Dual Spaces and Bilinear Forms in Supertropical Linear Algebra

0.21 MB·English

by  Zur Izhakian

DUAL SPACES AND BILINEAR FORMS
IN SUPERTROPICAL LINEAR ALGEBRA
ZURIZHAKIAN,MANFREDKNEBUSCH,ANDLOUISROWEN
2
Abstract. Continuing[5],thispaperinvestigatesfinerpointsofsupertropicalvectorspaces,including
1
dualbasesandbilinearforms,withsupertropicalversionsofstandardclassicalresultssuchastheGram-
0
SchmidttheoremandCauchy-Schwartzinequality,andchangeofbase. Wealsopresentthesupertropical
2
versionofquadraticforms,andseehowtheycorrespondtosymmetricsupertropicalbilinearforms.

n
a
J

1 1. Introduction
3
Thispaper,thecontinuationof[5],bringstheanalogofsomeclassicaltheoremsoflinearalgebratothe

]
supertropicalsetting. The majordifference of supertropicallinear algebrafromclassicallinear algebrais
C
that one can have proper subspaces of the same rank, which we call thick.
A
Wealsoconsiderlinearmapsinthesupertropicalcontext,forwhichtheequalityϕ(v+w)=ϕ(v)+ϕ(w)
.
h is replaced by the ghost surpassing relation ϕ(v+w) |=ϕ(v)+ϕ(w). Supertropical linear maps lead us
t gs
a to the notion of the supertropical dual space. The dual space depends on the choice of thick subspace
m
with s-base B, but there is a natural “dual s-base” of B, of the same rank (Theorem 2.21). This leads
[ to rather delicate considerations concerning dual spaces, including an identification of a space with its

1 double dual in Theorem 2.24.
v Tounderstandangles,westudy supertropicalbilinearforms. Asusual,inthe supertropicaltheorythe
1 zero element is replacedby the “ghostideal.” This complicates our approachto bilinear forms, since the
8
theorycanbedistortedbytheinnerproductoftwoelementsbeinga“large”ghost. Thus,weintroducea
4
condition(Definition4.3)tocontroltheν-valueoftheinnerproduct,topreventitfromobscuringtangible
6
. angles,whichfollowsfromananalogoftheCauchy-Schwartzinequality(cf.Definition4.6). Thenwealso
1
get a supertropical analog of the Gram-Schmidt process in Lemma 4.11 and Theorem 4.16.
0
As with the classical theory, one can pass back and forth from bilinear forms to quadratic forms.
2
1 Surprisingly, at times this is easier in the supertropical theory, as seen in Theorem 5.11, because many
: supertropical quadratic forms satisfy the quasilinear property of Definition 5.2.
v
i
X 1.1. Background. Letus brieflyreviewingbriefly the supertropicalfoundations. A semiring without
zero, which we notate as semiring , is a structure (R ,+,·,1 ) such that (R ,·,1 ) is a monoid and
r † † R † R
a (R ,+) is a commutative semigroup, with distributivity of multiplication overaddition on both sides. A
†
supertropical semiring is atriple(R ,G,ν), whereR isasemiring andG isasemiring ideal,called
† † † † †
the ghost ideal, together with an idempotent map
ν :R −→G
†
(preserving multiplication as well as addition) called the ghost map on R , satisfying the following
†
properties, where we write aν for ν(a):
(a) a+b=aν if aν =bν;
Date:February1,2012.
2010 Mathematics Subject Classification. Primary 15A03, 15A09, 15A15, 16Y60; Secondary 14T05, 15A33, 20M18,
51M20.
Keywordsandphrases. Tropicalalgebra,vectorspace,linearalgebra,d-base,s-base,dualbase,changeofbasesemirings,
bilinearform.
Theworkofthefirstandthirdauthors hasbeensupportedbytheIsraelScienceFoundation, grant448/09.
The second author was supported in part by the Gelbart Institute at Bar-Ilan University, the Minerva Foundation at
Tel-AvivUniversity,theMathematics Dept. ofBar-IlanUniversity,andtheEmmyNoetherInstitute.
1
2 Z.IZHAKIAN,M.KNEBUSCH,ANDL.ROWEN
(b) a+b∈{a,b}, ∀a,b∈R s.t. aν 6=bν.
†
(Equivalently, G is ordered, via aν ≤bν iff aν +bν =bν.)
