximal path Γ i` is unique. Secondly , the n um b er of facet defining equations for these cones are p oten tially as large as the n um b er of distinct paths from i to ` for eac h i ∈ [ n ]. In Theorem 2 w e shall show that there are only n 2 − n facets. Our pro of relies on the follo wing lemma, which is based on an argument w e learned from [3, Lemma 4.4.2]. 4 BERND STURMFELS AND NGOC MAI TRAN Lemma 4. Fix A in the interior of an eigenp air c one (6). F or e ach non-critic al vertex i , ther e is a unique critic al vertex i ∗ such that the p ath Γ ii ∗ uses no e dge in the critic al cycle. If j is any other non-critic al vertex on the p ath Γ ii ∗ , then j ∗ = i ∗ and Γ j j ∗ = Γ ii ∗ ( j → i ∗ ) . Pr o of. W e relab el vertices so that the critical cycle is (1 → 2 → . . . → k → 1). F or an y non-critical i and critical ` , the path Γ i` is unique, and b y the same argumen t as in the pro of of Prop osition 1, Γ i ( ` +1) = Γ i` + ( ` → ( ` + 1)) . Hence there exists a unique critical v ertex i ∗ suc h that Γ ii ∗ uses no edge in the critical cycle. F or the second statement, we note that Γ ii ∗ ( j → i ∗ ) uses no edge in the critical cycle. Supp ose that Γ j i ∗ 6 = Γ ii ∗ ( j → i ∗ ). The concatenation of Γ ii ∗ ( i → j ) and Γ j i ∗ is a path from i to i ∗ that is longer than Γ ii ∗ . This is a con tradiction and the pro of is complete. 2 Pr o of of The or em 2. W e define the critic al gr aph of A to b e the subgraph of G ( A ) consist- ing of all edges in the critical cycle and all edges in the sp ecial paths Γ ii ∗ ab o ve. Lemma 4 sa ys that the critical graph is the union of the critical cycle with trees ro oted at the critical v ertices. Each tree is directed tow ards its ro ot. Hence the critical graph is a connected function φ on [ n ], and this function φ determines the eigenpair type of the matrix A . W e next argue that every connected function φ : [ n ] → [ n ] is the critical graph of some generic matrix A ∈ R n × n . If φ is surjective then φ is a cycle and we take an y matrix A with the corresp onding eigenv alue type. Otherwise, we may assume that n is not in the image of φ . By induction w e can find an ( n − 1) × ( n − 1)-matrix A 0 with critical graph φ \{ ( n, φ ( n )) } . W e enlarge A 0 to the desired n × n -matrix A b y setting a n,φ ( n ) = 0 and all other entries very negativ e. Then A has φ as its critical graph. W e conclude that, for every connected function φ on [ n ], the set of all n × n -matrices that hav e the critical graph φ is a full-dimensional con vex polyhedral cone Ω φ in R n × n , and these are the op en cones, characterized in (6), on which the eigenpair map is linear. W e next sho w that these cones are linearly isomorphic to R n × R n ( n − 1) ≥ 0 . Let e ij denote the standard basis matrix of R n × n whic h is 1 in p osition ( i, j ) and 0 in all other p osi- tions. Let V n denote the n -dimensional linear subspace of R n × n spanned by the matrices P n i,j =1 e ij and P n j =1 e ij − P n k =1 e ki for i = 1 , 2 , . . . , n . Equiv alen tly , V n is the orthogonal complemen t to the affine span of the cycle p olytop e C n . The normal cone at each vertex of C n is the sum of V n and a p oin ted cone of dimension n ( n − 1). W e claim that the sub cones Ω φ inherit the same prop ert y . Let ¯ Ω φ denote the image