Norms on complex matrices induced by random vectors II: extension of weakly unitarily invariant norms

Abstract

We improve and expand in two directions the theory of norms on complex matrices induced by random vectors. We first provide a simple proof of the classification of weakly unitarily invariant norms on the Hermitian matrices. We use this to extend the main theorem in Chávez, Garcia, and Hurley (2023, Canadian Mathematical Bulletin 66, 808–826) from exponent $d\geq 2$ to $d \geq 1$. Our proofs are much simpler than the originals: they do not require Lewis’ framework for group invariance in convex matrix analysis. This clarification puts the entire theory on simpler foundations while extending its range of applicability.

1 Introduction

A norm $\| \cdot \|$ on $\mathrm {M}_n$ , the space of $n\times n$ complex matrices, is unitarily invariant if $\| UAV \|=\| A \|$ for all $A\in \mathrm {M}_n$ and unitary $U,V \in \mathrm {M}_n$ . A norm on $\mathbb {R}^n$ which is invariant under entrywise sign changes and permutations is a symmetric gauge function. A theorem of von Neumann asserts that any unitarily invariant norm on $\mathrm {M}_n$ is a symmetric gauge function applied to the singular values [10, Theorem 7.4.7.2]. For example, the Schatten norms are unitarily invariant and defined for $d\geq 1$ by

$$ \begin{align*} ||A||_{S_d}=\big( |\sigma_1|^d+|\sigma_2|^d+\cdots+ |\sigma_n|^d\big)^{1/d}, \end{align*} $$

in which $\sigma _1 \geq \sigma _2 \geq \cdots \geq \sigma _n \geq 0$ are the singular values of $A\in \mathrm {M}_n$ .

A norm $\| \cdot \|$ on the $\mathbb {R}$ -vector space $\mathrm {H}_n$ of $n\times n$ complex Hermitian matrices is weakly unitarily invariant if $\| U^*AU \|=\| A \|$ for all $A\in \mathrm {H}_n$ and unitary $U \in \mathrm {M}_n$ . For example, the numerical radius

$$ \begin{align*} r(A) = \sup_{\mathbf{x} \in {\mathbb{C}}^{n} \backslash \{\mathbf{0}\}} \frac{ \langle A \mathbf{x}, \mathbf{x} \rangle}{\langle \mathbf{x} , \mathbf{x} \rangle}\end{align*} $$

is a weakly unitarily invariant norm on $\mathrm {H}_n$ [12]. Lewis proved that any weakly unitarily invariant norm on $\mathrm {H}_n$ is a symmetric vector norm applied to the eigenvalues [11, Section 8].

Our first result is a short proof of Lewis’ theorem that avoids his theory of group invariance in convex matrix analysis [11], the wonderful but complicated framework that underpins [1, 7]. Our new approach uses more standard techniques, such as Birkhoff’s theorem on doubly stochastic matrices [6].

Theorem 1.1 A norm $\| \cdot \|$ on $\mathrm {H}_n$ is weakly unitarily invariant if and only if there is a symmetric norm $f:\mathbb {R}^n\to \mathbb {R}$ such that $\| A \|=f( \lambda _1, \lambda _2, \ldots , \lambda _n)$ for all $A\in \mathrm {H}_n$ . Here, $\lambda _1 \geq \lambda _2 \geq \cdots \geq \lambda _n$ are the eigenvalues of A.

The random-vector norms of the next theorem are weakly unitarily invariant norms on $\mathrm {H}_n$ that extend to weakly unitarily invariant norms on $\mathrm {M}_n$ (see Theorem 1.3). They appeared in [7], and they generalize the complete homogeneous symmetric polynomial norms of [1, Theorem 1]. The original proof of [7, Theorem 1.1(a)] requires $d \geq 2$ and relies heavily on Lewis’ framework for group invariance in convex matrix analysis [11]. However, Theorem 1.2 now follows directly from Theorem 1.1. Moreover, Theorem 1.2 generalizes [7, Theorem 1.1(a)] to the case $d\geq 1$ .