In particular, aν = a+a. We write a > b if aν > bν; we stipulate that a and b are ν-matched,
ν
written a∼=ν b, if aν =bν. We say that a dominates b if a>ν b; a weakly dominates b if a≥ν b.
Recall that any commutative supertropical semiring satisfies the Frobenius formula from [7, Re-
mark 1.1]:
(a+b)m =am+bm (1.1)
for any m∈N+.
A supertropical semifield is a supertropical semiring F for which
† † †
T :=F \G
†
is a group, such that the map ν| : T → G (defined as the restriction from ν to T) is onto. T is called
the set of tangible elements ofTF . Thus, G is also a group.
†
Asupertropical vector spaceoverasupertropicalsemifield F isjustasemiring module(satisfy-
† † †
ingtheusualmoduleaxioms,cf.[5,Definition2.8]). V hasthedistinguishedstandard ghost subspace
H0 :=eV, as well as the ghost map ν :V →H0, given by ν(v):=v+v =ev. We write vν for ν(v).
When dealing with vector spaces, we will assume for convenience of notation the existence of a zero
element0 ∈F. Moreprecisely,onecouldstartwithasemifield F andthenconsidertheformalvector
F † †
space F := F ∪{0 }. A nonzero vector v of F(n) is called tangible if each of its components is in
† F
T ∪{0 }.
F
Definition 1.1. Wedefinethe ghost surpasses relation |= on any supertropical semifield F (resp. on
† †
gs
a supertropical vector space V), by
b |=a iff b=a+c for some ghost element c,
gs
where a,b,c∈F (resp. a,b,c∈V).
†
This relationis antisymmetric,by [9, Lemma 1.5]. Inthis notation, by writing a =| 0F we meana∈H0.
gs
1.2. Matrices. Assume that A is a nonsingular matrix. We define the matrices
1
F
A = adj(A), A :=A AA ,
∇ ∇ ∇ ∇
|A|
cf. [9,Remark2.14],andrecallthatI =AA andI =A Aarequasi-identities,inthesensethatthey
A ∇ A′ ∇
are multiplicatively idempotent matrices having determinant 1 , and ghost surpass the identity matrix.
F
Then the matrices
IA, IA′ , A∇ =A∇AA∇ =A∇IA, and IAA
are nonsingular, since I AA =I2 =I is nonsingular.
A ∇ A A
1.3. Bases. In[8],wedefinedvectorsinV tobetropically independentifnolinearcombinationwith
tangible coefficients is in H0, and proved that a set of n vectors is tropically independent iff its matrix
has rank n.
Definition 1.2. A d-base (for dependence base) of a supertropical vector space V is a maximal set of
tropically independent elementsofV. Althoughd-bases couldhavedifferent numberofelements,wedefine
rank(V) to be the maximal possible cardinality of a d-base.
A subspace W of a supertropical vector space V is thick if rank(W)=rank(V).
An s-base of V (when it exists) is a minimal spanning set.
A d,s-base is a d-base which is an s-base. A vector v ∈/ H0 in V is critical if we cannot write
v |=v +v for v ,v ∈V \Fv.
1 2 1 2
gs
In [5, Theorem 5.24] we prove that the s-base (if it exists) is unique up to multiplication by scalars.
Example 1.3. The standard d,s-base for F(n) is the set of vectors
{(1 ,0 ,0 ...,0 ),(0 ,1 ,0 ,...,0 ),...,(0 ,...,0 ,1 )}.
F F F F F F F F F F F
DUAL SPACES AND BILINEAR FORMS 3
Given the plethora of thick subspaces, one would expect the theory of dual spaces to be rather com-
plicated, and one of our basic aims in this paper is to make sense of duality.
Bilinear forms are introduced in [5, Section 6] in order to treat orthogonality of vectors. We review
them in this paper as they are needed, in §3.
2. Supertropical linear maps and the dual space
In this section we introduce supertropical linear maps, and use these to define the dual space with
respect to a d,s-base B, showing that it has the canonical dual s-base to be given in Theorem 2.21. (A
version of a dual space for idempotent semimodules, in the sense of dual pairs, given in [1], leads to a
Hahn-Banach type-theorem.)
2.1. Supertropical maps. Recall that a linear map ϕ : V → V of vector spaces over a semifield F
′
satisfies
ϕ(v+w)=ϕ(v)+ϕ(w), ϕ(av)=aϕ(v), ∀a∈R, v,w ∈V.
We weaken this a bit in the supertropical theory.
