The p-norm of hypermatrices with symmetries

TL;DR

This paper investigates the properties of the p-norm of symmetric nonnegative hypermatrices, showing that symmetry simplifies the computation of spectral radius and norm, with implications for tensor analysis.

Contribution

It establishes that for symmetric nonnegative hypermatrices, the p-norm is achieved by vectors with certain symmetries, linking spectral radius and p-norm for p≥2.

Findings

p-norm is attained by vectors with identical components for symmetric indices

Spectral radius equals p-norm for symmetric nonnegative hypermatrices when p≥2

Symmetry reduces complexity in computing hypermatrix norms and spectral properties

Abstract

The $p$-norm of $r$-matrices generalizes the $2$-norm of $2$-matrices. It is shown that if a nonnegative $r$-matrix is symmetric with respect to two indices $j$ and $k$, then the $p$-norm is attained for some set of vectors such that the $i$th and the $j$th vectors are identical. It follows that the $p$-spectral radius of a symmetric nonnegative $r$-matrix is equal to its $p$-norm for any $p\geq2$.

Equations

$$L_A\left(\mathbf{x},\mathbf{y}\right):=\sum _{i,j}{a}_{i,j}{x}_i{y}_j$$

$${\Vert A\Vert }_p:=\underset{{|\mathbf{x}|}_p=1,{|\mathbf{y}|}_p=1}{\max }|L_A\left(\mathbf{x},\mathbf{y}\right)|.$$

$${\Vert A\Vert }_2=\underset{{|\mathbf{x}|}_2=1}{\max }|L_A\left(\mathbf{x},\mathbf{x}\right)|,$$

$${\Vert A\Vert }_p=\underset{{|\mathbf{x}|}_p=1}{\max }{L}_A\left(\mathbf{x},\mathbf{x}\right).$$

$$L_A:{\mathbb{R}}^{n_1}\times \cdots \times {\mathbb{R}}^{n_r}\to \mathbb{R}$$

$${\mathbf{x}}^{\left(1\right)}=(x_1^{\left(1\right)},\dots ,x_{n_1}^{\left(1\right)})\in {\mathbb{R}}^{n_1},\dots ,{\mathbf{x}}^{\left(r\right)}=(x_1^{\left(r\right)},\dots ,x_{n_r}^{\left(r\right)})\in {\mathbb{R}}^{n_r}$$

$$L_A({\mathbf{x}}^{\left(1\right)},\dots ,{\mathbf{x}}^{\left(r\right)}):=\sum _{i_1\in \left[n_1\right],\dots ,i_r\in \left[n_r\right]}{a}_{i_1,\dots ,i_r}{x}_{i_1}^{\left(1\right)}\cdots x_{i_r}^{\left(r\right)}.$$

$${\Vert A\Vert }_p:=\underset{|{\mathbf{x}}^{\left(1\right)}{|}_p=1,\dots ,|{\mathbf{x}}^{\left(r\right)}{|}_p=1}{\max }|L_A({\mathbf{x}}^{\left(1\right)},\dots ,{\mathbf{x}}^{\left(r\right)})|.$$

$${\Vert A\Vert }_2=\underset{|{\mathbf{x}}^{\left(1\right)}{|}_2=1,\dots ,|{\mathbf{x}}^{\left(r\right)}{|}_2=1,{\mathbf{x}}^{\left(j\right)}={\mathbf{x}}^{\left(k\right)}}{\max }|L_A\left({\mathbf{x}}^{\left(1\right)},\dots ,{\mathbf{x}}^{\left(r\right)}\right)|.$$

$${\Vert A\Vert }_p=\underset{|{\mathbf{x}}^{\left(1\right)}{|}_p=1,\dots ,|{\mathbf{x}}^{\left(r\right)}{|}_p=1,{\mathbf{x}}^{\left(j\right)}={\mathbf{x}}^{\left(k\right)}}{\max }{L}_A\left({\mathbf{x}}^{\left(1\right)},\dots ,{\mathbf{x}}^{\left(r\right)}\right).$$

$$b_{i_1,\dots ,i_r}=a_{i_1,\dots ,i_r}{s}_{i_j}{s}_{i_k}.$$

$${\Vert B\Vert }_p=\underset{|{\mathbf{x}}^{\left(1\right)}{|}_p=1,\dots ,|{\mathbf{x}}^{\left(r\right)}{|}_p=1,{\mathbf{x}}^{\left(j\right)}={\mathbf{x}}^{\left(k\right)}}{\max }{L}_B\left({\mathbf{x}}^{\left(1\right)},\dots ,{\mathbf{x}}^{\left(r\right)}\right).$$

