Norms on Complex Matrices Induced by Complete

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We may use λ and λ1, λ2,..., λn if the matrix A is clear from context. Let diag(x1, x2,..., xn) ∈ Mn(C) denote the n × n diagonal matrix with diagonal en- tries x1, x2,..., xn, in that order. If x =(x1, x2,..., xn) is understood from context, we may write diag(x) for brevity.

1.2. Complete homogeneous symmetric polynomials. The complete homogeneous symmetric (CHS) polynomial of degree d in the n variables x1, x2,..., xn is

hd(x1, x2,..., xn) = ∑ xi1 xi2 ··· xid , (1) 1≤i1≤···≤id≤n the sum of all degree d monomials in x1, x2,..., xn [28, Sec. 7.5]. For example,

h0(x1, x2) = 1,

h1(x1, x2) = x1 + x2, 2 2 h2(x1, x2) = x1 + x1x2 + x2, and 3 2 2 3 h3(x1, x2) = x1 + x1x2 + x1x2 + x2. n+d−1 Elementary combinatorics conﬁrms that there are precisely ( d ) summands in Rn the deﬁnition (1). We often write hd(x), in which x =(x1, x2,..., xn) ∈ , when the number of variables is clear from context. Rn For d even and x ∈ , Hunter proved that hd(x) ≥ 0, with equality if and only if x = 0 [10]. This is not obvious because some of the summands that comprise hd(x) (for d even) may be negative. Hunter’s theorem has been reproved many times; see [2], [4, p. 69 & Thm. 3], [6, Cor. 17], [25, Thm. 2.3], and [29, Thm. 1].

1.3. Partitions and traces. A partition of d ∈ N is an r-tuple π =(π1, π2,..., πr) ∈ r N such that π1 ≥ π2 ≥ ··· ≥ πr and π1 + π2 + ··· + πr = d; the number of terms r depends on the partition π. We write π ⊢ d if π is a partition of d. For π ⊢ d, deﬁne

pπ(x1, x2,..., xn) = pπ1 pπ1 ··· pπr , k k k in which pk(x1, x2,..., xn) = x1 + x2 + ··· + xn. If the length of x =(x1, x2,..., xn) is clear from context, we often write pπ(x) and pk(x), respectively. Another expression for (1) is

pπ(x1, x2,..., xn) hd(x1, x2,..., xn) = ∑ , (2) π ⊢ d zπ in which the sum runs over all partitions π =(π1, π2,..., πr) of d and

mi zπ = ∏ i mi!, (3) i≥1 where mi is the multiplicity of i in π [28, Prop. 7.7.6]. For example, if π = 3 1 2 (4,4,2,1,1,1), then zπ =(1 3!)(2 1!)(4 2!) = 384 [28, (7.17)]. C If A ∈ Hn( ) has eigenvalues λ =(λ1, λ2,..., λn), then

π1 π2 πr pπ(λ) = pπ1 (λ)pπ2(λ) ··· pπr (λ)=(tr A )(tr A ) ··· (tr A ). (4) This connects eigenvalues, traces, and partitions to symmetric polynomials. NORMS ON COMPLEX MATRICES 3

1.4. Main results. The following theorem provides a family of novel norms on the space Hn(C) of n × n Hermitian matrices. Some special properties of these norms are discussed in Section 4.

Theorem 1. For even d ≥ 2, the following is a norm on Hn(C): 1/d |||A|||d = hd λ1(A), λ2(A),..., λn(A) . For example, equations (2) and (4) yield trace-polynomial representations 1 |||A|||2 = tr(A2) + (tr A)2 , 2 2 1 |||A|||4 = (tr A)4 + 6(tr A)2 tr(A2)+ 3(tr(A2))2 + 8(tr A) tr(A3)+ 6 tr(A4) . (5) 4 24 Theorem 1 is nontrivial for several reasons.

(a) The sums (1) and (2) that characterize hd(λ(A)) may contain negative summands. For example, (tr A) tr(A3) in (5) can be negative for Hermitian A. (b) The relationship between the spectra of (Hermitian) A, B, and A + B, conjectured by A. Horn in 1962 [9], was only established in 1998-9 by Klyachko [13] and Knutson–Tao [14]. Therefore, the triangle inequality is difﬁcult to establish. Even if A and B are diagonal, the result is not obvious; see (11). (c) The sums that deﬁne these norms do not involve the absolute values of the eigenvalues of A. Theorem 1 does not follow from standard considerations, but rather from delicate properties of multivariate symmetric polynomials. Example 2. Because CHS norms do not rely upon the absolute values of the eigenvalues of a Hermitian matrix (that is, its singular values), they can sometimes distinguish singularly cospectral graphs (graphs with the same singular values) that are not cospectral. This feature is not enjoyed by many standard norms (e.g., operator, Frobenius, Schatten–von Neumann, Ky Fan). For example, 0 1 1 K = 1 0 1 ,   1 1 0 which has eigenvalues 2, −1, −1, is the adjacency matrix for the complete graph on three vertices. The graphs with adjacency matrices K 0 0 K A = and B = 0 K K 0 are singularly cospectral but not cospectral: their eigenvalues are −1, −1, −1, −1, 6 6 2,2 and −2, −1, −1, 1, 1, 2, respectively. However, |||A|||6 = 120 6= 112 = |||B|||6. The norms of Theorem 1 extend in a natural and nontrivial fashion to the set Mn(C) of all n × n complex matrices.

Theorem 3. Let d ≥ 2 be even and let π =(π1, π2,..., πr) be a partition of d. Deﬁne C R d d Tπ :Mn( ) → by setting Tπ(A) to be 1/(d/2) times the sum over the (d/2) possible locations to place d/2 adjoints ∗ among the d copies of A in (tr AA ··· A)(tr AA ··· A) ··· (tr AA ··· A).