of Ω φ in the quotient space R n × n /V n . This is an n ( n − 1)-dimensional pointed con v ex p olyhedral cone, so it has at least n ( n − 1) facets. T o show that it has precisely n ( n − 1) facets, we claim that Ω φ =  A ∈ R n × n : b ij ≤ B ( φ ij ∗ ) − B ( φ j j ∗ ) : ( i, j ) ∈ [ n ] 2 \ φ  . (7) In this form ula, φ ii ∗ denotes the directed path from i to i ∗ in the graph of φ , and B = ( b ij ) = ( a ij − λ φ ( A )), where λ φ ( A ) is the mean of the cycle in the graph of φ with edge w eigh ts ( a ij ). The inequality representation (7) will imply that the cone Ω φ is linearly isomorphic to R n × R n ( n − 1) ≥ 0 b ecause there are n ( n − 1) non-edges ( i, j ) ∈ [ n ] 2 \ φ . Let A b e an y matrix for which the n ( n − 1) inequalities in (7) hold strictly for non- edges of φ . Let ψ denote the connected function corresp onding to the critical graph of A . T o pro ve the claim, we must sho w that ψ = φ . First w e show that ψ and φ ha v e the same cycle. Without loss of generalit y , let (1 → 2 → . . . → k → 1) b e the cycle in φ , and ( i 1 → i 2 → . . . → i m → i 1 ) the cycle in ψ . Assuming they are differen t, the inequalit y in (7) holds strictly for at least one edge in ψ . Using the iden tities COMBINA TORIAL TYPES OF TROPICAL EIGENVECTORS 5 B ( φ i j i ∗ j +1 ) = B ( φ i j i ∗ j ) + B ( φ i ∗ j i ∗ j +1 ), we find b i 1 i 2 + b i 2 i 3 + · · · + b i m i 1 < B ( φ i 1 i ∗ 2 ) − B ( φ i 2 i ∗ 2 ) + B ( φ i 2 i ∗ 3 ) − B ( φ i 3 i ∗ 3 ) + · · · + B ( φ i m i ∗ 1 ) − B ( φ i 1 i ∗ 1 ) = B ( φ i 1 i ∗ 1 )+ B ( φ i ∗ 1 i ∗ 2 ) − B ( φ i 2 i ∗ 2 ) + B ( φ i 2 i ∗ 2 )+ B ( φ i ∗ 2 i ∗ 3 ) − · · · + B ( φ i m i ∗ m )+ B ( φ i ∗ m i ∗ 1 ) − B ( φ i 1 i ∗ 1 ) = B ( φ i ∗ 1 i ∗ 2 ) + B ( φ i ∗ 2 i ∗ 3 ) + · · · + B ( φ i ∗ m i ∗ 1 ) = 0 = b 12 + b 23 + · · · + b k 1 . This contradicts that ψ has maximal cycle mean, hence ψ and φ ha v e the same unique critical cycle. It remains to sho w that other edges agree. Supp ose for con tradiction that there exists a non-critical v ertex i in whic h ψ ( i ) 6 = φ ( i ). Since ( i, ψ ( i )) is a non-edge in [ n ] 2 \ φ , the inequalit y (7) holds strictly b y the assumption on the choice of A , and w e get B ( φ iψ ( i ) ∗ ) > B ( φ ψ ( i ) ψ ( i ) ∗ ) + b iψ ( i ) = B (( i → ψ ( i )) + φ ψ ( i ) ψ ( i ) ∗ ) . This sho ws that the path ( i → ψ ( i )) + φ ψ ( i ) ψ ( i ) ∗ is not critical, that is, it is not in the graph of ψ . Hence, there exists another v ertex i 2 along the path φ ψ ( i ) ψ ( i ) ∗ suc h that ψ ( i 2 ) 6 = φ ( i 2 ). Pro ceeding b y induction, we obtain a sequence of v ertices i, i 2 , i 3 , . . . , with this prop ert y . Hence even tually we obtain a cycle in ψ that consists en tirely of non-edges in [ n ] 2 \ φ . But this contradicts the earlier statemen t that the unique critical cycle in ψ agrees with that in φ . This completes the pro of of the first sentence in Theorem 2. F or the second sentence w e note that the num b er of connected functions in (2) is the sequence A001865 in [10]. Finally , it remains to b e seen that our simplicial cones do not form a fan in R n 2 /V n for n ≥ 3. W e shall demonstrate this explicitly in Example 7. 