Theorem 1.2 Let $d\geq 1$ be real and $\mathbf {X}$ be an independent and identically distributed (iid) random vector in $\mathbb {R}^n$ , that is, the entries of $\mathbf {X}=(X_1,X_2, \ldots , X_n)$ are nondegenerate iid random variables. Then

(1.1) $$ \begin{align} ||A||_{{\mathbf{X}},d} = \left(\frac{{\mathbb{E}} |\langle{\mathbf{X}}, \boldsymbol{\lambda}\rangle|^d}{\Gamma(d+1)}\right)^{1/d} \end{align} $$

is a weakly unitarily invariant norm on $\mathrm {H}_n$ . Here, $\Gamma (\cdot )$ denotes the gamma function and $\boldsymbol {\lambda }=(\lambda _1,\lambda _2, \ldots , \lambda _n)$ denotes the vector of eigenvalues $\lambda _1 \geq \lambda _2 \geq \cdots \geq \lambda _n$ of A. Moreover, if the entries of $\mathbf {X}$ each have at least m moments, then for all $A\in \mathrm {H}_n$ the function $f:[1,m] \to \mathbb {R}$ defined by $f(d) =\| A \|_{\mathbf {X},d}$ is continuous.

The simplified proof of Theorem 1.1 and the extension of Theorem 1.2 from $d\geq 2$ to $d \geq 1$ permit the main results of [7], restated below as Theorem 1.3, to rest on simpler foundations while enjoying a wider range of applicability. The many perspectives offered in Theorem 1.3 explain the normalization in (1.1).

Theorem 1.3 Let $\mathbf {X}=(X_1, X_2, \ldots , X_n)$ , in which $X_1, X_2, \ldots , X_n \in L^d(\Omega ,\mathcal {F},\mathbb {P})$ are nondegenerate iid random variables. Let $\boldsymbol {\lambda }=(\lambda _1,\lambda _2, \ldots , \lambda _n)$ denote the vector of eigenvalues $\lambda _1 \geq \lambda _2 \geq \cdots \geq \lambda _n$ of $A \in \mathrm {H}_n$ .

1. (1) For real $d\geq 1$ , $\| A \|_{\mathbf {X},d}= \bigg (\dfrac { \mathbb {E} |\langle \mathbf {X}, \boldsymbol {\lambda }\rangle |^d}{\Gamma (d+1)} \bigg )^{1/d}$ is a norm on $\mathrm {H}_n$ (now by Theorem 1.2).

2. (2) If the $X_i$ admit a moment generating function $M(t) = \mathbb {E} [e^{tX}] = \sum _{k=0}^{\infty } \mathbb {E} [X^k] \frac {t^k}{k!}$ and $d \geq 2$ is an even integer, then $\| A \|_{\mathbf {X},d}^d$ is the coefficient of $t^d$ in $M_{\Lambda }(t)$ for all $A \in \mathrm {H}_n$ , in which $M_{\Lambda }(t) = \prod _{i=1}^n M(\lambda _i t)$ is the moment generating function for the random variable $\Lambda =\langle \mathbf {X}, \boldsymbol {\lambda }(A) \rangle =\lambda _1X_1+\lambda _2X_2+\cdots +\lambda _n X_n$ . In particular, $\| A \|_{\mathbf {X},d}$ is a positive definite, homogeneous, symmetric polynomial in the eigenvalues of A.

3. (3) Let $d\geq 2$ be an even integer. If the first d moments of $X_i$ exist, then

   $$ \begin{align*} \| A \|_{\mathbf{X},d}^d = \frac{1}{d!} B_{d}(\kappa_1\operatorname{tr} A, \kappa_2\operatorname{tr} A^2, \ldots, \kappa_d\operatorname{tr} A^d) =\sum_{\boldsymbol{\pi}\vdash d}\frac{\kappa_{\boldsymbol{\pi}}p_{\boldsymbol{\pi}} (\boldsymbol{\lambda})}{y_{\boldsymbol{\pi}}} \quad \text{for }A \in \mathrm{H}_n, \end{align*} $$
   in which:

   1. (a) $\boldsymbol {\pi }=(\pi _1, \pi _2, \ldots , \pi _r) \in \mathbb {N}^r$ is a partition of d; that is, $\pi _1 \geq \pi _2 \geq \cdots \geq \pi _r$ and $\pi _1+ \pi _2 + \cdots + \pi _r = d$ [13, Section 1.7]; we denote this $\boldsymbol {\pi } \vdash d$ ;

   2. (b) $p_{\boldsymbol {\pi }}(x_1, x_2, \ldots , x_n)=p_{\pi _1}p_{\pi _2}\cdots p_{\pi _r}$ , in which $p_k(x_1,x_2, \ldots , x_n)=x_1^k+x_2^k+\cdots +x_n^k$ is a power-sum symmetric polynomial;

   3. (c) $B_d$ is a complete Bell polynomial, defined by $\sum _{\ell =0}^{\infty } B_{\ell }(x_1, x_2, \ldots , x_{\ell }) \frac {t^{\ell }}{\ell !} =\exp ( \sum _{j=1}^{\infty } x_j \frac {t^j}{j!})$ [2, Section II];