π1 π2 πr | {z } | {z } | {z } 4 K. AGUILAR, Á. CHÁVEZ, S.R. GARCIA, AND J. VOLCIˇ Cˇ

Then T (A) 1/d |||A||| = ∑ π , (6) d z π π in which the sum runs over all partitions π of d and zπ is deﬁned in (3), is a norm on Mn(C) that restricts to the norm on Hn(C) given by Theorem 1.

Example 4. The two partitions of d = 2 satisfy z(2) = 2 and z(1,1) = 2. There are 2 ∗ (1) = 2 ways to place two adjoints in a string of two As. Therefore, 1 T (A) = tr(A∗ A) + tr(AA∗) = tr(A∗ A), and (2) 2 1 T (A) = (tr A∗)(tr A)+(tr A)(tr A∗) =(tr A∗)(tr A), (1,1) 2 so 1 1 |||A|||2 = tr(A∗ A) + (tr A∗)(tr A). (7) 2 2 2

Example 5. The five partitions of d = 4 satisfy z(4) = 4, z(3,1) = 3, z(2,2) = 8, 4 ∗ z(2,1,1) = 4, and z(1,1,1,1) = 24. There are (2) = 6 ways to place two adjoints in a string of four As. For example, ∗ ∗ ∗ ∗ ∗ ∗ 6T(3,1)(A)=(tr A A A)(tr A)+(tr A AA )(tr A)+(tr A AA)(tr A ) +(tr AA∗ A∗)(tr A)+(tr AA∗ A)(tr A∗)+(tr AAA∗)(tr A∗) = 3 tr(A∗2A)(tr A) + 3(tr A2 A∗)(tr A∗). Some computation and (6) reveal that 1 |||A|||4 = (tr A)2 tr(A∗)2 + tr(A∗)2 tr(A2) + 4 tr(A) tr(A∗) tr(A∗ A) 4 24 +2 tr(A∗ A)2 +(tr A)2 tr(A∗2) + tr(A2) tr(A∗2) + 4 tr(A∗) tr(A∗ A2) + 4 tr(A) tr(A∗2A) + 2 tr(A∗ AA∗ A) + 4 tr(A∗2A2) . (8) ∗ If A = A , this simplifies to the norm (5) on Hn(C), as expected. Because of their origins in terms of complete homogeneous symmetric polynomials, we sometimes refer to the norm ||| · |||d as the CHS norm of order d. The notation k · k is used, occasionally with subscripts, for other norms. In the Hermitian case, the norm |||· |||d can be directly extracted from the Taylor expansion of an explicit rational function (Theorem 19). The general situation is elegantly summarized in a determinantal formula. C d d d/2 d/2 Theorem 6. Let A ∈ Mn( ). For d even, (d/2)|||A|||d is the coefficient of z z in the Taylor expansion of det(I − zA − zA∗)−1 about the origin. Helton and Vinnikov showed that polynomials of the form p = det(I − zA − zA∗) ∈ C[z, z] are precisely the real-zero polynomials in C[z, z] [8]. That is, they are characterized by the conditions p(0) = 1 and that x 7→ p(αx) has only real zeros for every α ∈ C. Properties of such polynomials are studied within the framework of hyperbolic [23] and stable [32] polynomials. This paper is structured as follows. Sections 2 and 3 contain the proofs of Theorems 1 and 3, respectively. Section 4 surveys the remarkable properties of the CHS norms, such as Theorem 6. We pose several open questions in Section 5. NORMS ON COMPLEX MATRICES 5

2. Proof of Theorem 1

Let d ≥ 2 be even. We prove that H : Hn(C) → R defined by 1/d H(A) = hd λ1(A), λ2(A),..., λn(A) (9) is a norm. Hunter’s theorem ensures that H(A) ≥ 0 and, moreover, that H(A) = 0 if and only if A = 0 (in fact, the nonnegativity of H follows from (10) below). Since H(cA) = |c|H(A) for all A ∈ Hn(C) and c ∈ R, it suffices to prove that H satisfies the triangle inequality. This is accomplished by combining Lewis’ framework for group invariance in convex matrix analysis [16] with a probabilistic approach to the complete homogeneous symmetric polynomials [1, 25, 29].

2.1. Group invariance. Let V be a finite-dimensional R-inner product space. The adjoint ϕ∗ of a linear map ϕ : V →V satisfies hϕ∗(X), Yi = hX, ϕ(Y)i for all X, Y ∈V. We say that ϕ is orthogonal if ϕ∗ ◦ ϕ is the identity map on V. Let O(V) denote the set of all orthogonal linear maps on V. For a subgroup G ⊆ O(V), we say that f : V → R is G-invariant if f (ϕ(X)) = f (X) for all ϕ ∈ G and X ∈V. Definition 7 (Def. 2.1 of [16]). δ : V→V is a G-invariant normal form if (a) δ is G-invariant, (b) For each X ∈V, there is an ϕ ∈ O(V) such that X = ϕ δ(X) , and (c) hX, Yi≤hδ(X), δ(Y)i for all X, Y ∈V. In this case, (V, G, δ) is a normal decomposition system. Suppose that (V, G, δ) is a normal decomposition system and W⊆V is a subspace. The stabilizer of W in G is GW = {ϕ ∈ G: ϕ(W) = W}. For convenience, we restrict the domain of each ϕ ∈ GW and consider GW as a subset of O(W). Our interest in this material stems from the next result.

Lemma 8 (Thm. 4.3 of [16]). Let (V, G, δ) and (W, GW , δ|W ) be normal decomposition systems with ran δ ⊆ W. Then a G-invariant function f : V → R is convex if and only if its restriction to W is convex.