2 3. Eigenp air Cones and F ailure of the F an Proper ty Let ( x φ , λ φ ) : R n × n → TP n − 1 × R denote the unique linear map whic h takes any matrix A in the interior of the cone Ω φ to its eigenpair ( x ( A ) , λ ( A )). Of course, this linear map is defined on all of R n × n , not just on Ω φ . The follo wing lemma is a useful c haracterization of Ω φ in terms of the linear map ( x φ , λ φ ) whic h elucidates its pro duct structure as R n × R n ( n − 1) ≥ 0 . Lemma 5. F or a matrix A ∈ R n × n , we abbr eviate x := x φ ( A ) , λ := λ φ ( A ) , and we set C = ( c ij ) = ( a ij − x i + x j − λ ) . Then A is in the interior of the c one Ω φ if and only if C iφ ( i ) = 0 for al l i ∈ [ n ] and C ij < 0 otherwise. (8) Pr o of. Since the matrix ( x i − x j + λ ) is in the linear subspace V n , the matrices A and C lie in the same eigenpair cone Ω φ . Since C ij ≤ 0 for all i, j = 1 , . . . , n , the conditions (8) are thus equiv alen t to ( C  [0 , . . . , 0] T ) i = max k ∈ [ n ] C ik = C iφ ( i ) = 0 for all i ∈ [ n ]. In w ords, the matrix C is a normalized version of A which has eigenv alue λ ( C ) = 0 and eigen v ector x ( C ) = [0 , . . . , 0] T . The condition (8) is equiv alen t to that in (7), with strict inequalities for { ( i, j ) : j 6 = φ ( i ) } , and it holds if and only if C is in the interior of Ω φ . 2 The linear map A 7→ ( C ij : j 6 = φ ( i )) defined in Lemma 5 realizes the pro jection from the eigenpair cone Ω φ on to its p oin ted v ersion ¯ Ω φ . Thus, the simplicial cone ¯ Ω φ is spanned b y the images in R n × n /V n of the matrices − e ij that are indexed b y the n ( n − 1) non-edges: Ω φ = V n + R ≥ 0  − e ij : ( i, j ) ∈ [ n ] 2 \ φ  ' R n × R n ( n − 1) ≥ 0 . (9) 6 BERND STURMFELS AND NGOC MAI TRAN A t this p oint, w e find it instructiv e to work out the eigenpair cone Ω φ explicitly for a small example, and to v erify the equiv alent represen tations (7) and (9) for that example. Example 6 ( n = 3) . Fix the connected function φ = { 12 , 23 , 31 } . Its eigen v alue functional is λ := λ φ ( A ) = 1 3 ( a 12 + a 23 + a 31 ). The eigenpair cone Ω φ is 9-dimensional and is c haracterized by 3 · 2 = 6 linear inequalities, one for eac h of the six non-edges ( i, j ), as in (7). F or instance, consider the non-edge ( i, j ) = (1 , 3). Using the iden tities B ( φ 13 ∗ ) = b 12 + b 23 = a 12 + a 23 − 2 λ and B ( φ 33 ∗ ) = b 31 + b 12 + b 23 = a 31 + a 12 + a 23 − 3 λ , the inequality b 13 ≤ B ( φ 13 ∗ ) − B ( φ 33 ∗ ) in (7) translates in to a 13 ≤ 2 λ − a 31 and hence into a 13 ≤ 1 3 (2 a 12 + 2 a 23 − a 31 ) . Similar computations for all six non-edges of φ give the follo wing six linear inequalities: a 11 ≤ 1 3 ( a 12 + a 23 + a 31 ) , a 22 ≤ 1 3 ( a 12 + a 23 + a 31 ) , a 33 ≤ 1 3 ( a 12 + a 23 + a 31 ) , a 13 ≤ 1 3 (2 a 12 + 2 a 23 − a 31 ) , a 32 ≤ 1 3 (2 a 12 − a 23 + 2 a 31 ) , a 21 ≤ 1 3 ( − a 12 + 2 a 23 + 2 a 31 ) . The eigenpair cone Ω φ equals the set of solutions in R 3 × 3 to this system of inequalities. According to Lemma 5, these same inequalities can also deriv ed from (7). W e ha ve x : = x φ ( A ) =  a 12 + a 23 − 2 λ, a 23 − λ, 0  T . The equations c 12 = c 23 = c 31 = 0 in (9) are equiv alen t to a 12 = x 1 − x 2 + λ, a 23 = x 2 − x 3 + λ, a 31 = x 3 − x 1 + λ, and the constrain ts c 11 , c 13 , c 21 , c 22 , c 32 , c 33 < 0 translate in to the six inequalities ab o v e. 2 T o describ e the combinatorial structure of the eigenpair t yp es, we introduce a simplicial