   4. (d) The cumulants $\kappa _1, \kappa _2, \ldots , \kappa _d$ are defined by the recursion $\mu _r=\sum _{\ell =0}^{r-1}{r-1\choose \ell } \mu _{\ell }\kappa _{r-\ell }$ for $1 \leq r \leq d$ , in which $\mu _r = \mathbb {E}[X_1^r]$ is the rth moment of $X_1$ [5, Section 9]; and

   5. (e) $\kappa _{\boldsymbol {\pi }} = \kappa _{\pi _1} \kappa _{\pi _2} \cdots \kappa _{\pi _{r}}$ and $y_{\boldsymbol {\pi }}=\prod _{i\geq 1}(i!)^{m_i}m_i!$ , in which $m_i=m_i(\boldsymbol {\pi })$ is the multiplicity of i in $\boldsymbol {\pi }$ .

4. (4) For real $d\geq 1$ , the function $\boldsymbol {\lambda }(A) \mapsto \| A \|_{\mathbf {X},d}$ is Schur convex; that is, it respects majorization $\prec $ (see (3.1)).

5. (5) Let $d\geq 2$ be an even integer. Define $\mathrm {T}_{\boldsymbol {\pi }} : \mathrm {M}_{n}\to \mathbb {R}$ by setting $\mathrm {T}_{\boldsymbol {\pi }}(Z)$ to be $1/{d\choose d/2}$ times the sum over the $\binom {d}{d/2}$ possible locations to place $d/2$ adjoints ${}^*$ among the d copies of Z in $(\operatorname {tr} \underbrace {ZZ\cdots Z}_{\pi _1}) (\operatorname {tr} \underbrace {ZZ\cdots Z}_{\pi _2}) \cdots (\operatorname {tr} \underbrace {ZZ\cdots Z}_{\pi _r})$ . Then

   (1.2) $$ \begin{align} \| Z \|_{\mathbf{X},d}= \bigg( \sum_{\boldsymbol{\pi} \,\vdash\, d} \frac{ \kappa_{\boldsymbol{\pi}}\mathrm{T}_{\boldsymbol{\pi}}(Z)}{y_{\boldsymbol{\pi}}}\bigg)^{1/d}\end{align} $$
   is a norm on $\mathrm {M}_n$ that restricts to the norm on $\mathrm {H}_n$ above. In particular, $\| Z \|_{\mathbf {X},d}^d$ is a positive definite trace polynomial in Z and $Z^*$ .

The paper is structured as follows. Section 2 provides several examples afforded by the theorems above. The proofs of Theorems 1.1 and 1.2 appear in Sections 3 and 4, respectively. Section 5 concludes with some brief remarks.

2 Examples

The norm $\| \cdot \|_{\mathbf {X},d}$ defined in (1.1) is determined by its unit ball. This provides one way to visualize the properties of random vector norms. We consider a few examples hereand refer the reader to [7, Section 2] for further examples and details.

2.1 Normal random variables

Suppose $d\geq 2$ is an even integer and $\mathbf {X}$ is a random vector whose entries are independent normal random variables with mean $\mu $ and variance $\sigma ^2$ . The example in [7, equation (2.12)] illustrates

$$ \begin{align*} \| A \|_{\mathbf{X},d}^d=\sum_{k=0}^{\frac{d}{2}} \frac{\mu^{2k} (\operatorname{tr} A)^{2k}}{(2k)!} \cdot \frac{\sigma^{d-2k} \| A\|_{\operatorname{F}}^{d-2k}}{2^{\frac{d}{2}-k} (\frac{d}{2}-k)!} \quad \text{for }A \in \mathrm{H}_n, \end{align*} $$

in which $\| \cdot \|_{\operatorname {F}}$ is the Frobenius norm. For $d=2$ , the extension to $\mathrm {M}_n$ guaranteed by Theorem 1.3 is $\| Z \|_{\mathbf {X},2}^2= \tfrac {1}{2} \sigma ^2 \operatorname {tr}(Z^*\!Z) + \tfrac {1}{2} \mu ^2 (\operatorname {tr} Z^*)(\operatorname {tr} Z)$ [7, p. 816].

Now, let $n=2$ . If $\mu =0$ , the restrictions of $\| \cdot \|_{\mathbf {X},d}$ to $\mathbb {R}^2$ (whose elements are identified with diagonal matrices) reproduce multiples of the Euclidean norm. If $\mu \neq 0$ , then the unit circles for $\| \cdot \|_{\mathbf {X},d}$ are approximately elliptical (see Figure 1).