Definition 2.1. Given supertropical vector spaces V and V over a supertropical semifield F, a su-
′
pertropical map
ϕ:V → V
′
is a function satisfying
ϕ(v+w) |=ϕ(v)+ϕ(w), ϕ(αv)=αϕ(v), ∀α∈T, v,w∈V. (2.1)
gs
We write Hom (V,V ) for the set of supertropical maps from V to V , which is viewed as a vector
gs ′ ′
space over F in the usual way, given by pointwise operations. We write H0 :=eV and H0′ :=eV′.
Lemma 2.2. Any supertropical map ϕ:V →V satisfies
′
ϕ(av) =| aϕ(v)
gs
for any v ∈V and a∈F. In particular,
ϕ(H0)⊆H0′. (2.2)
Proof. The assertion holds by definition when a ∈ T, and when a ∈ G we take α ∈ T such that
a=αν =α+α and thus have
ϕ(av)=ϕ((α+α)v) =| ϕ(αv)+ϕ(αv)=αϕ(v)+αϕ(v)=(α+α)ϕ(v)=aϕ(v).
gs
The last assertion follows by taking a=e. (cid:3)
Remark 2.3. One may wonder why we have required ϕ(αv) = αϕ(v) and not just ϕ(αv) =| αϕ(v).
gs
In fact, these are equivalent when α ∈ T, since F is a supertropical semifield. Indeed, assume that
ϕ(αv) =| αϕ(v) for any α∈T and v ∈V. Then also α 1 ∈T. By hypothesis,
−
gs
α 1ϕ(αv) |=α 1αϕ(v)=ϕ(v)
− −
gs
and
ϕ(v)=ϕ(α 1αv) =| α 1ϕ(αv),
− −
gs
so by antisymmetry, α 1ϕ(αv)=ϕ(v), implying ϕ(αv)=αϕ(v).
−
Remark 2.4. Lemma 2.2 implies
ϕ(vν)=ϕ(ev) |=eϕ(v)=ϕ(v)ν;
gs
i.e., ϕ◦ν |=ν◦ϕ.
gs
Lemma 2.5. If v |=w then ϕ(v) =| ϕ(w).
gs gs
4 Z.IZHAKIAN,M.KNEBUSCH,ANDL.ROWEN
Proof. Write v =w+w′ where w′ ∈H0. Then
ϕ(v) |=ϕ(w)+ϕ(w ) =| ϕ(w)
′
gs gs
since ϕ(w)∈H . (cid:3)
0′
Remark 2.6. Hom (V,V ) has a supertropical vector space structure, under the natural operations
gs ′
(ϕ +ϕ )(v)=ϕ (v)+ϕ (v), (aϕ)(v)=aϕ(v), ν(ϕ)(v)=ϕ(v)ν,
1 2 1 2
for ϕ∈Hom (V,V ), a∈F, v ∈V.
gs ′
The ghost maps are {f ∈Hom (V,V ):f(V)⊆H }.
gs ′ 0′
Definition 2.7. Given a supertropical map ϕ:V →V , we define the ghost kernel
′
g-ker(ϕ):=ϕ-1(H )={v∈V :ϕ(v)∈H }.
0′ 0′
We say that ϕ is ghost monic if ϕ-1(H0′)=H0.
Remark 2.8. g-ker(ϕ) is an F-subspace of V.
Definition 2.9. A supertropical map ϕ : V → W of vector spaces is called tropically onto if ϕ(V)
contains a thick subspace of W. An iso is a supertropical map that is both ghost monic and tropically
onto. (Note that this need not be an isomorphism in the usual sense, since ϕ need not be onto.)
Remark 2.10. The composition of isos is an iso.
2.2. Linear functionals.
Definition2.11. SupposeV isavectorspaceoverasupertropicalsemifieldF. Thespaceofsupertropical
maps
V :=Hom (V,F),
∗ gs
is called the (supertropical) dual F-space of V, and its elements are called linear functionals; i.e.,
any linear functional ℓ∈V satisfies
∗
ℓ(v +v ) |=ℓ(v )+ℓ(v ), ℓ(αv )=αℓ(v )
1 2 1 2 1 1
gs
for any v ,v ∈V and α∈T.
1 2
The set H0(V∗) of ghost linear functionalsis the set of linear functionals that are ghost maps, i.e.,
{ℓ∈V∗ :ℓ(V)⊆G0}.
