Non-archimedean valuations of eigenvalues of matrix polynomials

0.61 MB·

by  Marianne Akian

NON-ARCHIMEDEAN VALUATIONS OF EIGENVALUES OF
MATRIX POLYNOMIALS
MARIANNEAKIAN,RAVINDRABAPAT,ANDSTE´PHANEGAUBERT
6
1
0
2
Abstract. We establish general weak majorization inequalities, relating the

n leadingexponentsoftheeigenvaluesofmatricesormatrixpolynomialsoverthe
a fieldofPuiseuxserieswiththetropicalanaloguesofeigenvalues. Wealsoshow
J thattheseinequalitiesbecomeequalitiesundergenericityconditions,andthat

4 theleadingcoefficientsoftheeigenvaluesaredeterminedastheeigenvaluesof
auxiliarymatrixpolynomials.

]
P
1. Introduction
S
. 1.1. Non-archimedean valuations and tropical geometry. A non-
h
t archimedean valuation ν on a field K is a map K → R ∪ {+∞} such that
a
m
(1a) ν(a)=+∞ ⇐⇒ a=0
[

(1b) ν(a+b)(cid:62)min(ν(a),ν(b))
1
v (1c) ν(ab)=ν(a)+ν(b) .
8
3 These properties imply that ν(a + b) = min(ν(a),ν(b)) for a,b ∈ K such that
4 ν(a) (cid:54)= ν(b). Therefore, the map ν is almost a morphism from K to the min-
0 plus or tropical semifield R , which is the set R ∪ {+∞}, equipped with the
min
0
addition (a,b) (cid:55)→ min(a,b) and the multiplication (a,b) (cid:55)→ a + b. A basic ex-
.
1 ample of field with a non-archimedean valuation is the field of complex Puiseux
0 series, with the valuation which takes the leading (smallest) exponent of a series.
6 The images by a non-archimedean valuation of algebraic subsets of Kn are known
1
as non-archimedean amoebas. The latter have a combinatorial structure which is
:
v studied in tropical geometry [IMS07, MS15]. For instance, Kapranov’s theorem
i
X showsthattheclosureoftheimagebyanon-archimedeanvaluationofanalgebraic
hypersurface over an algebraically closed field is a tropical hypersurface, i.e., the
r
a non-differentiability locus of a convex polyhedral function, see [EKL06]. This gen-
eralizes the characterization of the leading exponents of the different branches of
an algebraic curve in terms of the slopes of the Newton polygon, which is part of
the classical Newton-Puiseux theorem.
1.2. Main results. In the present paper, we consider the eigenproblem over the
field of complex Puiseux series and related fields of functions. Our aim is to relate
the images of the eigenvalues by the non-archimedean valuation with certain easily
computable combinatorial objects called tropical eigenvalues.
2000 Mathematics Subject Classification. 47A55,47A75,05C50,12K10.
Key words and phrases. Perturbation theory, max-plus algebra, tropical semifield, spectral
theory,Newton-Puiseuxtheorem,amoeba,majorization,graphs,optimalassignment.
1
2 MARIANNEAKIAN,RAVINDRABAPAT,ANDSTE´PHANEGAUBERT
Thefirstmainresultofthepresentpaper,Theorem4.4,showsthatthesequence
ofvaluationsoftheeigenvaluesofamatrixA∈Kn×nisweakly(super)majorizedby
the sequence of (algebraic) tropical eigenvalues of the matrix obtained by applying
the valuation to the entries of A. Next, we show that the same majorization
inequality holds under more general circumstances. In particular, we consider in
Theorem 5.2 and Corollary 5.3 a relaxed definition of the valuation, in the spirit of
large deviations theory, assuming that the entries of the matrix are functions of a
small parameter (cid:15). We do not require these functions to have a Puiseux series type
expansion, but assume that they have some mild form of first order asymptotics.
Moreover, the results apply to a lower bound of the valuation of the entries of A.
