Non-Metzler spectrahedra

k̸=l Q^(k)_ii , Q^(l)_jj∈T+

(Q^(k)_ii ⊙Q^(l)_jj)⊙(Xk⊙Xl) )

⊕

( ⊕

Q^(k)_ii , Q^(k)_jj ∈T+

or|Q^(k)_ij |̸=−∞

αk⊙X_k^⊙2 )

,

whereα_k is given by:

α_k :=





Q^(k)_ii ⊙Q^(k)_jj ifQ^(k)_ii , Q^(k)_jj ∈T+andQ^(k)_ii +Q^(k)_jj >2|Q^(k)_ij |,

⊖(Q^(k)_ij )^⊙2 otherwise.

Recall that any inequality of the formmax(x, α+y)>max(x^′, β+y)is equivalent tomax(x, α+

y) > x^′ ifα > β, and to x >max(x^′, β+y) ifβ > α. Therefore, Assumption A ensures that P_ij⁺(x)> P_ij⁻(x) is equivalent to Q⁺_ii(x)⊙Q⁺_jj(x)>^⊕_k(Q^(k)_ij ⊙x_k)^⊙2. The same applies to the nonstrict counterparts of these inequalities. We conclude that (4.4) and (4.5) are satisfied.

Theorem 4.20. Let S ={x ∈ Kⁿ_⩾0:Q(x) ≽ 0} be a spectrahedral cone described by Metzler matrices Q⁽¹⁾, . . . , Q⁽ⁿ⁾ such that sval(Q^(k)) = Q^(k). Suppose that Assumption A is fulfilled and that every stratum of the setS(Q⁽¹⁾, . . . , Q⁽ⁿ⁾) is regular. Then

val(S) =S(Q⁽¹⁾, . . . , Q⁽ⁿ⁾).

Proof. It it clear that val(S) and S(Q⁽¹⁾, . . . , Q⁽ⁿ⁾) contain the point −∞. Fix a nonempty subset K ⊂ [n]. Observe that the stratum of val(S) associated with K is equal to val(S^(K)∩ K^K>0), whereS^(K)is the spectrahedral cone described by(Q^(k))k∈K. Similarly, the stratum ofS associated withKis equal toS^(K)∩R^K, whereS^(K)denotes the tropical Metzler spectrahedral cone described by(Q^(k))_k∈K. Therefore, we obtain the claim by applying Theorem 4.19 to every stratum.

4.2 Non-Metzler spectrahedra

In this section, we abandon the Metzler assumption that was imposed in the previous section.

LetQ⁽¹⁾, . . . , Q⁽ⁿ⁾∈T^m_±^×m be symmetric tropical matrices.

Definition 4.21. We introduce the setS(Q⁽¹⁾, . . . , Q⁽ⁿ⁾) (or simply S) of points x∈ Tⁿ that fulfill the following two conditions:

• for all i∈[m],Q⁺_ii(x)⩾Q⁻_ii(x);

• for alli, j∈[m],i < j, we haveQ⁺_ii(x)⊙Q⁺_jj(x)⩾(Q⁺_ij(x)⊕Q⁻_ij(x))^⊙2 orQ⁺_ij(x) =Q⁻_ij(x).

4.2. Non-Metzler spectrahedra 69 If n ⩾ 2, then we denote by S(Q⁽¹⁾|Q⁽²⁾, . . . , Q⁽ⁿ⁾) the set of all points x ∈ Tⁿ⁻¹ such that (0, x) ∈ S(Q⁽¹⁾, . . . , Q⁽ⁿ⁾). We point out that this generalizes Definition 4.13 to the case of non-Metzler matrices.

We point out that we donot claim that the setS defined above is a tropical spectrahedron.

In this work we only show that this is true under some additional assumptions (which are generically fulfilled as shown in Section 4.3). Let Q⁽¹⁾, . . . ,Q⁽ⁿ⁾ ∈ K^m×m be any symmetric matrices such that sval(Q^(k)) = Q^(k), and S := {x ∈ Kⁿ_⩾0:Q(x) ≽ 0} be the associated spectrahedral cone. The next lemma shows that the objects given in Definition 4.21 satisfy the basic inclusionval(S^out)⊂ S.