Let V = Hn(C) denote the R-inner product space of complex Hermitian matrices, endowed with the inner product hX, Yi = tr(XY), and let Un(C) denote the group of n × n unitary matrices; see Remark 10 for more details about this inner product. For each U ∈ Un(C), deﬁne a linear map ϕU : V →V by ∗ ϕU(X) = UXU . Observe that ϕU ◦ ϕV = ϕUV and hence

G = {ϕU : U ∈ Un(C)} ∗ is a group under composition. Since ϕU = ϕU∗ , we conclude that G is a subgroup of O(V). Moreover, the function (9) is G-invariant. Let W = Dn(R) denote the set of real diagonal matrices. Then W inherits an inner product from V and GW = {ϕP : P ∈ Pn}, in which Pn denotes the set of n × n permutation matrices. Deﬁne δ : V→V by

δ(X) = diag λ1(X), λ2(X),..., λn(X) , the n × n diagonal matrix with λ1(X), λ2(X),..., λn(X) on its diagonal. Observe that ran δ ⊆ W since the eigenvalues of a Hermitian matrix are real (in fact, ran δ = W). We maintain all of this notation below. 6 K. AGUILAR, Á. CHÁVEZ, S.R. GARCIA, AND J. VOLCIˇ Cˇ

Lemma 9. (V, G, δ) and (W, GW , δ|W ) are normal decomposition systems. Proof. We ﬁrst show that (V, G, δ) is a normal decomposition system. (a) Since eigenvalues are invariant under similarity, δ is G-invariant. (b) For X ∈ V, the ∗ spectral theorem provides a U ∈ Un(C) such that X = Uδ(X)U = ϕU(δ(A)). (c) For X, Y ∈V, note that tr XY ≤ tr δ(X)δ(Y) [15, Thm. 2.2]; see Remark 10. We now show that (W, GW , δ|W ) is a normal decomposition system. (a) δ|W ∗ is GW -invariant since δ(ϕP(X)) = δ(PXP ) = δ(X) for every X ∈ W and P ∈ Pn. ∗ (b) Let X ∈ W. Since X is diagonal there exists a P ∈ Pn such that X = Pδ(X)P = ϕP(δ(X)). (c) The diagonal elements of a diagonal matrix are its eigenvalues. Consequently, this property is inherited from V; see Remark 11.

2.2. CHS polynomials as expectations. Let ξ = (ξ1, ξ2,..., ξn) be a vector of independent standard exponential random variables [3, (20.10)], and let x = Rn E k (x1, x2,..., xn) ∈ . Since [ξi ] = k! for i = 1,2,..., n [3, Ex. 21.3], we deduce that d d E[hξ, xi ] = E[(ξ1x1 + ξ2x2 ··· + ξnxn) ] d! E k1 k2 kn k1 k2 kn = ∑ ξ1 ξ2 ··· ξn x1 x2 ··· xn k1! k2! ··· kn! "k1+k2+···+kn=d # d! E k1 k2 kn k1 k2 kn = ∑ ξ1 ξ2 ··· ξn x1 x2 ··· xn k1! k2! ··· kn! k1+k2+···+kn=d h i E[ξk1 ]E[ξk2 ] ··· E[ξkn ] 1 2 n k1 k2 kn = d! ∑ x1 x2 ··· xn k1! k2! ··· kn! k1+k2+···+kn=d

k1 k2 kn = d! ∑ x1 x2 ··· xn k1+k2+···+kn=d

2.3. Conclusion. Recall the definition (9) of the function H : Hn(C) → R. The inequality (11) ensures that the restriction of H to Dn(R) satisfies the triangle inequality. For A, B ∈ Dn(R) and t ∈ [0,1], note that H(tA +(1 − t)B) ≤ H(tA) + H((1 − t)B) = tH(A)+(1 − t)H(B) by (11) and homogeneity. Thus, H is a convex function on Dn(R). Since H is G-invariant, we conclude from Lemma 8 that H is convex on Hn(C). It satisfies the triangle inequality on Hn(C) since it is convex and homogeneous: 1 H H 1 1 1 H 1 H 2 (A + B) = ( 2 A + 2 B) ≤ 2 (A) + 2 (B). NORMS ON COMPLEX MATRICES 7

Consequently, H(A) is a norm on Hn(C). 2.4. Remarks. We collect here a few remarks about the proof of Theorem 1.

Remark 10. Consider the inner product hX, Yi = tr(XY) on Hn(C); it is the restriction of the Frobenius inner product to Hn(C). The inequality

tr(XY) ≤ tr δ(X)δ(Y) for X, Y ∈ Hn(C) (12) is due to von Neumann [31] and has been reproved many times; see de Sá [5], Lewis [15, Thm. 2.2], Marcus [17], Marshall [18], Mirsky [20, Thm. 1], Richter [24, Satz. 1], Rendl and Wolkowicz [22, Cor. 3.1], and Theobald [30]. Remark 11. For diagonal matrices, the inequality (12) is equivalent to a classical rearrangement result: hx, yi≤hx, yi, in which where x ∈ Rn has the components of x =(x1, x2,..., xn) in decreasing order [7, Thm. 368]. Remark 12. For even d, (10) impliese e the nonnegativitye of the CHS polynomials. This probabilistic approach appears in the comments on the blog entry [29], and in [27, Lem. 12], which cites [1]. There are many other proofs of the nonnegativity of the even-degree CHS polynomials. Of course, there is Hunter’s inductive proof [10]. Roven¸ta and Temereanc˘aused divided differences [25, Thm. 3.5]. Re- cently, Böttcher, Garcia, Omar and O’Neill [4] employed a spline-based approach suggested by Olshansky after Garcia, Omar, O’Neill, and Yih obtained it as a byproduct of investigations into numerical semigroups [6, Cor. 17]. Remark 13. We stress that our inequality (11) permits x, y ∈ Rn; that is, with no positivity assumptions. For p ∈ N, the similar inequality 1/p 1/p 1/p Rn hp(x + y) ≤ hp(x) + hp(y) for x, y ∈ ≥0 (13) has been rediscovered several times. According to McLeod [19, p. 211] and White- ley [33, p. 49], it was ﬁrst conjectured by A.C. Aitken. Priority must be given to Whiteley [33, eq. (5)], whose paper appeared in 1958. McLeod’s paper was re- ceived on March 16, 1959, although he was unaware of Whiteley’s proof: “To the best of my knowledge, no proof of [(13)] exists so far in the literature.” For more exotic inequalities along the lines of (13), see [27].