Then, in Section 7, we assume that the entries of A satisfy
A =a (cid:15)Aij +o((cid:15)Aij) ,
ij ij
for some scalars a ∈ C and A ∈ R∪{+∞}, as (cid:15) tends to 0. When a = 0,
ij ij ij
this reduces to Aij =o((cid:15)Aij), so that valuations are partially known: only a lower
bound is known. Applying Corollary 5.3, majorization inqualities are derived in
Theorem 7.1. The assumption of Section 7 is satisfied of course if the entries
of A are absolutely converging Puiseux series, or more generally, if these entries
belongtoapolynomiallyboundedo-minimalstructure[vdD99,Ale13]. Weshowin
Theorem 7.4 that the majorization inequalities of Theorem 7.1 become equalities
for generic values of the entries a . The proof of the latter theorem relies on
ij
some variations of the Newton-Puiseux theorem, which we state as Theorems 6.1
and6.2. Thelatterresultsonlyrequireapartialinformationontheasymptoticsof
thecoefficientsofthepolynomial. Theparticularcasewherethispartialinformation
containsatleastthefirstorderasymptoticsofallthecoefficientsofthepolynomial
was considered in [Die68]. However, here we show that a partial information on
the first order asymptotics, giving an outer approximation of a Newton polytope,
allow one to derive a partial information on the roots. The latter idea goes back at
least to [Mon34] in the context of archimedean valuations.
The valuation only gives an information on the leading exponent of Puiseux
series. Our aim in Section 8 is to refine this information, by characterizing also the
coefficients λ ∈ C of the leading monomials of the asymptotic expansions of the
i
different eigenvalues Li of the matrix A,
Li ∼λ (cid:15)Λi, 1(cid:54)i(cid:54)n .
i
As a byproduct, we shall end up with an explicit form, easily checkable, of the
genericity conditions under which the majorization inequalities become equalities.
To this end, it is necessary to embed the standard eigenproblem in the wider class
ofmatrixpolynomialeigenproblems. Theorems8.2and8.3showinparticularthat
thecoefficientsλ aretheeigenvaluesofcertainauxiliarymatrixpolynomialswhich
i
aredeterminedonlybytheleadingexponentsandleadingcoefficientsoftheentries
of A. These polynomials are constructed from the optimal dual variables of an
optimal assignment problem, arising from the evaluation of the tropical analogue
of the characteristic polynomial.
1.3. Application to perturbation theory and discussion of related work.
The present results apply to perturbation theory [Kat95, Bau85], and specially, to
the singular case in which a matrix with multiple eigenvalues is perturbed. The
lattersituationistheobjectofthetheorydevelopedbyViˇsikandLjusternik[VL60]
NON-ARCHIMEDEAN VALUATIONS OF EIGENVALUES OF MATRIX POLYNOMIALS 3
and completed by Lidski˘ı [Lid65], see [MBO97] for a survey. The goal of this the-
ory is to give a direct characterization of the exponents, without computing the
Newton polytope of the characteristic polynomial. The theorem of [Lid65] solves
this problem under some genericity assumptions, requiring the non-vanishing of
certainSchurcomplements. ThequestionofsolvingdegenerateinstanceofLidsk˘ı’s
theorem has been considered in particular, by Ma and Edelman [ME98] and Na-
jman [Naj99], and also by Moro, Burke and Overton in [MBO97]. Theorems 8.2
and 8.3 generalize the theorem of Lidski˘ı, as they allow one to solve degenerate
instances in which the Schur complements needed in Lidski˘ı’s construction are no
longer defined.
The present train of thoughts originates from a work of Friedland [Fri86], who
showed that a certain deformation of the Perron root of a nonnegative matrix, in
terms of Hadamard powers, converges to the maximal circuit mean of the matrix,
a.k.a.,themaximaltropicaleigenvalue. Then,in[ABG98],weshowedthatthelim-
iting Perron eigenvector, along the same deformation, can also be characterized by
tropical means, under a nondegeneracy condition. An early version of the present
Theorems 7.1 and 7.4 appeared as Theorem 3.8 of the authors’ preprint [ABG04a].
There, we also gave a generalization of the theorem of Lidski˘ı, in which the ex-
ponents of the first order asymptotics of the eigenvalues are given by the tropical
eigenvalues of certain tropical Schur complements and their coefficients are given
bysomeassociatedusualSchurcomplementslikeinthetrueLidski˘ıtheorem. How-
ever, some singular cases remained, see for instance Example 8.4, motivating the
introduction of matrix polynomial eigenproblems in further works. The results
of Theorems 8.2 and 8.3 were announced without proof in the note [ABG04b].
Therefore, the present article is, for some part, a survey of results which have not
appeared previously in the form of a journal article. It also provides a general
presentation of eigenvalues in terms of valuation theory, with several new results
or refinements, like the general majorization inequality for the eigenvalues of ma-
trix polynomials, Theorem 8.1. The interest of this presentation is that it explains
better the relation between the results obtained here, or in eigenvalues perturba-
tion theory, for the non-archimedean valuations of matrix entries and eigenvalues,
withtheiranaloguesforarchimedeanvaluations,likethemodulusmap,orforsome
generalization of the notion of archimedean valuation which includes in particular
matrix norms.