Our next goal is to describe the linear functionals for thick subspaces V of F(n) (including the case
V = F(n)). Towards this end, we want a definition of linear functionals that respects a given d-base
B ={b ,...,b } of V. We define the matrix A(B) ofB,to be the matrix whose columns arethe vectors
1 n
comprising B. For the remainder of this section we set the matrix
A:=A(B).
Definition 2.12. A d-base B is closed if I B = B. A closed subspace of V is a subspace having a
A
closed d-base.
There is an easy way to get a closed d-base from an arbitrary d-base B.
Definition 2.13. Write A =I A, and let B denote the set of column vectors of A . Let
A
B B
V :={A v :v ∈V},
B B
the subspace of V spanned by B.
Remark 2.14. A = I A is a nonsingular matrix, implying B is a d-base. B is easier to compute
A
B
than B, since now we have
A A I =AA AA I =I3 =I ,
∇ A ∇ ∇ A A A
B
implying
I A =I I A=I A=A .
A A A A
B B
Lemma 2.15. V is a thick, closed subspace of V, and B is a closed d,s-base of V .
B B
DUAL SPACES AND BILINEAR FORMS 5
Proof. V contains n independent vectors. Clearly B is closed since I2 =I . (cid:3)
A A
B
Rather than dualizing all of V, we turn to the space
V∗ :=Homgs(V ,F).
B B
Define L ∈Hom (V,V) by
A gs
L (v):=A v, for every v ∈V.
A ∇
We also define the map L :V →V by
A
L (v):=I v, for every v ∈V.
A A
e
Remark 2.16. (L )2 =L , and L is the identity on V since
A A eA
B
I (I Av)=I2Av =I Av.
A A A A
e e e
Likewise, L (v)=v for all v ∈V .
A
B
Lemma 2.17. If ℓ∈V , then ℓ=(ℓ◦L ) on V . In other words,
B∗ A VB B
V =(cid:12){(ℓ◦L ) :ℓ∈V }.
e∗ (cid:12) A ∗
B B
Proof. Follows at once from the remark. (cid:12) (cid:3)
e (cid:12)
Lemma 2.18. H0(V∗)={f|VB :f ∈H0(V∗)}.
B
Proof. Suppose f′ ∈ H0(V∗). Let f = f′ ◦ LA ∈ H0(V∗). Then f′ = f|VB. The other inclusion is
obvious. B (cid:3)
e
Definition 2.19. Given a closed d-base B ={b ,...,b } of V, define ǫ :V →F by
1 n i
B
ǫ (v)=b tL (v)=btA v,
i i A i ∇
the scalar product of b and A v. Also, define B ={ǫ :i=1,...,n}.
i ∇ ∗ i
When v is tangible, we saw in [5, Remark 4.19] that
n
[
v g ǫ (v)b
i i
gd
i=1
X
is a saturated tropical dependence relation of v on the b ’s; this is the motivation behind our definition.
i
Remark 2.20. ǫ is a linear functional. Also, by definition, ǫ (b ) is the (i,j) position of AA =I , a
i i j ∇ A
quasi-identity, which implies
ǫi(bi)=1F; ǫi(bj)∈G0, ∀i6=j.
Hence,
n
α ǫ (b ) |=α ǫ (b )=α .
i i j j j j j
i=1 gs
X
Theorem 2.21. If F is a supertropical semifield and B is a closed d-base of V, then {ǫ : i= 1,...,n}
i
is a closed d,s-base of V .
∗
B
n
Proof. For any ℓ∈V , we write α =ℓ(b ), and then see from Remark 2.20 that α ǫ |=ℓ on V .
∗ i i i=1 i i
B gs B
It remains to show that the {ǫ : i = 1,...,n} are tropically independent. IfP n β ǫ were ghost
i i=1 i i
for some βi ∈ T0, we would have ni=1βibitA∇ ghost. Let D denote the diagonalPmatrix {β1,...,βn},
and let I = {i : β 6= 0 }, and assume there are k such tangible coefficients β . Then for any i ∈/ I we
i F i
have β =0 , implying the i row oPf the matrix DI is zero. But the sum of the rows of the matrix DI
i F A A
corresponding to indices from I would be n β b tA , which is ghost, implying that these k rows of
i=1 i i ∇
DI are dependent; hence DI has rank ≤k−1. On the other hand, the k rows of DI corresponding
A A A
P
to indices from I yield a k ×k submatrix of determinant β ∈ T, implying its rank ≥ k by [6,
i i
Theorem 3.4], a contradiction. ∈I (cid:3)
Q
6 Z.IZHAKIAN,M.KNEBUSCH,ANDL.ROWEN
Intheviewofthetheorem,wedenoteB ={ǫ :i=1,...,n},andcallitthe(tropical)dual d,s-base
∗ i
of V . Write V for (V ) . Define a map
∗∗ ∗ ∗
B B B
Φ:V →V ,
∗∗
B B
given by v 7→f , where
v
f (ℓ) = ℓ(v).