$${\Vert A\Vert }_p=\underset{|\mathbf{x}{|}_p=1}{\max }{L}_A\left(\mathbf{x},\dots ,\mathbf{x}\right).$$

$$S_i=\sum _{i_2\in \left[n_2\right],\dots ,i_r\in \left[n_r\right]}{a}_{i,i_2\dots ,i_r}.$$

$${\rho }^{\left(p\right)}\left(A\right)\ge n^{1-r/p}{\left(\frac{1}{n}\left(S_1^{p/\left(p-1\right)}+\cdots +S_n^{p/\left(p-1\right)}\right)\right)}^{\left(p-1\right)/p}.$$

$${\rho }^{\left(p\right)}\left(G\right)\ge \left(r-1\right)!n^{1-r/p}{\left(\frac{1}{n}\left(d_1^{p/\left(p-1\right)}+\cdots +d_n^{p/\left(p-1\right)}\right)\right)}^{\left(p-1\right)/p}.$$

$${\Vert A\Vert }_p=\left|L_A\left(\mathbf{x},\mathbf{y}\right)\right|$$

$$|\sum _{i,j}{a}_{i,j}{x}_i{y}_j|\le \sum _{i,j}{a}_{i,j}\left|x_i\right|\left|y_j\right|.$$

$$\lambda x_i^{p-1}=a_{i,1}{y}_1+\cdots +a_{i,n}{y}_n,i=1,\dots ,n$$

$$\lambda y_i^{p-1}=a_{i,1}{x}_1+\cdots +a_{i,n}{x}_n,i=1,\dots ,n$$

$$\lambda =\lambda \left(x_1^p+\cdots +x_n^p\right)=\sum _{i,j}{a}_{i,j}{x}_i{y}_j={\Vert A\Vert }_p.$$

$$z_i:={\left(\frac{x_i^p+y_i^p}{2}\right)}^{1/p},i=1,\dots ,n.$$

$$\sum _{i=1}^n\lambda x_i^p+\lambda y_i^p=\sum _{i=1}^n\sum _{j=1}^n{a}_{i,j}\left(x_i{y}_j+y_i{x}_j\right).$$

$$\frac{x_i{y}_j+y_i{x}_j}{2}\le {\left(\frac{x_i^2+y_i^2}{2}\right)}^{1/2}{\left(\frac{x_j^2+y_j^2}{2}\right)}^{1/2}\le {\left(\frac{x_i^p+y_i^p}{2}\right)}^{1/p}{\left(\frac{x_j^p+y_j^p}{2}\right)}^{1/p}=z_i{z}_j.$$

$${\Vert A\Vert }_p=\lambda =\sum _{i=1}^n\lambda \frac{x_i^p+y_i^p}{2}\le \sum _{i=1}^n\sum _{j=1}^n{a}_{i,j}{z}_i{z}_j=L_A\left(\mathbf{z},\mathbf{z}\right).$$

$${\Vert A\Vert }_2=|L_A({\mathbf{x}}^{\left(1\right)},\dots ,{\mathbf{x}}^{\left(r\right)})|.$$

$$b_{s,t}=\sum _{i_1\in \left[n_1\right],\dots ,i_{r-2}\in \left[n_{r-2}\right]}{a}_{i_1,\dots i_{r-2},s,t}{x}_{i_1}^{\left(1\right)}\cdots x_{i_{r-2}}^{\left(r-2\right)}.$$

$$L_A({\mathbf{x}}^{\left(1\right)},\dots ,{\mathbf{x}}^{\left(r\right)})=\sum _{i=1}^n\sum _{j=1}^n{b}_{i,j}{x}_i{y}_j=L_B(\mathbf{x},\mathbf{y}).$$

$${\Vert A\Vert }_2={\Vert B\Vert }_2.$$

$$\lambda x_i=b_{i,1}{y}_1+\cdots +b_{i,n}{y}_n,i=1,\dots ,n$$

Full text

Keywords: $p$-norm*; hypermatrix; $p$*-*spectral radius; nonnegative hypermatrix; hypergraph. *

**AMS classification: **05C50, 05C65, 15A18, 15A42, 15A60, 15A69.

1 Introduction and main results

In this note we study the $p$-norm of hypermatrices with partial symmetries. To put our results in a more familiar context, we start with ordinary matrices.

Let $A=\left[a_{i,j}\right]$ be a real $m\times n$ matrix and let

[TABLE]

for any $\mathbf{x}=\left(x_{1},\ldots,x_{m}\right)\in\mathbb{R}^{m}$ and $\mathbf{y}=\left(y_{1},\ldots,y_{n}\right)\in\mathbb{R}^{n}.$

Now, given $p\geq 1,$ define the $p$-norm $\left\|A\right\|_{p}$ of $A$ as

[TABLE]

where $\left|\cdot\right|_{p}$ stands for the $l^{p}$ vector norm.