Indeed, the latter works have motivated a more recent work by Akian, Gaubert
and Marchesini, who showed in [AGM14], that one form of the theorem of Fried-
land concerning the Perron or dominant root carries over to all eigenvalues: the
sequence of moduli of all eigenvalues of a matrix is weakly log-majorized by the
sequence of tropical eigenvalues, up to certain combinatorial coefficients. There-
fore, the result there can be thought of as a analogue of Theorem 4.4 or 7.1 for
the modulus archimedean valuation. Also, the results of [ABG04a, ABG04b] have
been at the origin of the application of tropical methods by Gaubert and Sharify
to the numerical computation and estimation of eigenvalues [GS09, Sha11], based
on the norms of matrix polynomial coefficients. This is a subject of current in-
terest, with work by Akian, Bini, Gaubert, Hammarling, Noferini, Sharify, and
Tisseur [BNS13, HMT13, NST14, AGS13].
The present results provide a further illustration of the role of tropical alge-
bra in asymptotic analysis, which was recognized by Maslov [Mas73, Ch. VIII].
4 MARIANNEAKIAN,RAVINDRABAPAT,ANDSTE´PHANEGAUBERT
He observed that WKB-type or large deviation type asymptotics lead to limiting
equations, like Hamilton-Jacobi equations, of a tropical nature. This observation
is at the origin of idempotent analysis [MS92, DKM92, KM97, LMS01]. The same
deformation has been identified by Viro [Vir01], in relation with the patchworking
method he developed for real algebraic curves.
Note that all the perturbation results described or recalled above study suffi-
cient conditions for the possible computation of first order asymptotics of some
roots when an information on the first order asymptotics of the data is only avail-
able. However,whenallthePuiseuxseriesexpansionofthedataisknown,Puiseux
theoremallowsonetocomputewithoutanyconditionallthePuiseuxseriesexpan-
sion. In the context of matrix polynomials, Murota [Mur90], gave an algorithm to
compute the Puiseux series expansions of the eigenvalues of a matrix polynomial
depending polynomially in the parameter (cid:15), avoiding the explicit computation of
the characteristic polynomial. As for Theorems 8.2 and 8.3, his algorithm relies on
a parametric optimal assignment problem.
The present work builds on tropical spectral theory. It has been inspired by the
analogy with nonnegative matrix theory, of which Hans Schneider was a master.
We gratefully acknowledge our debt to him.
2. Min-plus polynomials and Newton polygons
Wefirstrecallsomeelementaryfactsconcerningformalpolynomialsandpolyno-
mialfunctionsoverthemin-plussemifield,andtheirrelationwithNewtonpolygons.
The min-plus semifield, R , is the set R∪{+∞} equipped with the addition
min
(a,b) (cid:55)→ min(a,b), denoted a⊕b, and the multiplication (a,b) (cid:55)→ a+b, denoted
a⊗b or ab. We shall denote by 0 = +∞ and 1 = 0 the zero and unit elements of
R , respectively. The familiar algebraic constructions and conventions carry out
min
to the min-plus context with obvious changes. For instance, if A,B are matrices
of compatible dimensions with entries in R , we shall denote by AB the matrix
min
product with entries (AB) = (cid:76) A B = min (A +B ), we denote by Ak
ij k ik kj k ik kj
the kth min-plus matrix power of A, etc. Moreover, if x∈R \{0}, then we will
min
denote by x−1 the inverse of x for the ⊗ law, which is nothing but −x, with the
conventional notation. The reader seeking information on the min-plus semifield
may consult [CG79, MS92, BCOQ92, KM97, ABG13, But10].
We denote by R [Y] the semiring of formal polynomials with coefficients in
min
R in the indeterminate Y: a formal polynomial P ∈ R [Y] is nothing but a
min min
sequence (Pk)k∈N ∈ RNmin such that Pk = 0 for all but finitely many values of k.
Formal polynomials are equipped with the entry-wise sum, (P ⊕Q) = P ⊕Q ,
k k k
(cid:76)
and the Cauchy product, (PQ) = P Q . As usual, we denote a formal
k 0(cid:54)i(cid:54)k i k−i
polynomial P as a formal sum, P = (cid:76)∞ P Yk. We also define the degree and
k=0 k
valuation of P: degP = sup{k ∈ N | P (cid:54)= 0}, valP = inf{k ∈ N | P (cid:54)= 0}
k k
(degP = −∞ and valP = +∞ if P = 0). To any P ∈ R [Y], we associate the
min
polynomial function P(cid:98) : Rmin → Rmin, y (cid:55)→ P(cid:98)(y) = (cid:76)∞k=0Pkyk, that is, with the
usual notation:
(2) P(cid:98)(y)=min(Pk+ky) .