v
Example 2.22. The map Φ:F(n) →F(n)∗∗ is a vector space isomorphism when B is the standard base
(cf. Example 1.3).
Remark 2.23. Since AA =I is a quasi-identity matrix, we see that
∇ A
f (ǫ )=ǫ (b )=b tA b
bj i i j i ∇ j
Theorem 2.24. Suppose V is a thick closed subspace of F(n), with a d,s-base of tangible vectors. For
any v ∈ V, define v ∈ V by v (ℓ) = ℓ(v). The map Φ : V → V given by v 7→ v is an iso of
∗∗ ∗∗ ∗∗ ∗∗ ∗∗
supertropical vector spaces.
Proof. Applying Theorem2.21twice,we see thatΦ(B)is a d-baseofn elements. Φ is ghostmonic, since
g-kerΦ cannotcontain tangible vectors,in view of [5, Theorem3.4] (which says that the g-annihilatorof
a nonsingular matrix cannot be tangible). But by Example 2.22, taking the standard classical base, we
see that V has rank n, and thus is thick in F(n). (cid:3)
∗∗
3. Supertropical bilinear forms
Linearfunctionalsanddualspacescastmorelightonthesupertropicaltheoryofbilinearforms. Hereis
a more concise versionof [5, Definition 6.1]. Throughout, F denotes a supertropicalsemifield (although
†
we permit the possibility that 0∈F).
Definition 3.1. A (supertropical) bilinear form on supertropical vector spaces V and V is a function
′
B :V ×V →F that is a linear functional in each variable.
′
We write hv,v i for B(v,v ). Specifically, given w ∈ V , we can define the functional w : V → F by
′ ′ ′
w(v)=hv,wi. (Similarly we define v :V →F for v ∈V.)
e
Example 3.2. There is a natural bilinear form B :V ×V →F, given by hv,fi=f(v), for v ∈V and
∗
e e
f ∈V .
∗
Remark 3.3.
(i) Notation as in Definition 3.1, any bilinear form induces a natural map Φ : V → V , given by
′ ∗
w7→w. Likewise, there is a natural map Φ:V →V′∗, given by v 7→v.
(ii) For any bilinear form B, if v |= α v and w =| β w , for α ,β ∈F, then
i i i j j j i j
e gs gs e
P P
hv,wi |= α β hv ,w i. (3.1)
i j i j
gs i,j
X
Definition 3.4. When V = V, we say that B is a (supertropical) bilinear form on the vector
′
space V. The space V is nondegenerate (with respect to B) if hv,Vi6⊆G, ∀v ∈V.
Although this definition suffices to carry through much of the theory, we might want to compute the
bilinear form B in terms of its values on an s-base of V. To permit this, we tighten the definition a bit.
Definition 3.5. We say that a bilinear form B is strict if
hα v +α v ,β w +β w i=α β hv ,w i+α β hv ,w i+α β hv ,w i+α β hv ,w i,
1 1 2 2 1 1 2 2 1 1 1 1 1 2 1 2 2 1 2 1 2 2 2 2
for v ∈V and w ∈V .
i i ′
DUAL SPACES AND BILINEAR FORMS 7
Definition 3.6. The Gram matrix of the bilinear form with respect to vectors v ,...,v ∈ V = F(n)
1 k
is defined as the k×k matrix
hv ,v i hv ,v i ··· hv ,v i
1 1 1 2 1 k
hv ,v i hv ,v i ··· hv ,v i
2 1 2 2 2 k
G(v1,...,vk)= ... ... ... ... . (3.2)
 
e  hvk,v1i hvk,v2i ··· hvk,vki 
Definition 3.7. We write v⊥⊥w when hv,wi∈G0, that is hv,wi |=0F. In this case, we say that v is left
gs
ghost orthogonal to w, or left g-orthogonal for short. Likewise, a subspace W is left g-orthogonal
1
to W2 if hw1,w2i∈G0 for all wi ∈Wi.