complex Σ n on the vertex set [ n ] 2 . The facets (= maximal simplices) of Σ n are the complemen ts [ n ] 2 \ φ where φ runs ov er all connected functions on [ n ]. Th us Σ n is pure of dimension n 2 − n − 1, and the n um b er of its facets equals (2). T o each simplex σ of Σ n w e associate the simplicial cone R ≥ 0 { ¯ e ij : ( i, j ) ∈ σ } in R n × n /V n . W e hav e sho wn that these cones form a decomp osition of R n × n /V n in the sense that every generic matrix lies in exactly one cone. The last assertion in Theorem 2 states that these cones do not form a fan. W e shall no w sho w this for n = 3 by giving a detailed com binatorial analysis of Σ 3 . Example 7. [ n = 3] W e here presen t the pr o of of the thir d and final p art of The or em 2 . The simplicial complex Σ 3 is 5-dimensional, and it has 9 vertices, 36 edges, 81 trian- gles, etc. The f-vector of Σ 3 is (9 , 36 , 81 , 102 , 66 , 17). The 17 facets of Σ 3 are, by def- inition, the set complements of the 17 connected functions φ on [3] = { 1 , 2 , 3 } . F or instance, the connected function φ = { 12 , 23 , 31 } in Example 5 corresp onds to the facet { 11 , 13 , 21 , 22 , 32 , 33 } of Σ 3 . This 5-simplex can b e written as { 11 , 22 , 33 } ∗ { 21 , 32 , 13 } , the join of t w o triangles, so it app ears as the central triangle on the left in Figure 1. Figure 1 is a pictorial representation of the simplicial complex Σ 3 . Eac h of the drawn graphs represen ts its clique complex, and ∗ denotes the join of simplicial complexes. The eigh t connected functions φ whose cycle has length ≥ 2 corresp ond to the eight facets on the left in Figure 1. Here the triangle { 11 , 22 , 33 } is joined with the depicted cyclic triangulation of the b oundary of a triangular prism. The other nine facets of Σ 3 come in three groups of three, corresp onding to whether 1 , 2 or 3 is fixed b y φ . F or instance, if COMBINA TORIAL TYPES OF TROPICAL EIGENVECTORS 7 11 22 33 ∗ 21 13 32 12 31 23 11 22 ∗ 21 13 32 12 31 23 Figure 1. The simplicial complex Σ 3 of connected functions φ : [3] → [3]. Fixed-p oin t free φ are on the left and functions with φ (3) = 3 on the righ t. φ (3) = 3 then the facet [3] 2 \ φ is the join of the segment { 11 , 22 } with one of the three tetrahedra in the triangulation of the solid triangular prism on the right in Figure 1. In the geometric realization given by the cones Ω φ , the square faces of the triangular prism are flat. How ev er, we see that b oth of their diagonals app ear as edges in Σ 3 . This pro v es that the cones cov ering these diagonals do not fit together to form a fan. 2 Naturally , each simplicial complex Σ n for n > 3 con tains Σ 3 as a subcomplex, and this is compatible with the embedding of the cones. Hence the eigenpair t yp es fail to form a fan for an y n ≥ 3. F or the sak e of concreteness, we note that the 11-dimensional simplicial complex Σ 4 has f-vector (16 , 120 , 560 , 1816 , 4320 , 7734 , 10464 , 10533 , 7608 , 3702 , 1080 , 142). The failure of the fan prop erty is caused by the existence of matrices that hav e disjoint critical cycles. Such a matrix lies in a lo wer-dimensional cone in the normal fan of C n , and it has t w o or more eigen v ectors in TP n − 1 that eac h arise from the unique eigen v ectors on neigh b oring full-dimensional cones. These eigen