k∈N
Thus,P(cid:98) isconcave,piecewiseaffinewithintegerslopes. WedenotebyRmin{Y}the
semiring of polynomial functions P(cid:98). The morphism Rmin[Y] → Rmin{Y}, P (cid:55)→ P(cid:98)
NON-ARCHIMEDEAN VALUATIONS OF EIGENVALUES OF MATRIX POLYNOMIALS 5
isnotinjective,asitisessentiallyaspecializationoftheclassicalFencheltransform
over R, which reads:
R R
F :R →R , F(g)(y)=sup(xy−g(x)) ,
x∈R
where R := R∪{±∞}. Indeed, for all y ∈ R, P(cid:98)(y) = −F(P)(−y), where the
function k (cid:55)→P from N to R is extended to a function
k min
(cid:40)
P if x=k ∈N ,
(3) P :R→R , x(cid:55)→P(x), with P(x)= k
min
+∞ otherwise
ThefollowingresultofCuninghame-GreenandMeijergivesamin-plusanalogue
of the fundamental theorem of algebra.
Theorem 2.1 ([CGM80]). Any polynomial function P(cid:98) ∈Rmin{Y} can be factored
in a unique way as
(4) P(cid:98)(y)=Pn(y⊕c1)···(y⊕cn) ,
with c (cid:54)···(cid:54)c .
1 n
The ci will be called the roots of P(cid:98). The multiplicity of the root c is the
cardinality of the set {j ∈ {1,...,n} | cj = c}. We shall denote by R(P(cid:98)) the
sequence of roots: R(P(cid:98)) = (c1,...,cn). By extension, if P ∈ Rmin[Y] is a formal
polynomial, we will call roots of P the roots of P(cid:98), so R(P) := R(P(cid:98)). The next
properties also follow from [CGM80]; they show that the definition of the roots in
Theorem2.1isaspecialcaseofthenotionofatropicalhypersurfacedefinedasthe
nondifferentiability locus of a tropical polynomial [IMS07].
Proposition 2.2 (See [CGM80]). The roots c ∈ R of a formal polynomial P ∈
Rmin[Y] are exactly the points at which the function P(cid:98) is not differentiable. The
multiplicity of a root c ∈ R is equal to the variation of slope of P(cid:98) at c, P(cid:98)(cid:48)(c−)−
P(cid:98)(cid:48)(c+). Moreover, 0 is a root of P if, and only if, P(cid:98)(cid:48)(0−) := limc→+∞P(cid:98)(cid:48)(c) (cid:54)= 0.
In that case P(cid:98)(cid:48)(0−) is the multiplicity of 0, and it coincides with valP. (cid:3)
Legendre-Fenchel duality allows one to relate the tropical roots to the slopes of
Newtonpolygons. Toseethis,denotebyvexf theconvexhullofamapf :R→R,
and denote by P the formal polynomial whose sequence of coefficients is obtained
by restricting to N the map vexP: k (cid:55)→P :=(vexP)(k), for k ∈N. The function
k
vexP isfiniteontheinterval[valP,degP]. Also,itshouldbenotedthatthegraph
ofvexP isthestandardNewtonpolygonassociatedtothesequenceofpoints(k,P ),
k
k ∈[valP,degP].
Theorem 2.3 ([BCOQ92, Th. 3.43, 1 and 2]). A formal polynomial of degree n,
P ∈R [Y], satisfies P =P if, and only if, there exist c (cid:54)···(cid:54)c ∈R such
min 1 n min
that
P =P (Y⊕c )···(Y⊕c ) .
n 1 n
The c are unique and given, by:
i
(cid:40)
P (P )−1 if P (cid:54)=0
(5) c = n−i n−i+1 n−i+1 for i=1,...,n .
i
0 otherwise,
The following standard observation relates the tropical roots with the Newton
polygon.
6 MARIANNEAKIAN,RAVINDRABAPAT,ANDSTE´PHANEGAUBERT
Proposition 2.4. The roots c ∈ R of a formal polynomial P ∈ R [Y] coincide
min
with the opposites of the slopes of the affine parts of vexP :[valP,degP]→R. The
multiplicity of a root c∈R coincides with the length of the interval in which vexP
has slope −c. (cid:3)
Remark 2.5. The duality between tropical roots and slopes of the Newton polygon
in Proposition 2.4 is a special case of the Legendre-Fenchel duality formula for
subdifferentials: −c ∈ ∂(vexP)(x) ⇔ x ∈ ∂F(P)(−c) ⇔ x ∈ ∂+P(cid:98)(c) where ∂ and
∂+ denote the subdifferential and superdifferential, respectively [Roc70, Th. 23.5].