Figure 1 (Left) Unit circles for $\|\cdot \|_{\mathbf {X},d}$ with $d=1, 2, 4, 20$ , in which $X_1$ and $X_2$ are standard normal random variables. (Right) Unit circles for $\|\cdot \|_{\mathbf {X},10}$ , in which $X_1$ and $X_2$ are normal random variables with means $\mu =-2, -1, 0, 1, 6$ and variance $\sigma ^2=1$ .

2.2 Standard exponential random variables

If $d\geq 2$ is an even integer and $\mathbf {X}$ is a random vector whose entries are independent standard exponential random variables, then $\| A \|_{\mathbf {X},d}^d$ equals the complete homogeneous symmetric polynomial  $h_d(\lambda _1, \lambda _2, \ldots , \lambda _n)=\sum _{1\leq k_1\leq \cdots \leq k_d\leq n} \lambda _{k_1}\lambda _{k_2}\cdots \lambda _{k_d}$ in the eigenvalues $\lambda _1, \lambda _2, \ldots , \lambda _n$ [1]. For $d=4$ , the extension to $\mathrm {M}_n$ guaranteed by Theorem 1.3 is [1, equation (9)]

$$ \begin{align*} \| Z \|_4^4 &= \frac{1}{24} \big( (\operatorname{tr} Z)^2 \operatorname{tr}(Z^*)^2 + \operatorname{tr}(Z^*)^2 \operatorname{tr}(Z^2) + 4 \operatorname{tr}(Z)\operatorname{tr}(Z^*) \operatorname{tr}(Z^*Z) \nonumber \\ & \qquad + 2 \operatorname{tr}(Z^*Z)^2 + (\operatorname{tr} Z)^2 \operatorname{tr}(Z^{*2}) + \operatorname{tr}(Z^2) \operatorname{tr}(Z^{*2}) + 4\operatorname{tr}(Z^*) \operatorname{tr}(Z^*Z^2) \nonumber \\ & \qquad + 4 \operatorname{tr}(Z) \operatorname{tr}(Z^{*2}Z) + 2 \operatorname{tr}(Z^*ZZ^*Z) + 4 \operatorname{tr}(Z^{*2}Z^2) \big). \end{align*} $$

The unit balls for these norms are illustrated in Figure 2 (left).

Figure 2 (Left) Unit circles for $\|\cdot \|_{\mathbf {X},d}$ with $d=1, 2, 3, 4, 20$ , in which $X_1$ and $X_2$ are standard exponentials. (Right) Unit circles for $\| \cdot \|_{\mathbf {X},d}$ with $d=2, 4, 20$ , in which $X_1$ and $X_2$ are Bernoulli with $q=0.5$ .

2.3 Bernoulli random variables

A Bernoulli random variable is a discrete random variable X defined according to $\mathbb {P}(X=k)=q^k(1-q)^{1-k}$ for $k=0,1$ and $0<q<1$ . Suppose d is an even integer and $\mathbf {X}$ is a random vector whose entries are independental Bernoulli random variables with parameter q.

Remark 2.1 An expression for $\| A \|^d_{\mathbf {X},d}$ appears in [7, Section 2.7]. However, there is a missing multinomial coefficient. The correct expression for $\| A \|^d_{\mathbf {X},d}$ is given by

$$ \begin{align*} \| A \|^d_{\mathbf{X},d}=\frac{1}{d!}\sum_{i_1+i_2+\cdots+i_n=d} \binom{d}{i_1,i_2,\ldots,i_n}q^{|I|} \lambda_1^{i_1}\lambda_2^{i_2}\cdots \lambda_n^{i_n}, \end{align*}$$

in which $|I|$ is the number of nonzero $i_k$ ; that is, $I = \{ k : i_k \neq 0\}$ . We thank the anonymous referee for pointing out the typo in [7, Section 2.7]. Figures 2 (right) and 3 illustrate the unit balls for these norms in a variety of cases.

Figure 3 Unit circles for $\|\cdot \|_{\mathbf {X},d}$ , in which $X_1$ and $X_2$ are Bernoulli with varying parameter q and with $d=2$ (left) and $d=10$ (right).

2.4 Pareto random variables

Suppose $\alpha , x_m>0$ . A random variable X distributed according to the probability density function

$$ \begin{align*} f_X(t)= \begin{cases} \dfrac{\alpha x_m^{\alpha}}{t^{\alpha+1}}, & \text{if}\ t\geq x_m,\\ 0, & \text{if}\ t<x_m, \end{cases} \end{align*} $$

is a Pareto random variable with parameters $\alpha $ and $x_m$ . Suppose $\mathbf {X}$ is a random vector whose entries are Pareto random variables. Then $\| A \|_{\mathbf {X},d}$ exists whenever $\alpha>d$ [7, Section 2.10].