To continue, we need some notation. For every subset Σ⊂ {(i, j)∈[m]²:i < j}

Lemma 4.24. Every set SΣ,♢ is a tropical Metzler spectrahedral cone. More precisely, it is described by the block diagonal matricesQ⁽¹⁾_Σ,♢, . . . , Q⁽ⁿ⁾_Σ,♢ of the form

70 Chapter 4. Tropical spectrahedra where the intersection goes over every♢∈ {⩽,⩾}^Σ∁ and the union goes over everyΣ ⊂ {(i, j)∈ [m]²:i < j}. (As previously, if Σ^∁=∅, then the intersection contains one elementS_Σ,∅.) Proof. Obvious from Definitions 4.13 and 4.23.

In the sequel, we will use the following observation, which already appeared in the proof of [ABGJ15, Corollary 3.6] on the tropicalization of polyhedra. We denote byconv(X)the convex hull of the set X⊂Kⁿ.

Lemma 4.25 ([ABGJ15]). Let a⁽¹⁾, . . . ,a^(p) ∈ Kⁿ and b ∈ K^p. Suppose that for every sign pattern δ ∈ {+1,−1}^p there is a point x^δ∈Kⁿ such that for all s∈[p] we have δs(⟨a^(s),x^δ⟩ − b_s)⩾0. Then, there exists a pointy∈conv_δ{x^δ} such that for alls we have ⟨a^(s),y⟩=b_s. Proof. If p = 1, then we have two points x⁽¹⁾ and x⁽²⁾ such that ⟨a⁽¹⁾,x⁽¹⁾⟩ ⩾ b₁ and

⟨a⁽¹⁾,x⁽²⁾⟩⩽b1. Therefore, there existsλsuch that0⩽λ⩽1and⟨a⁽¹⁾,λx⁽¹⁾+ (1−λ)x⁽²⁾⟩= b₁. This completes the proof forp= 1.

Suppose that the claim is true for p. We will prove it for p+ 1. Take

∆⁺:={δ∈ {+1,−1}^p+1:last entry of δ is equal to +1} and

∆⁻:={δ ∈ {+1,−1}^p+1:last entry ofδ is equal to −1}.

By the induction hypothesis, there exists a pointx⁽¹⁾∈conv_δ∈∆⁺{x^δ}such that⟨a^(s),x⁽¹⁾⟩=b_s for alls⩽p. Moreover, we have⟨a^(p+1),x^δ⟩⩾bp+1for allδ∈∆⁺and therefore⟨a^(p+1),x⁽¹⁾⟩⩾ bp+1. Analogously, there exists a point x⁽²⁾ ∈conv_δ∈∆−{x^δ} such that ⟨a^(s),x⁽²⁾⟩=bs for all s⩽p and ⟨a^(p+1),x⁽²⁾⟩ ⩽b_p+1. Therefore, there is a point y ∈conv{x⁽¹⁾,x⁽²⁾} ⊂ conv_δ{x^δ} such that⟨a^(p+1),y⟩=b_p+1. Furthermore, since⟨a^(s),x⁽¹⁾⟩=⟨a^(s),x⁽²⁾⟩=b_sfor all s⩽p, we have ⟨a^(s),y⟩=bs for all s⩽p.

In Lemma 4.17 we showed that the set T(Q⁽¹⁾, . . . , Q⁽ⁿ⁾) is included in a tropical spec-trahedron val(S) if the matrices Q⁽¹⁾, . . . ,Q⁽ⁿ⁾ are Metzler. The following result generalizes Lemma 4.17 to the non-Metzler case.

Lemma 4.26. For every {Σ,♢}, we denote TΣ,♢ := T(Q⁽¹⁾_Σ,♢, . . . , Q⁽ⁿ⁾_Σ,♢) ⊂ Tⁿ, where the ma-trices Q^(k)_Σ,♢ are as in (4.6). Then, we have the inclusion

∪

Σ

∩

♢

cl_T⁽TΣ,♢)

⊂val(Sⁱⁿ).

Proof. Fix any Σ and take x ∈ ^∩_♢cl_T⁽TΣ,♢)

. By Lemma 4.17, for every ♢ ∈ {⩽,⩾}^Σ∁ there exists a lift x^♢∈Kⁿ_⩾0∩val⁻¹(x)such that we have the inequalities

∀i,Qii(x^♢)⩾0,

∀(i, j)∈Σ,Qii(x^♢)Qjj(x^♢)⩾(m−1)²(Q⁺_ij(x^♢) +Q⁻_ij(x^♢))²,

∀(i, j)∈Σ^∁,Q_ij(x^♢)♢(i,j)0.