3. Proof of Theorem 3 The first step in the proof of Theorem 3 is a general complexification result. Let V be a complex vector space with a conjugate-linear involution v 7→ v∗. Let ∗ k · k be a norm on the real subspace VR = {v ∈ V : v = v } of ∗-fixed points. For each v ∈ V and t ∈ R, we have eitv + e−itv∗ ∈ VR. Note that the path t 7→ keitv + e−itv∗k is continuous for each v ∈V. Proposition 14. For even d ≥ 2, the following is a norm on V that extends k · k:

2π 1/d 1 it −it ∗ d Nd(v) = ke v + e v k dt . (14) d 0 2π(d/2) Z it −it ∗ Proof. If v ∈VR, then ke v + e v k = |2 cos t|kvk. Moreover, Nd(v) = kvk since 2π d |2 cos t|d dt = 2π . d/2 Z0 8 K. AGUILAR, Á. CHÁVEZ, S.R. GARCIA, AND J. VOLCIˇ Cˇ

Next we verify that Nd is a norm on V.

Positive definiteness. The nonnegativity of k · k on VR and (14) ensure that Nd 1 ∗ 1 ∗ is nonnegative on V. If v ∈ V\{0}, then v = 2 (v + v ) + i 2 (−iv + iv ), in which v + v∗ and −iv + iv∗ belong to VR. Thus, kv + v∗k 6= 0 or k − iv + iv∗k 6= 0. Since it −it ∗ t 7→ ke v + e v k is continuous and nonnegative, it follows that Nd(v) 6= 0. R iθ iθ Absolute homogeneity. For r > 0 and θ ∈ , we have Nd((re )v) = rNd(e v) = R rNd(v) by the -homogeneity of k · k and the periodicity of the integrand in (14). Triangle inequality. For u, v ∈V, 2π 1/d keit(u + v) + e−it(u + v)∗kd dt Z0 1/d 2π d ≤ keitu + e−itu∗k + keitv + e−itv∗k dt Z0 2π 1/d 2π 1/d ≤ keitu + e−itu∗kd dt + keitv + e−itv∗kd dt , Z0 Z0 where the ﬁrst inequality holds by monotonicity of power functions and the triangle inequality for k · k, and the second inequality holds by the triangle inequality for the Ld norm on the space C[0,2π]. There are several natural complexiﬁcations of a real Banach space [21]. How- ever, the extensions Nd preserve some of the analytic and algebraic properties of the original norm. For example, if the extension Nd is applied to the norm ||| · |||d on Hn(C), one obtains a norm on Mn(C) that is given by a trace polynomial. Let hx, x∗i be the free monoid generated by x and x∗. Let |w| denote the length ∗ of a word w ∈hx, x i and let |w|x count the occurrences of x in w. For A ∈ Mn(C), ∗ 2 let w(A) ∈ Mn(C) be the natural evaluation of w at A. For example, if w = xx x , ∗ 2 then |w| = 4, |w|x = 3, and w(A) = AA A .

Lemma 15. Let d ≥ 2 be even and let π = (π1,..., πr) be a partition of d. If A ∈ Mn(C), then 1 2π tr(eit A + e−it A∗)π1 ··· tr(eit A + e−it A∗)πr dt 2π Z0 = ∑ tr w1(A) ··· tr wr(A) . (15) ∗ w1,...,wr∈hx,x i : |wj|=πj ∀j d |w1···wr|x= 2 −1 Proof. For every Laurent polynomial f ∈ C[z, z ] with the constant term f0 we 2π it have 0 f (e ) dt = 2π f0. Let us view R f = tr(zA + z−1 A∗)π1 ··· tr(zA + z−1 A∗)πr as a Laurent polynomial in z. Its constant term is

f0 = ∑ tr w1(A) ··· tr wr(A) w1,...,wr ∗ where the sum runs over all words w1, w2,..., wr in hx, x i with |wj| = πj such that the number of occurrences of x in w1w2 ··· wr equals the number of occur- ∗ rences of x in w1w2 ··· wr. Thus, (15) follows. NORMS ON COMPLEX MATRICES 9

C Given a partition π =(π1,..., πr) of d and A ∈ Mn( ) let 1 ( ) = ( ) ··· ( ) Tπ A d ∑ tr w1 A tr wr A . ( ) ∗ d/2 w1,...,wr∈hx,x i : |wj|=πj ∀j d |w1···wr|x= 2 We now complete the proof of Theorem 3. The conjugate transpose A 7→ A∗ is a real structure on Mn(C). The corresponding real subspace of ∗-fixed points C C is Hn( ). We apply Proposition 14 to the norm ||| · |||d on Hn( ) and obtain its C extension Nd(·) to Mn( ) defined by (14). The fact that Nd(A) admits a trace- polynomial expression as in (6) follows from (2) and Lemma 15. Remark 16. Proving that (6) is a norm relies crucially on Theorem 1, which states that its restriction to Hn(C) is a norm. On the other hand, demonstrating that (6) is a norm in a direct manner seems arduous. To a certain degree, this mirrors the current absence of general certificates for dimension-independent positivity of trace polynomials in x, x∗ (see [12] for the analysis in a dimension-fixed setting).

it −it ∗ Remark 17. For any A ∈ Mn(C) and t ∈ [0,2π], the matrices e A + e A are it −it ∗ Hermitian. Thus, |||e A + e A |||d can be deﬁned as in Theorem 1 and hence

2π 1/d 1 it −it ∗ d |||A|||d = |||e A + e A |||d dt . (16) d 0 2π(d/2) Z

Remark 18. Here is another way to restrict ||| · |||d to the Hermitian matrices. The d d d/2 d/2 proof of Lemma 15 shows that (d/2)|||A|||d is the coefﬁcient of z z¯ in ∗ d C |||zA + zA |||d ∈ [z, z].