Suppose $d=2$ and $\mathbf {X}$ is a random vector whose entries are independent Pareto random variables with $\alpha>2$ and $x_m=1$ . If $n=2$ , then

$$ \begin{align*} \| A \|_{\mathbf{X},2}^2=\frac{\alpha}{2}\Bigg( \frac{\lambda_1^2}{\alpha-2}+\frac{2\alpha \lambda_1\lambda_2}{(\alpha-1)^2}+\frac{\lambda_2^2}{\alpha-2} \Bigg).\end{align*} $$

Figure 4 (left) illustrates the unit circles for $\| \cdot \|_{\mathbf {X},2}$ with varying $\alpha $ . As $\alpha \to \infty $ , the unit circles approach the parallel lines at $\lambda _2=\pm \sqrt {2}-\lambda _1$ ; that is, $|\operatorname {tr} A|^2 = 2$ . Figure 4 (right) depicts the unit circles for $\| \cdot \|_{\mathbf {X},d}$ with fixed $\alpha $ and varying d.

Figure 4 (Left) Unit circles for $\| \cdot \|_{\mathbf {X},2}$ , in which $X_1$ and $X_2$ are independent Pareto random variables with $\alpha =2.1, 3, 4, 10$ and $x_m=1$ . (Right) Unit circles for $\| \cdot \|_{\mathbf {X},d}$ , in which $X_1$ and $X_2$ are independent Pareto random variables with $\alpha =5$ and $p=1, 2, 4$ .

3 Proof of Theorem 1.1

The proof of Theorem 1.1 follows from Propositions 3.1 and 3.5.

Proposition 3.1 If $\| \cdot \|$ is a weakly unitarily invariant norm on $\mathrm {H}_n$ , then there is a symmetric norm f on $\mathbb {R}^n$ such that $\| A \|=f( \boldsymbol {\lambda }(A))$ for all $A\in \mathrm {H}_n$ .

Proof Hermitian matrices are unitarily diagonalizable. Since $\| \cdot \|$ is weakly unitarily invariant, $\| A \|=\| D \|$ , in which D is a diagonalization of A. Consequently, $\| A \|$ must be a function in the eigenvalues of A. Moreover, any permutation of the entries in D is obtained by conjugating D by a permutation matrix, which is unitary. Therefore, $\| A \|$ is a symmetric function in the eigenvalues of A. In particular, $\| A \|=f( \boldsymbol {\lambda }(A) )$ for some symmetric function f. Given $\mathbf {a}=( a_1, a_2,\dots , a_n)\in \mathbb {R}^n$ , define the Hermitian matrix

$$ \begin{align*} \operatorname{diag}{\mathbf{a}} = \begin{bmatrix} a_1 & & &\\ & a_2 & &\\ & & \ddots & \\ & & & a_n \end{bmatrix}. \end{align*} $$

Then $\boldsymbol {\lambda }(\operatorname {diag}{\mathbf {a}}) = P\mathbf {a}$ for some permutation matrix P. Symmetry of f implies

$$ \begin{align*} f(\mathbf{a}) =f(P\mathbf{a}) = f\big(\boldsymbol{\lambda}(\operatorname{diag}{\mathbf{a}})\big) = \| \operatorname{diag}{\mathbf{a}} \|. \end{align*} $$

Consequently, f inherits the defining properties of a norm on $\mathbb {R}^n$ .

Let $\widetilde {\mathbf {x}}=(\widetilde {x}_1,\widetilde {x}_2, \ldots , \widetilde {x}_n)$ denote the nondecreasing rearrangement of $\mathbf {x}= (x_1, x_2, \ldots , x_n)\in \mathbb {R}^n$ . Then $\mathbf {y}$ majorizes  $\mathbf {x}$ , denoted $\mathbf {x}\prec \mathbf {y}$ , if

(3.1) $$ \begin{align} \sum_{i=1}^n \widetilde{x}_i=\sum_{i=1}^n \widetilde{y}_i \quad \text{and} \quad\sum_{i=1}^k \widetilde{x}_i\leq \sum_{i=1}^k \widetilde{y}_i \quad \text{for }1 \leq k\leq n-1. \end{align} $$

Recall that a matrix with nonnegative entries is doubly stochastic if each row and column sums to $1$ . The next result is due to Hardy, Littlewood, and Pólya [9].

Lemma 3.2 If $\mathbf {x}\prec \mathbf {y}$ , then there exists a doubly stochastic matrix D such that $\mathbf {y} = D \mathbf {x}$ .