Furthermore, we have(Q⁺_ij(x^♢) +Q⁻_ij(x^♢))²⩾(Qij(x^♢))². Observe that the set {y∈Kⁿ_⩾0:∀i,Q_ii(y)⩾0∧ ∀(i, j)∈Σ,Q_ii(y)Q_jj(y)⩾(m−1)²(Q_ij(y))²}

4.2. Non-Metzler spectrahedra 71 is convex. Indeed, it is a spectrahedron defined by some block diagonal matrices with blocks of size at most2. Therefore, by Lemma 4.25, there exists a pointz ∈conv_♢{x^♢} such that generalization to all strata can be obtained analogously to the proof of Theorem 4.20. Let (Σ,♢) be fixed. Since the matrices Q⁽¹⁾_Σ,♢, . . . , Q⁽ⁿ⁾_Σ,♢ fulfill Assumption A by Lemma 4.27, the

Remark 4.29. We note that if the hypotheses of Theorem 4.28 are fulfilled, then (by Lemma 4.7) the setS(Q⁽¹⁾|Q⁽²⁾, . . . , Q⁽ⁿ⁾)is a tropicalization of the spectrahedron S˜:={x∈Kⁿ_⩾⁻₀¹:Q⁽¹⁾+ this tropical spectrahedron is not regular for any choice ofa, b, c, d∈R.

72 Chapter 4. Tropical spectrahedra

x₁ x2

Figure 4.2: A nonregular tropical spectrahedron that fulfills the regularity conditions of Theo-rem 4.28.

Example4.31. In [Yu15], Yu characterized the image by the valuation of the positive semidefinite cone. Let us show how this result can be derived from Theorem 4.28. Let m^′ := m(m+ 1)/2 and let S ⊂K^m′×m′ be the cone of positive semidefinite matrices. We will compute the image by valuation ofS orthant-by-orthant. As in the proof of Lemma 3.11, for everyδ∈ {+1,−1}^m′, we denote by f_δ: K^m′ → K^m′ the involution which maps x ∈ K^m′ to the vector with entries δijxij. We want to characterize val(S ∩fδ(K^m_⩾0^′)). Note that we can restrict ourselves to the orthants such that δ_ii = 1 for all i ∈ [m] (because every point in S fulfills the inequalities x_ii ⩾ 0 for all i ∈ [m]). Moreover, we have val(S ∩f_δ(K^m_⩾0^′)) = val⁽f_δ(S∩f_δ(K^m_⩾0^′))⁾ and the set fδ(S ∩fδ(K^m_⩾0^′)) is included in K^m_⩾0^′. More precisely, the set Sδ := fδ(S ∩fδ(K^m_⩾0^′)) is a spectrahedral cone defined as the set of all points x∈K^m_⩾0^′×m′ such that the linear pencil

Qδ(x) :=









x11 δ12x12 . . . δ1mx1m

δ₁₂x₁₂ x₂₂ . . . δ_2mx_2m ... ... . .. ... δ_1mx_1m δ_2mx_2m . . . x_mm









is positive semidefinite. Let Q^(ij) ∈ K^m×m denote the matrix that has δij on the (i, j)th and (j, i)th positions and zeros otherwise (we allow i = j). Let Q^(ij) := sval(Q^(ij)). It is clear that the matrices (Q^(ij))_i⩽j satisfy Assumption A. Moreover, for every {Σ,♢}, the set SΣ,♢:=SΣ,♢(

(Q^(ij))i⩽j

)is given by

SΣ,♢={x∈T^m′×m′:∀(i, j)∈Σ, xii⊙xjj ⩾x^⊙_ij² ∧ ∀(i, j)∈Σ, x˜ ij =−∞},

where the set Σ˜ consists of the elements of Σ^∁ such that ♢ij is equal to ⩽ and δij = 1 or ♢ij

is equal to ⩾ and δ_ij =−1. This set has regular strata. Indeed, let K ⊂[m^′] be a nonempty set, and letSΣ,♢,K denote the stratum ofSΣ,♢ associated withK. IfSΣ,♢,K is nonempty, then Σ˜∩K=∅. Furthermore, for every(i, j)∈Σ∩K we have{(i, i),(j, j)} ⊂K. In particular, the set SΣ,♢,K is given by

SΣ,♢,K ={x∈R^K:∀(i, j)∈Σ∩K, x_ii⊙x_jj ⩾x^⊙_ij²}.