4. Properties of CHS norms We now establish several properties of the CHS norms. First, we show how the CHS norm of a Hermitian matrix can be computed rapidly and exactly from its characteristic polynomial and recursion (Subsection 4.1). This leads quickly to the determinantal interpretation presented in the introduction (Subsection 4.2). Next, we identify those CHS norms induced by inner products (Subsection 4.3). In Sub- section 4.4, we use Schur convexity to provide a lower bound on the CHS norms in terms of the trace seminorm on Mn(C). We discuss monotonicity properties in Subsection 4.5 and symmetric tensor powers in Subsection 4.6.

4.1. Exact computation via characteristic polynomial. The CHS norm of a Her- mitian matrix can be exactly computed from its characteristic polynomial. The following theorem involves only formal series manipulations. C Theorem 19. Let pA(x) denote the characteristic polynomial of A ∈ Hn( ). For d ≥ 2 d even, |||A|||d is the dth coefﬁcient in the Taylor expansion of 1 1 = n det(I − xA) x pA(1/x) about the origin. 10 K. AGUILAR, Á. CHÁVEZ, S.R. GARCIA, AND J. VOLCIˇ Cˇ

Proof. Let pA(x)=(x − λ1)(x − λ2) ··· (x − λn). For |x| small, [29, (1)] provides ∞ n n d 1 1 1 1 ∑ hd(λ1, λ2,..., λn)x = ∏ = n ∏ −1 = n ; d=0 k=1 1 − λkx x k=1 x − λk x pA(1/x) the apparent singularity at the origin is removable. Now observe that n 1 1 1 ∏ = = k=1 1 − λkx det diag(1 − λ1x,1 − λ2x,...,1 − λnx) det(I − xA) by the spectral theorem.

1 1 Example 20. Let A = . Then p (z) = x2 − x − 1 and 1 0 A ∞ 1 1 n 2 = 2 = ∑ fn+1x , x pA(1/x) 1 − x − x n=0 in which fn is the nth Fibonacci number; these are defined by fn+2 = fn+1 + fn d and f0 = 0 and f1 = 1. Thus, |||A|||d = fd for even d ≥ 2. C Remark 21. If A ∈ Hn( ) is fixed, the sequence hd(λ1, λ2,..., λn) satisfies a constant-coefficient recurrence of order n since its generating function is a rational function whose denominator has degree n. Solving such a recurrence is elementary, so one can compute kAkd for d = 2, 4, 6, . . . via this method.

Remark 22. For small d, there is a simpler method. Since pA(x) is monic, it n follows that pA(x) = x pA(1/x) has constant term 1. For small x, we have ∞ ∞ d 1 1 d ∑ hd(λf1, λ2,..., λn)x = = = ∑ (1 − pA(x)) d=0 pA(x) 1 − (1 − pA(x)) d=0 so the desired h (λ , λ ,..., λ ) can be computed by the expanding thef geometric d 1 2 n f f series to the appropriate degree. Remark 23. For d ≥ 1, the Newton–Gerard identities imply

1 d hd(x1, x2,..., xn) = ∑ hd−i(x1, x2,..., xn)pi(x1, x2,..., xn); d i=1 see [26, §10.12]. For A ∈ Hn(C) and d ≥ 2 even, it follows that d 1 i hd(λ(A)) = ∑ hd−i(λ(A)) tr(A ), d i=1 d which can be used to compute |||A|||d = hd(λ(A)) recursively.

Remark 24. If H, K ∈ Hn(C), then det(I − xH) = det(I − xK) if and only if they are unitarily similar. However, H = diag(1,0) and K = diag(1, −1) give ∞ ∞ 1 1 j 1 1 2k = = ∑ z and = 2 = ∑ z , det(I − xH) 1 − x j=0 det(I − xK) 1 − x k=0 so |||H|||d = |||K|||d for even d ≥ 2. Of course, the odd-indexed coefﬁcients (the complete homogeneous symmetric polynomials of odd degree) do not agree. NORMS ON COMPLEX MATRICES 11

4.2. Determinantal interpretation. The material of the previous subsection leads to the determinantal interpretation (Theorem 6) stated in the introduction. We restate (and prove) the result here for convenience: C d d d/2 d/2 Theorem 25. Let A ∈ Mn( ). For d even, (d/2)|||A|||d is the coefficient of z z in the Taylor expansion of det(I − zA − zA∗)−1 about the origin. C d −1 d Proof. If H ∈ Hn( ), the coefficient of x in det(I − xH) is |||H|||d by Theorem 19. By plugging in H = zA + zA∗ and treating the resulting expression as a series d/2 d/2 d d in z and z, Remark 18 implies that the coefficient of z z equals (d/2)|||A|||d. 0 1 Example 26. Let A = . Then 0 0 1 ∞ det(I − zA − zA∗)−1 = = ∑ znzn, 1 − zz n=0 d d −1 and hence kAkd = (d/2) for even d ≥ 2. Example 27. For 0 1 0 1 A = 0 0 1 , we have det(I − zA − zA∗)−1 = .   3 3 1 0 0 1 − z − 3zz − z   2 4 3 6 29 8 99 Computer algebra reveals that |||A|||2 = |||A|||4 = 2 , |||A|||6 = 20 , and |||A|||8 = 70 . Example 28. The matrices 00 0 0 0 1 A = 0 1 i and B = 0 0 i     0 i −1 1 i 0 satisfy     1 det(I − zA − zA∗)−1 = = det(I − zB − zB∗)−1, 1 − 4zz so |||A|||d = |||B|||d for even d ≥ 2. These matrices are not similar (let alone unitarily similar) since A is nilpotent of order two and B is nilpotent of order three. ∆ ∂2 Remark 29. In terms of the Laplace operator = ∂z ∂z, Theorem 6 states that for even d, 1 d!|||A|||d = ∆d/2 (0) . d det(I − zA − zA∗) C 4.3. Inner products. Theorem 3 says that ||| · |||d is a norm on Mn( ) for even d ≥ 2. It is natural to ask when these norms are induced by an inner product. C C Theorem 30. The norm ||| · |||d on Mn( ) (and its restriction to Hn( )) is induced by an inner product if and only if d = 2 or n = 1