TheabovenotionsareillustratedinFigure1,whereweconsidertheformalmin-
plus polynomial P =Y3⊕5Y2⊕6Y⊕13. The map j (cid:55)→P , together with the map
j
vexP, are depicted at the left of the figure, whereas the polynomial function P(cid:98) is
depictedattherightofthefigure. WehaveP =Y3⊕3Y2⊕6Y⊕13=(Y⊕3)2(Y⊕7).
Thus, the roots of P are 3 and 7, with respective multiplicities 2 and 1. The roots
arevisualizedattherightofthefigure,oralternatively,astheoppositeoftheslopes
of the two line segments at the left of the figure. The multiplicities can be read
either on the map P(cid:98) at the right of the figure (the variation of slope of P(cid:98) at points
3 and 7 is 2 and 1, respectively), or on the map vexP at the left of the figure (as
the respective horizontal widths of the two segments). We conclude this section by
13 =P 13
=P¯ 5+2y
6+y
9 =vexP 9
6
5 5
3y
0 0
0 1 2 3 0 3 7
Figure 1. TheNewtonpoygonoftheformalmin-pluspolynomial
P =Y3⊕5Y2⊕6Y⊕13(left)andtheassociatedpolynomialfunction
P(cid:98) (right).
two technical results.
Lemma 2.6. Let P = (cid:76)n P Yi ∈ R [Y] be a formal polynomial of degree n.
i=0 i min
Then, R(P)=(c (cid:54)···(cid:54)c ) if, and only if, P (cid:62)P (Y⊕c )···(Y⊕c ) and
1 n n 1 n
(6) P =P c ···c for all i∈{0,n}∪{i∈{1,...,n−1}| c <c } .
n−i n 1 i i i+1
Inparticular, P =P holdsforalliasin (6), andP =P (Y⊕c )···(Y⊕c ).
n−i n−i n 1 n
Proof. We first prove the “only if” part. If R(P) = (c (cid:54) ··· (cid:54) c ), then P =
1 n
P (Y⊕c )···(Y⊕c ) and, by Theorem 2.3, P =P c ···c for all i=1,...n.
n 1 n n−i n 1 i
Recall that P defines a map x(cid:55)→P(x) by (3). By definition of vexP, the epigraph
of vexP, epivexP, is the convex hull of the epigraph of P, epiP. By a classical
result [Roc70, Cor 18.3.1], if S is a set with convex hull C, any extreme point
of C belongs to S. Let us apply this to S = epiP and C = epivexP. Since
c = P (P )−1, the piecewise affine map vexP changes its slope at any
i n−i n−i+1
NON-ARCHIMEDEAN VALUATIONS OF EIGENVALUES OF MATRIX POLYNOMIALS 7
pointn−isuchthatc <c . Thus, anypoint(n−i,vexP(n−i))withc <c
i i+1 i i+1
isanextremepointofepivexP,whichimpliesthat(n−i,vexP(n−i))∈epiP,i.e.,
P (cid:54) vexP(n−i) = P . Since the other inequality is trivial by definition of
n−i n−i
the convex hull, we have P =P . Obviously, P and P have the same degree,
n−i n−i
which is equal to n, and they have the same valuation, k. Then, (n,vexP(n))
and (k,vexP(k)) are extreme points of epivexP, and by the preceding argument,
P = P , and P = P . Hence, P = P , if k = 0, and P = P = +∞, if k > 0.
n n k k 0 0 0 0
We have shown (6), together with the last statement of the lemma. Since P =P
n n
and P (cid:62)P, we also obtain P (cid:62)P (Y⊕c )···(Y⊕c ).
n 1 n
For the “if” part, assume that P (cid:62) P (Y⊕c )···(Y⊕c ) and that (6) holds.
n 1 n
Since Q = P (Y⊕c )···(Y⊕c ) is convex, and the convex hull map P (cid:55)→ P is
n 1 n
monotone, we must have P (cid:62) Q = Q. Hence, P (cid:62) P (cid:62) Q and since P = Q
n−i n−i
foralliasin(6),wemusthaveP =P =Q ,thusvexP(n−i)=Q(n−i)at
n−i n−i n−i
thesei. SincevexP andQareconvex,QispiecewiseaffineandQ(j)=vexP(j)for
j attheboundaryofthedomainofQandatallthej whereQchangesofslope,we
must have vexP =Q. Hence P =Q and R(P)=R(P)=R(Q)=(c ,...,c ). (cid:3)
1 n
Corollary 2.7. Let P =(cid:76)n P Yi ∈R [Y] be a formal polynomial of degree n.
i=0 i min
Let c∈R be a finite root of P with multiplicity m, and denote by m(cid:48) the sum of the
multiplicities of all the roots of P greater than c (+∞ comprised). Then, P = P
i i
for both i = m(cid:48) and i = m+m(cid:48), P(cid:98)(c) = Pm(cid:48)cm(cid:48) = Pm+m(cid:48)cm+m(cid:48) and P(cid:98)(c) < Pici
for all 1(cid:54)i<m(cid:48) and m+m(cid:48) <i(cid:54)n.