A subset S of V is g-orthogonal (with respect to a given bilinear form) if any pair of distinct vectors
from S is g-orthogonal.
Inthispaperweusuallyrequire⊥⊥tobeasymmetricrelation. Thiswasstudiedingreaterdetailin[5,
Definition 6.12],but we take the simpler definition here since we focus on strict bilinear forms, for which
the two notions coincide in view of [5, Lemma 6.15]).
3.1. Isotropic vectors.
Definition 3.8. A vector v ∈ V is g-isotropic if hv,vi ∈ G0; v is g-nonisotropic if hv,vi ∈ T. A
subset S ⊂V is g-nonisotropic if each vector of S is g-isotropic.
For any supertropical semifield F and k ∈N, we have the sub-semifield
† †
Fk ={ak :a∈F}.
For example, when F is the supertropical semifield built from the ordered group (R,·), then F2 6= F,
†
since we only get the positive elements. However, when F is the supertropical semifield built from the
†
ordered group (R,+), or from (R+,·), then F2 =F.
Definition 3.9. A vector v ∈V is called normal if hv,vi=1 .
F
Remark 3.10. Suppose F2 =F. If hv,vi=a∈T, then h v , v i=1 , so v is normal. Thus, in this
√a √a F √a
case, any g-nonisotropic vector has a scalar multiple that is normal.
The bilinear formB is supertropically alternate if eachvector is g-isotropic;i.e., hv,vi∈G0 for all
v ∈ V. B is supertropically symmetric if hv,wi+hw,vi ∈ G0 for all v,w ∈ V. A special case: B is
symmetric if hv,wi=hw,vi for all v,w∈V.
Lemma 3.11. If B is supertropically symmetric on the vector space Fv +Fv and v and v are both
1 2 1 2
g-isotropic, then B is supertropically alternate.
Proof. hγ1v1+γ2v2,γ1v1+γ2v2i=γ12hv1,v1i+γ1γ2(hv1,v2i+hv2,v1i)+γ22hv2,v2i∈G0. (cid:3)
Proposition 3.12. If B is supertropically symmetric on a vector space with an s-base of g-isotropic
vectors, then B is supertropically alternate.
Proof. Apply induction to the lemma. (cid:3)
We recall another way of verifying tropical dependence, in terms of bilinear forms.
Theorem 3.13 ([5,Theorem6.7]). If the vectors w ...,w ∈V span a nondegenerate subspace W of V
1 k
with |G(w1...,wk)|∈G0, then w1,...,wk are tropically dependent.
3.2. The radical with respect to a bilinear form.
e
Definition 3.14. The (left) orthogonal ghost complement of S ⊆V is defined as
S⊥ :={v∈V :hv,Si∈G0}.
The radical, rad(V), with respect to a given bilinear form B, is defined as V . Vectors w are radically
⊥ i
dependent if iαiwi ∈rad(V) for suitable αi ∈T0, not all 0F.
Clearly, H0P⊆rad(V).
8 Z.IZHAKIAN,M.KNEBUSCH,ANDL.ROWEN
Remark 3.15.
(i) rad(V)=H0 when V is nondegenerate, in which case radical dependence is the same as tropical
dependence.
(ii) Any ghost complement V of rad(V) is obviously left g-orthogonal to rad(V), and nondegenerate
′
since
rad(V′) ⊆ V′∩rad(V) ⊆H0.
This observation enables us to reduce many proofs to nondegenerate subspaces, especially when
the Gram-Schmidt procedure is applicable (described below in Remark 4.10).
Proposition 3.16. If
hv,wihw,vi |=hv,vihw,wi, (3.3)
gs
and the vector space Fv +Fw is nondegenerate, then v,w are tropically dependent on rad(Fv+Fw).
Conversely, if v,w are tropically dependent on rad(Fv +Fw) and hv,vi and hw,wi are tangible, then
(3.3) holds.