The next lemma is Birkhoff’s [6]; $n^2-n+1$ works in place of $n^2$ [10, Theorem 8.7.2].

Lemma 3.3 If $D \in \mathrm {M}_n$ is doubly stochastic, then there exist permutation matrices $P_1,P_2,\ldots ,P_{n^2} \in \mathrm {M}_n$ and nonnegative numbers $c_1,c_2,\ldots ,c_{n^2}$ satisfying $\sum _{i=1}^{n^2} c_i = 1$ such that $D = \sum _{i=1}^{n^2} c_i P_i$ .

For each $A \in \mathrm {H}_n$ , recall that $\boldsymbol {\lambda }(A)=(\lambda _1(A),\lambda _2(A), \ldots , \lambda _n(A))$ denotes the vector of eigenvalues $\lambda _1(A) \geq \lambda _2(A) \geq \cdots \geq \lambda _n(A)$ . We regard $\boldsymbol {\lambda }(A)$ as a column vector for purposes of matrix multiplication.

Lemma 3.4 If $A, B\in \mathrm {H}_n$ , then there exist permutation matrices $P_1,P_2,\ldots ,P_{n^2} \in \mathrm {M}_n$ and $c_1,c_2,\ldots ,c_{n^2}\geq 0$ such that

$$ \begin{align*} \boldsymbol{\lambda}(A+B) = \sum_{i = 1}^{n^2} c_{i} P_i(\boldsymbol{\lambda}(A) + \boldsymbol{\lambda}(B)) \quad \text{and}\quad \sum_{i = 1}^{n^2} c_i = 1. \end{align*}$$

Proof The Ky Fan eigenvalue inequality [8] asserts that

(3.2) $$ \begin{align} \sum_{i=1}^{k} \lambda_i(A+B) \leq \sum_{i=1}^{k} \big(\lambda_i(A) + \lambda_i(B)\big) \quad \text{for all }1\leq k\leq n. \end{align} $$

The sum of the eigenvalues of a matrix is its trace. Consequently,

$$ \begin{align*} \sum_{i=1}^{n} \lambda_i(A+B) = \operatorname{tr} (A+B) = \operatorname{tr} A + \operatorname{tr} B = \sum_{i=1}^{n} \big(\lambda_i(A) + \lambda_i(B)\big), \end{align*}$$

so equality holds in (3.2) for $k=n$ . Thus, $\boldsymbol {\lambda }(A+B) \prec \boldsymbol {\lambda }(A) + \boldsymbol {\lambda }(B)$ . Lemma 3.2 provides a doubly stochastic matrix D such that $\boldsymbol {\lambda }(A+B) = D(\boldsymbol {\lambda }(A) + \boldsymbol {\lambda }(B))$ . Lemma 3.3 provides the desired permutation matrices and nonnegative scalars.

The following proposition completes the proof of Theorem 1.1.

Proposition 3.5 If f is a symmetric norm on $\mathbb {R}^n$ , then $\| A \|=f(\boldsymbol {\lambda }(A))$ defines a weakly unitarily invariant norm on $\mathrm {H}_n$ .

Proof The function $\| A \|=f(\boldsymbol {\lambda }(A))$ is symmetric in the eigenvalues of A, so it is weakly unitarily invariant. It remains to show that $\| \cdot \|$ defines a norm on $\mathrm {H}_n$ .

Positive definiteness. A Hermitian matrix $A = 0$ if and only if $\boldsymbol {\lambda }(A) = 0$ . Thus, the positive definiteness of f implies the positive definiteness of $\| \cdot \|$ .

Homogeneity. If $c\geq 0$ , then $\boldsymbol {\lambda }(cA) = c\boldsymbol {\lambda }(A)$ . If $c<0$ , then

$$ \begin{align*} \boldsymbol{\lambda}(cA) = c \begin{bmatrix} && 1 \\ & \unicode{x22F0} & \\ 1 & & \end{bmatrix} \boldsymbol{\lambda}(A). \end{align*} $$

Then the homogeneity and symmetry of f imply that

$$ \begin{align*} \| cA \| = f\big(\boldsymbol{\lambda}(cA)\big) = f\big(c\boldsymbol{\lambda}(A)\big) = |c| f\big(\boldsymbol{\lambda}(A)\big) = |c|\| A \|. \end{align*} $$