Proof. If n = 1 and d ≥ 2 is even, then |||A|||d is a ﬁxed positive multiple of |a| for C C each A = [a] ∈ M1( ). Thus, ||| · |||d on M1( ) is induced by a positive multiple of the inner product hA, Bi = ba, in which A =[a] and B =[b]. 2 1 ∗ 1 ∗ If d = 2 and n ≥ 1, then |||A|||2 = 2 tr(A A) + 2 (tr A) tr(A ), which is induced 1 ∗ 1 ∗ C by the inner product hA, Bi = 2 tr(B A) + 2 (tr B )(tr A) on Mn( ). 12 K. AGUILAR, Á. CHÁVEZ, S.R. GARCIA, AND J. VOLCIˇ Cˇ

1/d It suffices to show that in all other cases the norm |||A|||d = (hd(λ(A))) on Hn(R) does not arise from an inner product. For n ≥ 2, let A = diag(1,0,0,...) R 2 2 and B = diag(0,1,0,...,0) ∈ Hn( ). Then |||A|||d = |||B|||d = 1. Next observe that 2 2/d |||A + B|||d = (d + 1) since there are exactly d + 1 nonzero summands, each equal to 1, in the evaluation of hd(λ(A + B)). Because of cancellation, a similar 2 argument shows that |||A − B|||d = 1. A result of Jordan and von Neumann says that a vector space norm k · k arises from an inner product if and only if it satisfies the parallelogram identity kx + yk2 + kx − yk2 = 2 kxk2 + kyk2) for all x, y [11]. 2/d If ||| · |||d satisfies the parallelogram identity, then (d + 1) + 1 = 2(1 + 1); that is, (d + 1)2 = 3d. The solutions are d = 0 (which does not yield an inner product) and d = 2 (which, as we showed above, does). Thus, for n ≥ 2 and d ≥ 2, the C norm ||| · |||d on Hn( ) does not arise from an inner product.

4.4. A tracial lower bound. Each CHS norm on Mn(C) is bounded below by an explicit positive multiple of the trace seminorm. That is, the CHS norms of a matrix can be related to its mean eigenvalue.

Theorem 31. For A ∈ Mn(C) and d ≥ 2 even, n + d − 1 1/d | tr A| |||A||| ≥ d d n with equality if and only if A is a multiple of the identity. Rn Proof. Let d ≥ 2 be even. For x = (x1, x2,..., xn) ∈ , let x = (x1, x2,..., xn) denote its decreasing rearrangement (the notation x↓ is frequently used in the literature). Then x majorizes y, denoted x y, if e e e f k k n n ∑ xi ≥ ∑ yi for k = 1,2,..., n, and ∑ xi = ∑ yi. i=1 i=1 i=1 i=1 The even-degreee completee homogeneous symmetric polynomials are Schur convex [29, Thm. 1]. That is, hd(x) ≥ hd(y) whenever x y, with equality if and only if x is a permutation of y. it −it ∗ Let A ∈ Mn(C) and deﬁne B(t) = e A + e A for t ∈ R. Then λ(B(t)) majorizes µ(t)=(µ(t), µ(t),..., µ(t)) ∈ Rn, in which µ(t) = tr B(t)/n. Thus, n + d − 1 |||B(t)|||d = h λ(B(t)) ≥ h µ(t) = µ(t)d d d d d with equality if and only if B(t) = µ(t)I. It follows from (16) that 1/d n+d−1 2π ( d ) d |||A|||d ≥ µ(t) dt . (17) d 0 2π(d/2) Z ! Combine this with d 2π 2π tr B(t) 1 2π d µ(t)d dt = dt = eit tr A + e−it tr(A∗) dt 0 0 n nd 0 Z Z Z 1 d d 2π = ∑ (tr A∗)d−k(tr A)k ei(2k−d)t dt nd k 0 k=0 Z NORMS ON COMPLEX MATRICES 13

2π d = | tr A|d nd d/2 and get the desired inequality. The continuity of the integrand ensures that equality occurs in (17) if and only if B(t) = µ(t)I for all t ∈ R. An operator-valued it −it ∗ int Fourier expansion reveals that e A + e A = (∑n∈Z µ(n)e )I, so A = µˆ(1)I. Conversely, equality holds in (17) if A is a multiple of the identity. b n+d−1 1/d Remark 32. For each ﬁxed n ≥ 1, the constant ( d ) in Theorem 31 tends to ∞ 1 C 1 from above as d → . Therefore, |||A|||d ≥ n | tr A| for all A ∈ Mn( ). 4.5. Monotonicity. The next result shows how CHS norms relate to each other. For Hermitian matrices, the ﬁrst inequality below is superior to the second. Theorem 33. Let 2 ≤ p < q be even. 1/p 1/q (a) If A ∈ Hn(C), then (p!) |||A|||p ≤ (q!) |||A|||q. C p 1/p q 1/q (b) If A ∈ Mn( ), then (p/2)p! |||A|||p ≤ (q/2)q! |||A|||q. C Proof. (a) Let A ∈ Hn( ) have eigenvalues λ =(λ1, λ2,..., λn), listed in decreasing order, and let ξ = (ξ1, ξ2,..., ξn) be a random vector, in which ξ1, ξ2,..., ξn are independent standard exponential random variables. Let d ≥ 2 be even and consider the random variable X = hξ, λi. Then (10) ensures that 1/d 1/d E d 1/d E d 1/d (d!) |||A|||d = d! hd(λ) = |hξ, λi| = [|X| ] = kXkLd .