Proof. If hv,wihw,vi |=hv,vihw,wi, then
gs
hv,vi hv,wi
∈G0,
hw,vi hw,wi
(cid:12)(cid:18) (cid:19)(cid:12)
(cid:12) (cid:12)
so the vectors v and w are tropically d(cid:12)ependent by Theor(cid:12)em 3.13. Conversely, if hv,vi,hw,wi ∈ T and
(cid:12) (cid:12)
hv,vi hv,wi
∈G0, then necessarily hv,wihw,vi |=hv,vihw,wi. (cid:3)
hw,vi hw,wi
(cid:12)(cid:18) (cid:19)(cid:12) gs
(cid:12) (cid:12)
(cid:12) (cid:12)
(cid:12) (cid:12) 4. Cauchy-Schwartz spaces
For convenience, we assume throughout this section that that B is supertropically symmetric, i.e.,
hv,wi+hw,vi∈G0 for all v,w ∈V. This assumption is justified by the following result from [5]:
Theorem4.1([5,Theorem6.19]). Ifg-orthogonalityisasymmetricrelationforthesupertropicalbilinear
form B, then B is supertropically symmetric.
Then we have:
hv+w,v+wi =| hv,vi+hw,wi+(hv,wi+hw,vi) =| hv,vi+hw,wi. (4.1)
gs gs
4.1. Compatible vectors. Since we cannot subtract vectors, we introduce a notion that plays a key
role in the supertropical theory.
Remark 4.2. If
hv,wi+hw,vi≥ hv,vi+hw,wi,
ν
then v+w is g-isotropic. This is clear from the first =| relation in (4.1).
gs
To avoid such g-isotropic vectors, we formulate the following definition:
Definition 4.3. Vectors v and w are weakly compatible if
hv,vi+hw,wi≥ hv,wi+hw,vi; (4.2)
ν
weakly compatible vectors v and w are called compatible if either hv,vi ∼=ν hw,wi or we have strict
ν-inequality in (4.2).
Example 4.4. If v+w is g-nonisotropic, then v and w are compatible. Indeed,
hv+w,v+wi =| hv,vi+hw,wi+(hv,wi+hw,vi)
gs
is presumed tangible. But hv,wi+hw,vi is ghost, which means hv,wi+hw,vi< hv,vi+hw,wi.
ν
DUAL SPACES AND BILINEAR FORMS 9
Lemma 4.5. Compatible vectors satisfy hv+w,v+wi |= hv,vi+hw,wi, equality holding when B is
gs
strict.
Proof. hv+w,v+wi =| hv,vi+hw,wi+hv,wi+hw,vi |=hv,vi+hw,wi. To prove equality when B is
gs gs
strict, note that this is clear unless hv,vi∼=ν hw,wi, in which case both sides are hv,viν. (cid:3)
4.2. Cauchy-Schwartz spaces. We are ready for the main kind of vector space.
Definition 4.6. A subset S ⊆ V is weakly Cauchy-Schwartz if every pair of elements of S satisfies
the condition
hv,vihw,wi≥ hv,wi2+hw,vi2. (4.3)
ν
S is Cauchy-Schwartz if strict ν-inequality holds in Equation (4.3). A space V is Cauchy-Schwartz
if it has a Cauchy-Schwartz s,d-base.
For example, the standard base of F(n) (cf. Example 1.3) with respect to the scalar bilinear form is
Cauchy-Schwartz.
Lemma 4.7. If {v,w} is Cauchy-Schwartz (resp. weakly Cauchy-Schwartz), then v and w are compatible
(resp. weakly compatible).
Proof. Clearly hv,vihw,wi≤ (hv,vi+hw,wi)2. Hence,
ν
hv,wi2 ≤ (hv,vi+hw,wi)2,
ν
implying hv,wi ≤ hv,vi+hw,wi. Analogously, hw,vi ≤ hv,vi+hw,wi. The same argument works for
ν ν
ν-inequality. (cid:3)
4.3. Bilinear forms on a space of rank 2. Muchofthetheoryreducestotherank2situation. We fix
some notation for this subsection. We say that {v ,v } is a corner singular pair if the Gram matrix
1 2
α αβ
of the bilinear form B with respect to v and v is ν-matched to a matrix of the form .
1 2 αβ αβ2
(cid:18) (cid:19)
Givenvectorsv1,v2 inV,letαij =hvi,vji,andputα=α12+α21 ∈G0. Bysymmetrywemayassume
that α11 ≤ν α22. Take α∈T such that α∼=ν α, and α22 ∈T such that α22 ∼=ν α22.
In contrast to the classical situation, any space V of rank ≥ 2 must have g-isotropic vectors, as seen
in the following computation.
b b b b
Example 4.8. Given β ∈T, we define w=v +βv and have
1 2
hw,wi = α +αβ+α β2;
11 22
hw,v i = α +α β;
2 12 22
hv ,wi = α +α β.