Triangle inequality. Suppose that $A,B \in \mathrm {H}_n$ . Lemma 3.4 ensures that there exist permutation matrices $P_1,P_2,\ldots ,P_{n^2} \in \mathrm {M}_n$ and nonnegative numbers $c_1,c_2,\ldots ,c_{n^2}$ satisfying $\sum _{i=1}^{n^2} c_i = 1$ such that $D = \sum _{i=1}^{n^2} c_i P_i$ . Thus,

$$ \begin{align*} \| A + B \| = f\big(\boldsymbol{\lambda}(A+B)\big) = f\Bigg(\sum_{i = 1}^{n^2} c_{i} P_i\big(\boldsymbol{\lambda}(A) + \boldsymbol{\lambda}(B)\big)\Bigg). \end{align*}$$

The triangle inequality and homogeneity of f yield

(3.3) $$ \begin{align} \| A+B \|\leq \sum_{i=1}^{n^2} c_{i} f\big( P_i(\boldsymbol{\lambda}(A) + \boldsymbol{\lambda}(B))\big). \end{align} $$

Since f is permutation invariant and $\sum _{i = 1}^{n^2} c_i = 1$ ,

$$ \begin{align*} \sum_{i=1}^{n^2} c_{i} f\big( P_i(\boldsymbol{\lambda}(A) + \boldsymbol{\lambda}(B))\big) = \sum_{i=1}^{n^2} c_{i} f\big(\boldsymbol{\lambda}(A) +\boldsymbol{\lambda}(B)\big)= f\big(\boldsymbol{\lambda}(A) + \boldsymbol{\lambda}(B)\big). \end{align*} $$

Thus, the triangle inequality for f and (3.3) yield

$$ \begin{align*} \| A+B \| \leq f\big(\boldsymbol{\lambda}(A) + \boldsymbol{\lambda}(B)\big) \leq f\big(\boldsymbol{\lambda}(A) \big)+ f\big(\boldsymbol{\lambda}(B)\big) =\| A \|+\| B \|.\\[-35pt] \end{align*} $$

4 Proof of Theorem 1.2

Let $\mathbf {X}$ be an iid random vector and define $f_{\mathbf {X},d}:\mathbb {R}^n\to \mathbb {R}$ by

(4.1) $$ \begin{align} f_{\mathbf{X},d}(\boldsymbol{\lambda}) = \left( \frac{\mathbb{E} |\langle\mathbf{X}, \boldsymbol{\lambda}\rangle|^d}{\Gamma(d+1)}\right)^{1/d} \quad \text{for }d \geq 1.\end{align} $$

Since the entries of $\mathbf {X}$ are iid, $f_{\mathbf {X},d}$ is symmetric. In light of Theorem 1.1, it suffices to show that $f_{\mathbf {X},d}$ is a norm on $\mathbb {R}^n$ ; the continuity remark at the end of Theorem 1.2 is Proposition 4.2.

Proposition 4.1 The function $f_{\mathbf {X},d}$ in (4.1) defines a norm on $\mathbb {R}^n$ for all $d\geq 1$ .

Proof The proofs for homogeneity and the triangle inequality in [7, Section 3.1] are valid for $d\geq 1$ . However, the proof for positive definiteness in [7, Lemma 3.1] requires $d\geq 2$ . The proof below holds for $d\geq 1$ and is simpler than the original.

Positive definiteness. If $f_{\mathbf {X},d}(\boldsymbol {\lambda })=0$ , then $\mathbb {E}|\langle \mathbf {X},\boldsymbol {\lambda }\rangle |^d=0$ . The nonnegativity of $|\langle \mathbf {X},\boldsymbol {\lambda }\rangle |^d$ ensures that

(4.2) $$ \begin{align} \lambda_1X_1+\lambda_2X_2+\cdots+\lambda_nX_n=0 \end{align} $$

almost surely. Assume (4.2) has a nontrivial solution $\boldsymbol {\lambda }$ with nonzero entries $\lambda _{i_1}, \lambda _{i_2}, \ldots , \lambda _{i_k}$ . If $k=1$ , then $X_{i_k}=0$ almost surely, which contradicts the nondegeneracy of our random variables. If $k>1$ , then (4.2) implies that

(4.3) $$ \begin{align} X_{i_1}=a_{i_2}X_{i_2}+a_{i_3}X_{i_3}+\cdots +a_{i_k}X_{i_k} \end{align} $$

almost surely, in which $a_{i_j}=-\lambda _{i_j}/\lambda _{i_1}$ . The independence of $X_{i_1}, X_{i_2}, \ldots , X_{i_k}$ contradicts (4.3). Relation (4.2) therefore has no nontrivial solutions.