vectors ha v e distinct critical graphs φ and φ 0 and the cones Ω φ and Ω φ 0 do not intersect along a common face. In other words, the failure of the fan prop erty reflects the discontin uit y in the eigenv ector map A 7→ x ( A ). F or concrete example, consider the edge connecting 13 and 23 on the left in Figure 1 and the edge connecting 31 and 32 on the right in Figure 1. These edges in tersect in their relativ e interiors, th us violating the fan prop erty . In this in tersection we find the matrix A =   0 0 − 1 0 0 − 1 − 1 − 1 0   , (10) whose eigenspace is a tropical segmen t in TP 2 . Any nearby generic matrix has a unique eigen v ector, and that eigen vector lies near one of the tw o endp oints of the tropical segment. A diagram lik e Figure 1 c haracterizes the com binatorial structure of such discon tin uities. 4. Skew-Symmetric Ma trices This pro ject arose from the application of tropical eigen vectors to the statistics problem of inferring rankings from pairwise comparison matrices. This application w as pioneered b y Elsner and v an den Driessche [5, 6] and further studied in [11, § 3]. W orking on the additiv e scale, an y pairwise comparison matrix A = ( a ij ) is skew-symmetric , i.e. it satisfies a ij + a j i = 0 for all 1 ≤ i, j ≤ n . The set ∧ 2 R n of all skew-symmetric matrices is a linear subspace of dimension  n 2  in R n × n . The input of the tr opic al r anking algorithm is a generic matrix A ∈ ∧ 2 R n and the output is the p erm utation of [ n ] = { 1 , . . . , n } giv en by sorting the en tries of the eigen v ector x ( A ). See [11] for a comparison with other ranking methods. 8 BERND STURMFELS AND NGOC MAI TRAN In this section w e are in terested in the com binatorial t yp es of eigenpairs when restricted to the space ∧ 2 R n of sk ew-symmetric matrices. In other w ords, we shall study the de- comp osition of this space in to the conv ex polyhedral cones Ω φ ∩ ∧ 2 R n where φ runs ov er connected functions on [ n ]. Note that, λ ( A ) ≥ 0 for all A ∈ ∧ 2 R n , and the equalit y λ ( A ) = 0 holds if and only if A ∈ V n . Hence the intersection Ω φ ∩ ∧ 2 R n is trivial for all connected functions φ whose cycle has length ≤ 2. This motiv ates the following definition. W e define a kite to b e a connected function φ on [ n ] whose cycle has length ≥ 3. By restricting the sum in (2) accordingly , we see that the num ber of kites on [ n ] equals n X k =3 n ! ( n − k )! · n n − k − 1 . (11) Th us the num ber of kites for n = 3 , 4 , 5 , 6 , 7 , 8 equals 2 , 30 , 444 , 7320 , 136590 , 2873136. The following result is the analogue to Theorem 2 for sk ew-symmetric matrices. Theorem 8. The op en c ones in ∧ 2 R n on which the tr opic al eigenp air map for skew- symmetric matric es is r epr esente d by distinct and unique line ar functions ar e Ω φ ∩ ∧ 2 R n wher e φ runs over al l kites on [ n ] . Each c one has n ( n − 3) fac ets, so it is not simplicial, but it is line arly isomorphic to R n − 1 times the c one over the standar d cub e of dimension n ( n − 3) / 2 =  n 2  − n . This c ol le ction of c ones do es not form a fan for n ≥ 6 . Pr o of. It follo ws from our results in Section 2 that eac h cone of linearit y of the map ∧ 2 R n → R × TP n − 1 , A 7→ ( λ ( A ) , x ( A )) has the form Ω φ ∩ ∧ 2 R n for some kite φ . Con v ersely , let φ b e any kite on [ n ] with cycle (1 → 2 → . . . → k → 1). W e m ust show that Ω φ ∩ ∧ 2 R n has non-empty in terior (inside ∧ 2 R n ). W e shall prov e the statemen t b y induction on n − k . Note that this w ould pro v e distinctiv eness, for the matrices constructed in the induction step lie strictly in the in terior of each cones. The base case n − k = 0 is easy: here the skew-symmetric matrix A = P n i =1 ( e iφ ( i ) − e φ ( i ) i ) lies in the interior of Ω φ . F or the induction step, supp ose that A lies in the interior of Ω φ ∩ ∧ 2 R n , and fix an extension of φ to [ n + 1] by setting φ ( n + 1) = 1. Our task is to construct a suitable matrix A ∈ ∧ 2 R n +1 that extends the old matrix and realizes the new φ . T o do this, we need to solve for the n unknown en tries a i,n +1 = − a n +1 ,i , for i = 1 , 2 , . . . , n . By (7), the necessary and sufficient conditions for A to satisfy φ ( n + 1) = 1 are a ( n +1) j ≤ λ ( A ) + B ( φ ( n +1) j ∗ ) − B ( φ j j ∗ ) , a j ( n +1) ≤ λ ( A ) + B ( φ j j ∗ ) − B ( φ 1 j ∗ ) − B ( φ ( n +1) , 1 ) . Let | φ j j ∗ | denote the n um b er of edges in the path φ j j ∗ Since a ij = − a j i , rearranging gives a 1 ( n +1) + a ( n +1) j ≤ A ( φ 1 j ∗ ) − A ( φ j j ∗ ) + ( | φ j j ∗ | − | φ 1 j ∗ | ) λ ( A ) , a 1 ( n +1) + a ( n +1) j ≥ A ( φ 1 j ∗ ) − A ( φ j j ∗ ) + ( | φ j j ∗ | − | φ 1 j ∗ | ) λ ( A ) − 2 λ ( A ) . The quan tities on the righ t hand side are constan ts that do not dep end on the new matrix en tries w e seek to find. They specify a solv able system of upper and low er b ounds for the quan tities a 1 ( n +1) + a ( n +1) j for j = 2 , . . . , n . Fixing these n − 1 sums arbitrarily in their re- quired in terv als yields n − 1 linear equations. W orking modulo the 1-dimensional subspace of V n +1 spanned by P n j =1 ( e n +1 ,j − e j,n +1 ), w e add the extra equation P n j =1 a j ( n +1) = 0. F rom these n linear equations, the missing matrix entries a 1( n +1) , a 2( n +1) , . . . , a n ( n +1) can b e computed uniquely . The resulting matrix A ∈ ∧ 2 R n +1 strictly satisfies all the necessary inequalities, so it is in the in terior of the required cone Ω φ . COMBINA TORIAL TYPES OF TROPICAL EIGENVECTORS 9 The quotien t of ∧ 2 R n mo dulo its n -dimensional subspace V n ∩ ∧ 2 R n has dimension n ( n − 3) / 2. The cones we are interested in, one for eac h kite φ , are all p ointed in this quotien t space. F rom the inductiv e construction abov e, w e see that each cone Ω φ ∩ ∧ 2 R n is characterized by upp er and low er b ounds on linearly indep endent linear forms. This pro v es that this cone is linearly isomorphic to the cone o v er a standard cube of dimension n ( n − 3) / 2. If n = 4 then the cub es are squares, as shown in [11, Figure 1]. F ailure to form a fan stems from the existence of disjoint critical cycles, as discussed at the end of Section 3. F or n ≥ 6, w e can fix tw o disjoint triangles and their adjacen t cones in the normal fan of C n . By an analogous argumen t to that giv en in Example 6, w e conclude that the cones Ω φ ∩ ∧ 2 R n , as φ runs o v er kites, do not form a fan for n ≥ 6. 