Since we are in a probability space (in particular, a ﬁnite measure space), kXkLp ≤ kXkLq for 0 < p < q < ∞. For 2 ≤ p < q even, this yields the desired inequality.

(b) Let A ∈ Mn(C) and let 2 ≤ p < q be even. For t ∈ [0,2π], (a) ensures that p/q p (q!) p |||eit A + e−it A∗||| ≤ |||eit A + e−it A∗||| . p p! q it −it ∗ p Consider f (t) = |||e A + e A |||q as an element of L [0,2π]. Hölder’s inequality and (16) imply the desired inequality:

2π 1/p 1 it −it ∗ p |||A|||p = p |||e A + e A |||p dt 2π( ) 0 p/2 Z 1/p 1/p p/q 2π 1 (q!) it −it ∗ p ≤ p |||e A + e A |||q dt 2π( ) p! 0 p/2 Z ! 1/p 1/p ( )1/q ( )1/q 1 1 q! 1 q! 1 p − q ≤ k f kLp ≤ (2π) k f kLq (p!)1/p 2π( p ) (p!)1/p 2π( p ) p/2 p/2 1/p 1/q ( )1/q 1 1 2π q! 1 p − q it −it ∗ q ≤ p (2π) |||e A + e A |||q dt (p!)1/p 2π( ) 0 p/2 Z ! 1/p 1/q ( )1/q 1 q! 1 − q q q ≤ (2π) 2π |||A|||q (p!)1/p ( p ) q/2 p/2 14 K. AGUILAR, Á. CHÁVEZ, S.R. GARCIA, AND J. VOLCIˇ Cˇ

1/q ( q )q! q/2 q ≤ |||A|||q. p 1/p (p/2)p!

Remark 34. The previous result suggests that suitable constant multiples of the CHS norms may be preferable in some circumstances. However, the beneﬁts appear to be outweighed by the cumbersome nature of these constants.

Remark 35. For A, B ∈ Mn(C), 2 ∗ ∗ 2|||AB|||2 = tr(AB) tr((AB) ) + tr((AB) AB) ≤ 2 tr(A∗ A) tr(B∗B) ≤ 2 tr(A) tr(A∗) + tr(A∗ A) tr(B) tr(B∗) + tr(B∗B) 2 2 = 8||| A|||2|||B|||2, so 2|||· |||2 is submultiplicative. Actually, 2 is the smallest constant independent of n with this property, since 0 1 J = 0 0 ∗ ∗ satisﬁes |||JJ |||2 = 1 = 2|||J|||2|||J |||2. 4.6. Symmetric Tensor Powers. Let V denote an n-dimensional R-inner product k space with orthonormal basis v1, v2,..., vn. The kth tensor power of V is the n - dimensional R-inner product space V ⊗k spanned by the simple tensors

vi1 ⊗ vi2 ⊗···⊗ vik . (18) These simple tensors form an orthonormal basis of V ⊗k. An operator A : V→V lifts to an operator on V ⊗k as follows. Deﬁne ⊗k A (vi1 ⊗ vi2 ⊗···⊗ vik ) = Avi1 ⊗ Avi2 ⊗···⊗ Avik and use the linearity of A and ⊗ to write this in terms of the basis vectors (18). An important fact is that V ⊗k is basis-independent, in the sense of abstract nonsense. n+k−1 The kth symmetric tensor power of V is the ( k )-dimensional vector space ⊗k Symk V⊂V spanned by the symmetric tensors: 1 v ⊙ v ⊙···⊙ v = ∑ v ⊗ v ⊗···⊗ v , i1 i2 ik k! σ(i1) σ(i2) σ(ik) σ∈Sk where Sk denotes the symmetric group on k letters.

Proposition 36. If d ≥ 2 is even and A ∈ Hn(C), then

d Symd |||A|||d = tr(A ).

Proof. Let A : V →V be selfadjoint with eigenvalues λ1, λ2,..., λn and cor- Symk responding orthonormal eigenbasis v1, v2,..., vn. Deﬁne A : Symk V → Symk Symk V by linear extension. Then vi1 ⊙ vi2 ⊙···⊙ vik is an eigenvector of A n+k−1 with eigenvalue λi1 λi2 ··· λik . Sum over these ( k ) eigenvectors and conclude Sym that tr(A k ) = hk(λ1, λ2,..., λn). NORMS ON COMPLEX MATRICES 15

If A is the adjacency matrix of a graph Γ, then |||A|||d concerns the dth symmetric tensor power of Γ, a weighted graph obtained from Γ in a straightforward (but tedious) manner by computing the matrix representation of ASymd with respect to the normalization of the orthogonal basis of symmetrized tensors.

4.7. Equivalence constants. Any two norms on a ﬁnite-dimensional vector space C are equivalent. Thus, each norm |||· |||d on Hn( ) (with d ≥ 2 even) is equivalent to the operator norm k · kop. We compute admissible equivalence constants below.

Theorem 37. For A ∈ Hn(C) and even d ≥ 2,

1 1/d n + d − 1 1/d d kAkop ≤ |||A|||d ≤ kAkop 2 d d 2 ( 2 )! The upper inequality is sharp if and only if A is a multiple of the identity.