Proof. Let us denote R(P) = (c (cid:54) ··· (cid:54) c ). By definition of c, m and m(cid:48) we
1 n
have m (cid:62) 1, m(cid:48) (cid:62) 0, m+m(cid:48) (cid:54) n, c = c = ··· = c , c < c
n−m(cid:48)−m+1 n−m(cid:48) n−m(cid:48)−m
if n−m(cid:48) −m > 0, and c < c if n−m(cid:48) < n. By Lemma 6, this implies
n−m(cid:48)+1
that for both i = m(cid:48) and i = m+m(cid:48), P = P = P c ···c . We also have
i i n 1 n−i
Since P = (vexP)(i) (cid:54) P , we have P (cid:62) (vexP)(i) = P = P c ···c for
i i i i n 1 n−i
all i = 0,...,n. Moreover, by Theorem 2.1, we have P(cid:98)(c) = Pn(c⊕c1)···(c⊕
cn)=Pnc1···cn−m(cid:48)−mcm+m(cid:48) =Pnc1···cn−m(cid:48)cm(cid:48). Hence,P(cid:98)(c)=Pm+m(cid:48)cm+m(cid:48) =
Pm(cid:48)cm(cid:48), and P(cid:98)(c)<Pnc1···cn−ici (cid:54)Pici for i<m(cid:48) and for i>m+m(cid:48). (cid:3)
3. Tropical eigenvalues
We now recall some classical results on tropical eigenvalues and characteristic
polynomials.
Thepermanentofamatrixwithcoefficientsinanarbitrarysemiring(S,⊕,⊗)is
defined by
n
(cid:77) (cid:79)
per(A)= A ,
iσ(i)
σ∈Sn i=1
where S is the set of permutations of [n] := {1,...,n}. In particular, for any
n
matrix A∈Rn×n,
min
per(A)= min |σ| ,
A
σ∈Sn
where for any permutation σ ∈S , we define the weight of σ with respect to A as
n
|σ| :=A +···+A .
A 1σ(1) nσ(n)
To any min-plus n×n matrix A, we associate the (directed) graph G(A), which
has set of nodes [n] and an arc (i,j) if A (cid:54)= 0, and the weight function which
ij
8 MARIANNEAKIAN,RAVINDRABAPAT,ANDSTE´PHANEGAUBERT
associates the weight A to the arc (i,j) of G(A). In the sequel, we shall omit
ij
the word “directed” as all graphs will be directed. Then, per(A) is the value of
an optimal assignment in this weighted graph. It can be computed in O(n3) time
using the Hungarian algorithm [Sch03, § 17]. We refer the reader to [BR97, § 2.4]
or [Sch03, § 17] for more background on the optimal assignment problem and a
discussion of alternative algorithms.
We define the formal characteristic polynomial of A,
n
(cid:77) (cid:79)
P :=per(YI⊕A)= (Yδ ⊕A )∈R [Y] ,
A iσ(i) iσ(i) min
σ∈Sn i=1
whereI istheidentitymatrix,andδ =1ifi=jandδ =0otherwise. Theformal
ij ij
polynomial P has degree n and its coefficients are given by (P ) =trmin(A), for
A A k n−k
k =0,...,n−1 and (P ) =1, where trmin(A) is the min-plus k-th trace of A:
A n k
 
(cid:77) (cid:77) (cid:79)
(7) trmkin(A):=  Ajσ(j) ,
J⊂{1,...,n},#J=k σ∈SJ j∈J
where S is the set of permutations of J. The associated min-plus polynomial
J
function will be called the characteristic polynomial function of A, and its roots
will be called the (algebraic) eigenvalues of A.
The algebraic eigenvalues of A (and so, its characteristic polynomial function)
can be computed in O(n4) time by the method of Burkard and Butkoviˇc [BB03].
Gassner and Klinz [GK10] showed that this can be reduced to a O(n3) time, using
parametric optimal assignment techniques. However, it is not known whether the
sequenceofcoefficientsoftheformalcharacteristicpolynomialP canbecomputed
A
in polynomial time.