It is known that if $A$ is symmetric, then

[TABLE]

but this equality turns out to be truly exceptional and fails for any $p\neq 2$ and some appropriate matrix $A$. Our first theorem gives a condition on $A$ that is sufficient to preserve the equality for all $p\geq 2.$

Theorem 1

If $A$ is a symmetric nonnegative matrix and $p\geq 2,$ then

[TABLE]

The main goal of this note is to extend Theorem 1 to $r$-matrices, which we introduce next.

Let $r\geq 2,$ and let $n_{1},\ldots,n_{r}$ be positive integers. An $r$-matrix of order $n_{1}\times\cdots\times n_{r}$ is a real function defined on the Cartesian product $\left[n_{1}\right]\times\cdots\times\left[n_{r}\right].$

Thus, hereafter, *matrix *means an $r$-matrix with unspecified $r$, and ordinary matrices are referred to as $2$-matrices.

As usual, matrices are denoted by capital letters, and their values are denoted by the corresponding lowercase letter with the variables listed as subscripts, i.e., if $A$ is an $r$-matrix of order $n_{1}\times\cdots\times n_{r}$, we write $a_{i_{1},\ldots,i_{r}}$ for $A\left(i_{1},\ldots,i_{r}\right)$ whenever $i_{1}\in\left[n_{1}\right],\ldots,i_{r}\in\left[n_{r}\right].$

The linear form** **of an $r$-matrix $A$ of order $n_{1}\times\cdots\times n_{r}$ is a function

[TABLE]

defined for any $r$ vectors

[TABLE]

as

[TABLE]

Now, the $p$-norm111The idea of $p$-norm for hypermatrices comes from Hardy, Littlewood and Polya [5]. In the present form it was introduced for integral $p$ by Lek-Heng Lim [7].** **of $A$ is defined as

[TABLE]

So far we have extended all of the setup of Theorem 1 to $r$-matrices, except for the symmetry property, which is less straightforward for $r\geq 3$.

Definition 2

An $r$-matrix $A=\left[a_{i_{1},\ldots,i_{r}}\right]$ of order $n_{1}\times\cdots\times n_{r}$ is called $\left(j,k\right)$-symmetric if $i\neq k,$ $n_{j}=n_{k},$ and $A$ is invariant under the swap of $i_{j}$ and $i_{k}$.

If $A$ is $\left(j,k\right)$-symmetric for every $1\leq j<k\leq r,$ then it is called symmetric. If an $r$-matrix $A$ of order $n_{1}\times\cdots\times n_{r}$ is symmetric, then $n_{1}=\cdots=n_{r},$ and $n_{1}$ is called its order.

We are ready now to state our main results.

Theorem 3

If $A$ is a $\left(j,k\right)$-symmetric $r$-matrix, then

[TABLE]

Note that statements similar to Theorem 3 have been studied for almost nine decades by now, mostly in abstract normed spaces. In particular, motivated by problem 73 of Mazur and Orlicz in [12], Banach [1] proved a general result that implies Theorem 3 for symmetric matrices222For newer proofs of Banach’s result, see [3] and [11], and for further results, see [2], [11] and their references.. Our proof is much simpler, almost an observation, because we use tools which may not available in general normed spaces.

If $A$ is nonnegative, Theorem 3 can be extended similarly to Theorem 1.

Theorem 4

If $A$ is a $\left(j,k\right)$-symmetric nonnegative $r$-matrix and $p\geq 2,$ then

[TABLE]

The nonnegativity of $A$ in Theorem 4 is far from necessary. Indeed let $A$ be a $\left(j,k\right)$-symmetric nonnegative $r$-matrix of order $n_{1}\times\cdots\times n_{r}$ and set $n=n_{j}=n_{k}$. Let $\mathbf{s}=\left(s_{1},\ldots,s_{n}\right)$ be a $\pm 1$ vector. Define an $r$-matrix $B$ of order $n_{1}\times\cdots\times n_{r}$ by letting

[TABLE]

Clearly $B$ may have both positive and negative entries, but it is easy to see that

[TABLE]

Another simple consequence of Theorem 4 is the following corollary, which was proved for $p\geq r$ in [10], by rather involved methods..