Proof. For A ∈ Hn(C) and even d ≥ 2, the triangle inequality yields d |||A|||d = hd(λ1(A), λ2(A),..., λn(A))

= hd(λ1(A), λ2(A),..., λn(A)) ≤ h d(|λ1(A)|, |λ2(A)|,..., |λn(A )|)

≤ hd(kAkop, kAkop,..., kAkop) d = kAkophd(1,1,...,1) n + d − 1 = kAkd . op d Equality occurs if and only if λi(A) = |λi(A)| = kAkop for 1 ≤ i ≤ n; that is, if and only if A is a multiple of the identity. Hunter [10] established that 1 h (x) ≥ kxk2p, 2p 2p p! in which kxk denotes the Euclidean norm of x ∈ Rn. Let d = 2p and conclude

1 1/d 1 1/d |||A|||d ≥ d kAkF ≥ d kAkop, 2 d 2 d 2 ( 2 )! 2 ( 2 )! in which kAkF denotes the Frobenius norm of A ∈ Hn(C).

Remark 38. For A ∈ Mn(C), we may apply the upper bound in Theorem 37 to eit A + e−it A∗ and use (14) to deduce that 1/d 1/d n+d−1 2π n+d−1 ( d ) it −it ∗ d ( d ) |||A|||d ≤ ke A + e A kop dt ≤ 2 kAkop. d 0 d 2π(d/2) Z ! (d/2) ! Remark 39. Hunter’s lower bound was improved by Baston [2], who proved that

n p n 2p 1 2 h (x) ≥ ∑ x + λp ∑ x 2p 2p p! i i i=1 i=1 16 K. AGUILAR, Á. CHÁVEZ, S.R. GARCIA, AND J. VOLCIˇ Cˇ

Rn for x =(x1, x2,..., xn) ∈ , where 1 n + 2p − 1 1 1 λ = − > 0. p np 2p np 2p p! Equality holds if and only if p = 1 or p ≥ 2 and all the xi are equal. However, Baston’s result does not appear to yield a signiﬁcant improvement in the lower bound of Theorem 37.

5. Open Questions The answers to the following questions have eluded us.

Problem 1. What are the best constants cd, independent of n, such that cdk · kd is submultiplicative? Do such constants exists? See Remark 35. Problem 2. What is the best complexified version of Theorem 37? Can the upper bound it −it ∗ be improved (the estimate ke A + e A kop ≤ 2kAkop seems wasteful on average)? Can we get a sharp lower bound? d Problem 3. If one uses (6) to evaluate |||A|||d, there are many repeated terms. For example, (tr A∗ A)(tr A)(tr A∗)=(tr AA∗)(tr A∗)(tr A) because of the cyclic invariance of the trace and the commutativity of multiplication. If one chooses a single representative for each such class of expressions and simplifies, one gets expressions such as (7) and (8). Is there a combinatorial interpretation of the resulting coefficients? For motivation, the reader is invited to consider 1 |||A|||6 = (tr A)3 tr(A∗)3 + 3 tr(A) tr(A∗)3 tr(A2) 6 720 + 9(tr A)2 tr(A∗)2 tr(A∗ A)+ 9 tr(A∗)2 tr(A2) tr(A∗ A) + 18 tr(A) tr(A∗) tr(A∗ A)2 + 6 tr(A∗ A)3 + 3(tr A)3 tr(A∗) tr(A∗2) + 9 tr(A) tr(A∗) tr(A2) tr(A∗2)+ 9(tr A)2 tr(A∗ A) tr(A∗2) + 9 tr(A2) tr(A∗ A) tr(A∗2)+ 2 tr(A∗)3 tr(A3) + 6 tr(A∗) tr(A∗2) tr(A3)+ 18 tr(A) tr(A∗)2 tr(A∗ A2) + 36 tr(A∗) tr(A∗ A) tr(A∗ A2)+ 18 tr(A) tr(A∗2) tr(A∗ A2) + 18(tr A)2 tr(A∗) tr(A∗2 A)+ 18 tr(A∗) tr(A2) tr(A∗2 A) + 36 tr(A) tr(A∗ A) tr(A∗2 A)+ 36 tr(A∗ A2) tr(A∗2 A) + 2(tr A)3 tr(A∗3)+ 6 tr(A) tr(A2) tr(A∗3) + 4 tr(A3) tr(A∗3)+ 18 tr(A∗)2 tr(A∗ A3) + 18 tr(A∗2) tr(A∗ A3)+ 18 tr(A) tr(A∗) tr(A∗ AA∗ A) + 18 tr(A∗ A) tr(A∗ AA∗ A)+ 36 tr(A) tr(A∗) tr(A∗2 A2) + 36 tr(A∗ A) tr(A∗2 A2)+ 18(tr A)2 tr(A∗3 A) + 18 tr(A2) tr(A∗3 A)+ 36 tr(A∗) tr(A∗ AA∗ A2) + 36 tr(A∗) tr(A∗2 A3)+ 36 tr(A) tr(A∗2 AA∗ A) + 36 tr(A) tr(A∗3 A2)+ 12 tr(A∗ AA∗ AA∗ A)+ 36 tr(A∗2 A2 A∗ A)

+ 36 tr(A∗2 AA∗ A2)+ 36 tr(A∗3 A3) . NORMS ON COMPLEX MATRICES 17

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Department of Mathematics, Pomona College, 610 N. College Ave., Claremont, CA 91711 Email address: [email protected] URL: https://aguilar.sites.pomona.edu/

Email address: [email protected]

Email address: [email protected] URL: http://pages.pomona.edu/~sg064747

Department of Mathematics, Texas A&M University, TAMU 3368, College Station, TX 77843-3368 Email address: [email protected]