The term algebraic eigenvalue is used here since unlike for matrices with real or
complexcoefficients,arootλ∈R ofthecharacteristicpolynomialofan×nmin-
min
plusmatrixAmaynotsatisfyAu=λuforsomeu∈Rn . Toavoidanyconfusion,
min
weshallcallascalarλwiththelatterpropertyageometric eigenvalue. Thefollow-
ing statement and remarks collect some results in tropical spectral theory, which
have been developed by several authors [CG79, Vor67, Rom67, GM77, CDQV83,
MS92, BSvdD95, AGW05, BCGG09]. We refer the reader to [BCOQ92, But10]
for more information. We say that a matrix A is irreducible if G(A) is strongly
connected.
Theorem 3.1 (See e.g. [But10]). The minimal algebraic eigenvalue of a matrix
A∈Rn×n is given by
min
n
(8) ρmin(A)=(cid:77) (cid:77) (Ai1i2···Aiki1)k1 ,
k=1i1,...,ik
or equivalently, by the following expression called minimal circuit mean,
|c|
(9) min A ,
ccircuitinG(A) |c|
where for all paths p = (i ,i ,...,i ) in G(A), we denote by |p| = A +···+
0 1 k A i0i1
A the weight of p, and by |p| = k its length, and the minimum is taken over
ik−1ik
all elementary circuits of G(A). (cid:3)
NON-ARCHIMEDEAN VALUATIONS OF EIGENVALUES OF MATRIX POLYNOMIALS 9
An important notion to be used in the sequel is the one of critical circuit, i.e.,
of circuit c=(i ,i ,...,i ,i ) of G(A) attaining the minimum in (9). The critical
1 2 k 1
graph of A is the union of the critical circuits, that is the graph whose nodes and
arcs belong to critical circuits. It is known that ρ (A) is the minimal geometric
min
eigenvalueofAandthatthemultiplicityotρ (A)asageometriceigenvalue(i.e.,
min
the“dimension”oftheassociatedeigenspace)concideswiththenumberofstrongly
connected components of the critical graph. Note also that, if A is irreducible,
ρ (A) is the unique geometric eigenvalue of A.
min
Remark 3.2. Themultiplicityofρ (A),asanalgebraiceigenvalue,coincideswith
min
the term rank (i.e., the maximal number of nodes of a disjoint union of circuits) of
the critical graph of A. This follows from the arguments of proof of Theorem 4.7
in [ABG04a].
4. Majorization inequalities for valuations of eigenvalues
The inequalities that we shall establish involve the notion of weak majorization,
see [MO79] for background.
Definition 4.1. Let u,v ∈ Rn . Let u (cid:54) ··· (cid:54) u (resp. v (cid:54) ··· (cid:54) v )
min (1) (n) (1) (n)
denote the components of u (resp. v) in increasing order. We say that u is weakly
(super) majorized by v, and we write u≺w v, if the following conditions hold:
u ···u (cid:62)v ···v ∀k =1,...,n .
(1) (k) (1) (k)
Theweakmajorizationrelationisonlydefinedin[MO79]forvectorsofRn. Here,
it is convenient to define this notion for vectors with infinite entries. We used the
min-plusnotationforhomogeneitywiththerestofthepaper. Thefollowinglemma
statesausefulmonotonicitypropertyofthemapwhichassociatestoaformalmin-
plus polynomial P its sequence of roots, R(P).
Lemma 4.2. Let P,Q∈R [X] be two formal polynomials of degree n. Then,
min
(10) P (cid:62)Q and P =Q =⇒ R(P)≺w R(Q) .
n n
Proof. From P (cid:62) Q, we deduce P (cid:62) Q. Let R(P) = (c (P) (cid:54) ··· (cid:54) c (P)) and
1 n
R(Q) = (c (Q) (cid:54) ··· (cid:54) c (Q)) denote the sequence of roots of P and Q, respec-
1 n
tively. Using P (cid:62) Q, P = P = Q = Q and (5), we get c (P)···c (P) =
n n n n 1 k
P (P )−1 (cid:62) Q (Q )−1 = c (Q)···c (Q), for all k = 1,...,n, that is
n−k n n−k n 1 k
R(P)≺w R(Q). (cid:3)
Let ν be a (non-archimedean) valuation on a field K, i.e., a map ν : K →
R∪{+∞}satisfyingtheconditions(1)recalledintheintroduction. Weshallthink
of the images of ν as elements of the min-plus semifield, writing ν(ab)=ν(a)ν(b).