2 21 22
CASE I: α = 0 . Then hw,wi = αβ, hw,v i = α , and hv ,wi = α . Thus, v and w are
22 F 2 12 2 21 2
weakly compatible, and we have reduced to the next case.
CASE II: α 6=0 .
22 F
Take β large enough; i.e., β > α +1+ α11. (We discard the last summand when α=0 .)
ν αb22 αb F
Then
hw,wi=α β2; hw,v i=α β; hv ,wi=α β,
22 2 22 2 22
so {w,v } is a corner singular pair. In particular, when v is g-nonisotropic, replacing v by
2 2 1
w for large β gives us an independent pair of g- nonisotropic vectors, but at the cost of corner
singularity. Next, we look for g-isotropic vectors.
CASE II.a: α2 > α α . At any rate, taking β = α yields
ν 11 22 αb22
α2 ν
hw,wi=αβν = ; hw,v i=α=hv ,wi
2 2
α
22
α α
22
Thus, theGram matrix ofthebilinearbform B with respect tov2 andw is α α2 ν , soagain
αb22 !
we have corner singularity.
10 Z.IZHAKIAN,M.KNEBUSCH,ANDL.ROWEN
On the other hand, taking β = α11 yields
αb
α α α α
hw,wi=αν ; hw,v i=α + 11 22; hv ,wi=α + 11 22.
11 2 12 α 2 21 α
α γ
Thus, the Gram matrix of the bilinear form B with respect to v and w is 22 , with
b 2 b δ αν
γ+δ ∼=ν α. (cid:18) 11(cid:19)
The pair {v,w} is not corner singular, since α α < α2.
11 22 ν
For the situation α∈G0, we see that w is g-isotropic when αβ ≥ν α11 and αβ ≥ν α22β2, i.e.,
α α
11
≥ β ≥ .
ν ν
α α
22
We call this range of β the g-isotropic strip of the plane.
CASE II.b: α2 ≤ν α11α22. Then hw,wbi=α11+α22βb2. Thus, w is g-isotropic for β2 ∼=ν ααb2121.
Lemma 4.9. When F2 =F, any space of tangible rank at least two contains an g-isotropic vector.
Proof. By Example 4.8, since we have α∈G0. (cid:3)
4.4. The Gram-Schmidt procedure. We start with a standard sort of calculation.
Remark 4.10. (The Gram-Schmidt procedure) Suppose W ⊂ V is supertropically spanned by an
g-orthogonal set B ={b1,...,bm} for which each hbj,bji=6 0F. We take βj ∈T for which βj ∼=ν hbj,bji.
Then for each j =1,...,m, and for any v ∈V, the vector
m
hv,b i
j
v = b ∈W
j
B βj
j=1
X
satisfies
hv,b i2 hv,b ihb ,vi
j j j
hv,v i |= and hv ,vi |= . (4.4)
B gs j βj B gs j βj
X X
The vector v =v+v satisfies
′
B B
hv,b i2
hv′ ,bii |=hv,biiν + j hbj,bii∈G0. (4.5)
B gs βj
j=i
X6
Hence, v ⊥⊥b for each i, implying v ⊥⊥W. Furthermore,
′ i ′
B B
hv,b i(hv,b i+hb ,vi) hv,b ihv,b ihb ,b i
j j j j k j k
hv ,v i |=hv,vi+ + . (4.6)
′ ′
B B gs j βj j,k βj βk
X X
Equality holds in Equations (4.4), (4.5), and (4.6) when the bilinear form B is strict.
Lemma 4.11. Notation as in Remark 4.10, suppose that B is weakly Cauchy-Schwartz. Then
hv,b i(hv,b i+hb ,vi)
j j j
hv ,v i |=hv,vi+ ,
′ ′
B B gs βj
j
X
equality holding when the bilinear form B is strict.
Proof. The first assertion is clear unless hv ,v i is tangible, which means that there is one term in the
′ ′
right of (4.6) which dominates all others, aBndBagain we have the first assertion unless this term comes
from j,k hv,bjihβvj,bβkkihbj,bki.
Now note that v is g-orthogonalto each b , as observed above.
P ′ i
Also, by hypotheBsis,
β β ≥ hb ,b i2, (4.7)
j k ν j k
so multiplying both sides by hv,bjβi22hβv2,bki2 yields
j k
hv,b i2hv,b i2 hv,b i2hv,b i2hb ,b i2
j k j k j k
≥ ;
β β ν β2β2
j k j k