Homogeneity. This follows from the bilinearity of the inner product and linearity of expectation:

$$ \begin{align*} f_{\mathbf{X},d}(c\boldsymbol{\lambda}) =\left( \frac{\mathbb{E} |c\langle\mathbf{X}, \boldsymbol{\lambda}\rangle|^d}{\Gamma(d+1)}\right)^{1/d} =\left(\frac{|c|^d\mathbb{E} |\langle\mathbf{X}, \boldsymbol{\lambda}\rangle|^d}{\Gamma(d+1)}\right)^{1/d} =|c|f_{\mathbf{X},d}(\boldsymbol{\lambda}). \end{align*} $$

Triangle inequality. For $\boldsymbol {\lambda }, \boldsymbol {\mu }\in \mathbb {R}^n$ , define random variables $X=\langle \mathbf {X},\boldsymbol {\lambda }\rangle $ and $Y=\langle \mathbf {X},\boldsymbol {\mu }\rangle $ . Minkowski’s inequality implies

$$ \begin{align*} \big(\mathbb{E}\vert \langle \mathbf{X}, \boldsymbol{\lambda}+\boldsymbol{\mu}\rangle\vert^d\big)^{1/d} =\big(\mathbb{E}|X+Y|^d\big)^{1/d} \leq\big(\mathbb{E}|X|^d\big)^{1/d}+\big(\mathbb{E}|Y|^d\big)^{1/d}. \end{align*} $$

The triangle inequality for $f_{\mathbf {X},d}$ follows.

Proposition 4.2 Suppose $\mathbf {X}$ is an iid random vector whose entries have at least m moments. The function $f:\left [1,m\right ] \to \mathbb {R}$ defined by $f(d) =\| A \|_{\mathbf {X},d}$ is continuous for all $A\in \mathrm {H}_n$ .

Proof Define the random variable $Y = \langle \mathbf {X}, \boldsymbol {\lambda }\rangle $ , in which $\boldsymbol {\lambda }$ denotes the vector of eigenvalues of A. The random variable Y is a measurable function defined on a probability space $(\Omega , \mathcal {F}, \mathbb {P})$ . The pushforward measure of Y is the probability measure $\mu _{Y}$ on $\mathbb {R}$ defined by $\mu _Y(E)=\mathbb {P} (Y^{-1}(E) )$ for all Borel sets E. Consequently,

$$ \begin{align*} \Gamma(d+1)\big(f(d)\big)^d = \mathbb{E}|Y|^d = \int |x|^d \, d\mu_{Y}. \end{align*} $$

The bound $|x|^d \leq |x| + |x|^m$ holds for all $x\in \mathbb {R}$ and $1 \leq d \leq m$ . Therefore,

$$ \begin{align*} \int |x|^d \, d\mu_{Y} \leq \int |x| \, d\mu_Y + \int |x|^m \, d\mu_Y. \end{align*} $$

If $d_i\to d$ , then $\int |x|^{d_i}d\mu _Y\to \int |x|^{d}d\mu _Y$ by the dominated convergence theorem. Consequently, $ \Gamma (d_i+1) (f(d_i) )^{d_i}\to \Gamma (d+1) (f(d) )^d $ whenever $d_i\to d$ . The function $\Gamma (d+1) (f(d) )^d$ is therefore continuous in d. The continuity of the gamma function establishes continuity for $f^d$ and f.

5 Remarks

Remark 5.1 A norm $\| \cdot \|$ on $\mathrm {M}_n$ is weakly unitarily invariant if $\| A \|=\| U^*AU \|$ for all $A\in \mathrm {M}_n$ and unitary $U \in \mathrm {M}_n$ . A norm $\Phi $ on the space $C(S)$ of continuous functions on the unit sphere $S\subset \mathbb {C}^n$ is a unitarily invariant function norm if $\Phi (f\circ U)=\Phi (f)$ for all $f\in C(S)$ and unitary $U \in \mathrm {M}_n$ . Every weakly unitarily invariant norm $\| \cdot \|$ on $\mathrm {M}_n$ is of the form $\| A \|=\Phi (f_A)$ , in which $f_A\in C(S)$ is defined by $f_A(\mathbf {x})=\langle A\mathbf {x},\mathbf {x}\rangle $ and $\Phi $ is a unitarily invariant function norm [4], [3, Theorem 2.1].

Remark 5.2 Remark 3.4 of [7] is somewhat misleading. We state there that the entries of $\mathbf {X}$ are required to be identically distributed but not independent. To clarify, the entries of $\mathbf {X}$ being identically distributed guarantee that $\| \cdot \|_{\mathbf {X},d}$ satisfies the triangle inequality on $\mathrm {H}_n$ . The additional assumption of independence guarantees that $\| \cdot \|_{\mathbf {X},d}$ is also positive definite.