The main example of valuation considered here is obtained by considering the field
of complex Puiseux series, with the valuation which takes the smallest exponent of
a series. Recall that this field consists of the series of the form (cid:80)∞ a (cid:15)k/s with
k=K k
a ∈ C, K ∈ Z and s ∈ N\{0}, in which case the smallest exponent is equal to
k
K/s as soon as a (cid:54)= 0. The results of the present paper apply as well to formal
K
series or to series that are absolutely convergent for a sufficiently small positive (cid:15).
The following proposition formulates in terms of tropical roots a well known
propertyusuallystatedintermsofNewtonpolygons,seeforinstance[Bou89,Exer.
VI.4.11]. It is a special case of a result proved in [EKL06] for non-archimedean
10 MARIANNEAKIAN,RAVINDRABAPAT,ANDSTE´PHANEGAUBERT
amoebas of hypersurfaces. We include a proof relying on Lemma 2.6 for the con-
venience of the reader, since we shall use the same argument in the sequel.
Proposition 4.3 (See [EKL06, Th. 2.1.1]). Let K be an algebraically closed field
witha(non-archimedean)valuationν andletP =(cid:80)n P Yk ∈K[Y],withP =1.
k=0 k n
Then, the valuations of the roots of P (counted with multiplicities) coincide with
the roots of the min-plus polynomial ν(P):=(cid:76)n ν(P )Yk.
k=0 k
Proof. Let Y ,...,Y denote the roots of P, ordered by nondecreasing valu-
1 n
ation, c := ν(Y ), so that c (cid:54) ··· (cid:54) c , Q := (cid:76)n c ...c Yn−k, and
i i 1 n k=0 1 k
P := ν(P). Observe that P = Q = 1, and Q = (Y ⊕c )...(Y ⊕c ). Since
n n 1 n
P =(−1)k(cid:80) Y ···Y , we get ν(P )(cid:62)ν(Y ···Y )=c ···c , and
son−Pk(cid:62) Q, and Pi1<=···<Qik .i1Moreoivker, if c < cn−kor k =1n, Y k···Y 1is thke only
n n k k+1 1 k
term in the sum (−1)k(cid:80) Y ···Y , having a minimal valuation, and so,
i1<···<ik i1 ik
P = ν(P ) = c ···c = P c ···c . Then, it follows from Lemma 2.6 that
n−k n−k 1 k n 1 k
R(P)=(c (cid:54)···(cid:54)c )=(ν(Y ),··· ,ν(Y )). (cid:3)
1 n 1 n
We now establish majorization inequalities for the valuations of the eigenvalues
of matrices.
Theorem 4.4. Let K be an algebraically closed field with a (non-archimedean)
valuation ν. Let A = (A ) ∈ Kn×n. Then, the sequence of valuations of the
ij
eigenvalues of A (counted with multiplicities) is weakly majorized by the sequence
of (algebraic) eigenvalues of the matrix A:=(ν(A ))∈Rn×n.
ij min
Proof. Let Q:=det(YI−A)∈K[Y] be the characteristic polynomial of A, and let
P := per(YI ⊕A) ∈ R [Y] be the min-plus characteristic polynomial of A. Let
min
Q:=ν(Q). ObservethatthecoefficientsofQaregivenbyQ =(−1)n−ktr (A),
k n−k
for k =0,...,n−1 and Q =1, where tr (A) is the k-th trace of A:
n k
 
(cid:88) (cid:88) (cid:89)
(11) trk(A):=  sgn(σ) Ajσ(j) .
J⊂{1,...,n},#J=k σ∈SJ j∈J
Similarly, the coefficients of P are given by P = trmin(A), for k = 0,...,n−1
k n−k
and P = 1, where trmin(A) is the min-plus k-th trace of A (7). It follows that
n k
Q = ν(Q) (cid:62) P, and Q = P = 0. Hence, by Lemma 4.2, R(Q) ≺w R(P). By
n n
Proposition 4.3, R(Q) coincides with the sequence of valuations of the eigenvalues
of A, which establishes the result. (cid:3)
Iftheminimumineveryexpression(7)isattainedbyonlyoneproduct, wehave
ν(Q) = P in the previous proof, and so, the majorization inequality becomes an
equality. However, this condition is quite restrictive (it requires each of a family of
a combinatorial optimization problem to have a unique solution). We shall see in
the next section that the same conclusion holds under milder assumptions.
5. Large Deviation Type Asymptotics and Quasivaluations
The results of the previous section apply to the field of complex Puiseux series.
However, in some problems of asymptotic analysis, we need to deal with complex
functions f of a small positive parameter (cid:15) which may not have Puiseux series