If we take any x ∈ SΣ,♢,K, then for every ε > 0, the point x^(ε) defined as x^(ε)_ii := x^(ε)_ii +ε for all (i, i) ∈ K and x^(ε)_ij =x_ij for all (i, j) ∈K, i < j, belongs to the interior of SΣ,♢,K. Hence

4.2. Non-Metzler spectrahedra 73 x ∈cl_R(int(SΣ,♢,K)). Therefore, by Theorem 4.28, we obtain the equality val(Sδ) =S, where the set S ⊂T^m′×m′ is given by

S={x∈T^m′×m′:∀i < j, xii⊙xjj ⩾x^⊙_ij²}.

This set does not depend onδ. Therefore, by taking the union over δ, we getval(S) =S. Example 4.32. In Definition 4.13 we introduced the class of tropical Metzler spectrahedral cones, and we have shown that these objects are, indeed, tropical spectrahedra. One can ask if the valuation of every spectrahedral cone is in fact a tropical Metzler spectrahedral cone.

This is, however, not true. The following example shows a spectrahedral cone that fulfills the conditions of Theorem 4.28 but whose image by valuation cannot be expressed as a tropical Metzler spectrahedral cone. Take the matrices

It is easy to check thatS satisfies the conditions of Theorem 4.28. One can also verify by hand thatval(S) =S. Suppose thatS is a tropical Metzler spectrahedral cone,S =S(Q⁽¹⁾, . . . , Q⁽⁴⁾) for some symmetric tropical Metzler matricesQ⁽¹⁾, . . . , Q⁽⁴⁾ ∈T^m×m. As noted in Remark 4.18, we may suppose that Q⁽¹⁾, . . . , Q⁽⁴⁾ satisfy Assumption A. Let u = (5,−3,−2,1) ∈ R⁴ and observe that the half-line{λu:λ∈R_⩾0} is included in the boundary ofS. Therefore, for every λ, at least one of the nontrivial inequalities that describeSbecomes an equality when evaluated atλu.

First, suppose thatQ⁺_ii(λ_su) =Q⁻_ii(λ_su)for someiand a sequenceλ_s>0,λ_s →0. Consider the function f: R → R defined as f(λ) := Q⁺_ii(λu)−Q⁻_ii(λu). The function f is continuous and piecewise affine. Therefore, f(λ) = 0 on some interval λ ∈ [0, ε]. We can suppose that ε is so small that the functions Q⁺_ii(λu), Q⁻_ii(λu) are affine on [0, ε]. In other words, we have Q⁺_ii(λu) =a+λu_k and Q⁻_ii(λu) =b+λu_l for somea, b∈Rand k̸=l. Since f(λ) is equal to 0 on [0, ε], we have a=b and u_k=u_l, which gives a contradiction with our choice of u.

Second, suppose that Q⁺_ii(λ_su)⊙Q⁺_jj(λ_su) = (Q_ij(λ_su))^⊙2 for some i, j and a sequence λs > 0, λs → 0. As previously, the function f(λ) := Q⁺_ii(λu)⊙Q⁺_jj(λu)−(Qij(λu))^⊙2 is continuous and piecewise affine, there isε >0such thatf(λ) = 0on[0, ε], and all the functions Q⁺_ii(λu), Q⁺_jj(λu), (Q_ij(λu))^⊙2 are affine on [0, ε], Q⁺_ii(λu) = a+λu_k, Q⁺_jj(λu) = b+λu_l, (Qij(λu))^⊙2 = 2c+ 2λup. Sincef(λ)is equal to0on[0, ε], we havea+b= 2canduk+ul= 2up. Thanks to our choice of vectoru, this is possible only ifk=l=por(k, l, p)∈ {(1,2,4),(2,1,4)}. The first case does not occur because the matricesQ⁽¹⁾, . . . , Q⁽⁴⁾ satisfy Assumption A (and we have a=Q^(k)_ii , b=Q^(l)_jj, c=|Q^(p)_ij |). Suppose that(k, l, p) = (1,2,4)(the case (k, l, p) = (2,1,4)

74 Chapter 4. Tropical spectrahedra Qij(ε^′v) ⩾ 2c. Therefore (Qij(ε^′v))^⊙2 ⩾ 2c > a+b−ε^′ ⩾ Q⁺_ii(ε^′v)⊙Q⁺_jj(ε^′v), which gives a contradiction with the fact that ε^′v∈ S.