2 In this note w e examined the division of the space of all (sk ew-symmetric) n × n -matrices in to op en p olyhedral cones that represen t distinct com binatorial types of tropical eigen- pairs. Since that partition is not a fan, in teresting phenomena happen for special matrices A , i.e. those not in an y of the op en cones Ω φ . F or such matrices A , the eigenv alue λ is still unique but the p olyhedral set Eig( A ) = { x ∈ TP n − 1 : A  x = λ  x } ma y hav e dimension ≥ 1. Let B ∗ = B ⊕ B  2 ⊕ · · · ⊕ B  n and let B ∗ 0 b e the submatrix of B ∗ giv en b y all columns i such that B ∗ ii = 0. It is well kno wn (see e.g. [3, § 4.4]) that Eig( A ) = Eig ( B ) = Eig ( B ∗ ) = Image( B ∗ 0 ) . Th us, Eig ( A ) is a tropical p olytop e in the sense of Develin and Sturmfels [4], and w e refer to Eig ( A ) as the eigenp olytop e of the matrix A . This p olytop e has ≤ n tropical vertices. Eac h tropical v ertex of an eigenp olytop e Eig ( A ) can b e represented as the limit of eigen v ectors x ( A  ) where ( A  ) is a sequence of generic matrices lying in the cone Ω φ for some fixed connected function φ . This means that the com binatorial structure of the eigenp olytop e Eig( A ) is determined by the connected functions φ that are adjacen t to A . F or example, let us revisit the (inconsistently sub divided) square { 13 , 32 , 23 , 31 } in Figure 1. The 3 × 3-matrices that corresp ond to the p oin ts on that square ha v e the form A =   0 0 a 0 0 b c d 0   , where a, b, c, d < 0 . One particular instance of this was featured in (10). The eigenp olytop e Eig ( A ) of the ab o ve matrix is the tropical line segment spanned by the columns of A , and its t wo v ertices are limits of the eigenv ectors coming from the t wo adjacen t facets of Σ 3 . It would b e worth while to study this further in the skew-symmetric case. Using kites, can one classify all tropical eigenp olytop es Eig ( A ) where A ranges ov er matrices in ∧ 2 R n ? References [1] F. Baccelli, G. Cohen, G.J. Olsder and J.-P. Quadra t : Synchr onization and Line arity: An Al gebr a for Discr ete Event Systems , Wiley Interscience, 1992. [2] R. Bap a t : A max version of the Perr on-Fr ob enius the or em , Linear Algebra and Its Applications, 275-276 (1997) 3–18. [3] P. Butko vi ˇ c : Max-line ar Systems: The ory and A lgorithms , Springer, 2010. [4] M. Develin and B. Sturmfels : T r opic al c onvexity , Do cumen ta Mathematica 9 (2004) 1–27. [5] L. Elsner and P. v an den Driessche : Max-algebr a and p airwise c omp arison matric es , Linear Algebra and its Applications 385 (2004) 47–62. [6] , Max-algebr a and p airwise c omp arison matric es, ii , Linear Algebra and its Applications 432 (2010) 927–935. 10 BERND STURMFELS AND NGOC MAI TRAN [7] R. Karp : A char acterization of the minimum cycle me an in a digr aph , Discrete Mathematics 23 (1978) 309–311. [8] R. A. Cuninghame-Green : Minimax A lgebr a , Springer-V erlag, 1979. [9] D. Macla gan and B. Sturmfels : Intr o duction to Tr opic al Ge ometry . Manuscript, 2009. [10] S. Plouffe and N. Slo ane : The On-Line Encyclop e dia of Inte ger Se quenc es , published electron- ically at http://oeis.org , 2010. [11] N.M. Tran : Pairwise r anking: choic e of metho d c an pr o duc e arbitr arily differ ent r ank or der , arXiv:1103.1110 . Dep ar tment of St a tistics, University of California, Berkeley, CA 94720-3860, USA E-mail addr ess : bernd@math.berkeley.edu, tran@stat.berkeley.edu URL : www.math.berkeley.edu/~bernd/, www.stat.berkeley.edu/~tran/