Corollary 5

If $A$ is a symmetric nonnegative $r$-matrix and $p\geq 2,$ then

[TABLE]

For symmetric $r$-matrices and $p\geq 1$, the value $\max_{|\mathbf{x}|_{p}=1}$ $|L_{A}\left(\mathbf{x},\ldots,\mathbf{x}\right)|$ is known as the $p$-spectral radius of $A$ and is denoted by $\rho^{\left(p\right)}\left(A\right).$ The $p$-spectral radius333The idea of the $p$-spectral radius can be traced back to Lusternik and Schnirelman [9], later revived by Friedman and Wigderson [4]. For hypergraphs it was introduced by Keevash, Lenz, and Mubayi [6]. has been studied in some detail, particularly for the adjacency matrix of uniform hypergraphs (see [10] and its references). Corollary 5 can be used to obtain various lower bounds on $\rho^{\left(p\right)}\left(A\right),$ by choosing vectors $\mathbf{x}^{\left(1\right)},\ldots,\mathbf{x}^{\left(r\right)}$ with $|\mathbf{x}^{\left(1\right)}|_{p}=1,$ $\ldots$ , $|\mathbf{x}^{\left(r\right)}|_{p}=1$ and calculating $L_{A}\left(\mathbf{x}^{\left(1\right)},\ldots,\mathbf{x}^{\left(r\right)}\right).$ We give an illustration next.

Suppose that $A$ is an $r$-matrix of order $n_{1}\times\cdots\times n_{r}$ and for every $i\in\left[n_{1}\right],$ set

[TABLE]

Apparently the values $S_{1},\ldots,S_{n_{1}}$ generalize the row-sums of $2$-matrices, so we call them the slice-sums of $A.$

Combining Theorem 21 of  [10] with Theorem 4, one comes up with the following useful lower bound on the $p$-spectral radius:

Corollary 6

Let $A$ be a symmetric nonnegative $r$-matrix of order $n$ with slice-sums $S_{1},\ldots,S_{n}.$ If $p\geq 2,$ then

[TABLE]

When restated for $r$-uniform graphs (see, e.g., [10] for the basics), the corollary reads as:

Corollary 7

Let $G$ be an $r$-uniform graph of order $n$ with degrees $d_{1},\ldots,d_{n}.$ If $p\geq 2,$ then

[TABLE]

Let us note that the case $p=r$ of this corollary has been proved in [8], and the case $p\geq r$ has been proved in [10].

The remaining part of the note is split into two sections: in Section 2, we present the proofs of Theorems 1, 3, 4 and in Section 3, we state two open problems.

2 Proofs of Theorems 1, 3, and 4

Proof of Theorem 1 Let $A$ be a symmetric nonnegative matrix of order $n,$ and

[TABLE]

where $\left|\mathbf{x}\right|_{p}=\left|\mathbf{y}\right|_{p}=1.$ We assume that $\mathbf{x}$ and $\mathbf{y}$ are nonnegative because

[TABLE]

and the $l^{p\text{ }}$norm of both $\left(\left|x_{1}\right|,\ldots,\left|x_{n}\right|\right)$ and $\left(\left|y_{1}\right|,\ldots,\left|y_{n}\right|\right)$ is $1.$

We see that $\mathbf{x}$ and $\mathbf{y}$ maximize $L_{A}\left(\mathbf{x},\mathbf{y}\right)$ under the constraints $x_{1}^{p}+\cdots+x_{n}^{p}=1$ and $y_{1}^{p}+\cdots\cdots+y_{n}^{p}=1.$ Using Lagrange’s multipliers, it follows that there exists $\lambda$ such that

[TABLE]

and

[TABLE]

It is easy to see that $\lambda=\left\|A\right\|_{p}:$ indeed, multiplying the $i$th equation (1) by $x_{i}$ and adding the results, we get

[TABLE]

Now, let

[TABLE]

Multiplying the $i$th equation (1) by $x_{i}$ and the $i$th equation (2) by $y_{i}$ and adding the results, we get

[TABLE]

On the other hand, using the Cauchy-Schwarz inequality and the Power Mean inequality, we find that

[TABLE]

Therefore,

[TABLE]

Since $\left|\mathbf{z}\right|_{p}=1,$ the proof is completed. $\Box$

Proof of Theorem 3 Let $A$ be a $\left(j,k\right)$-symmetric $r$-matrix of order $n_{1}\times\cdots\times n_{r}$. For convenience, let us reindex the variables so that $j=r-1,$ $k=r,$ and let $n=n_{r-1}=n_{r}.$

Suppose that $\mathbf{x}^{\left(1\right)}\in\mathbb{R}^{n_{1}},$ $\ldots$ $,\mathbf{x}^{\left(r\right)}\in\mathbb{R}^{n_{r}}$ are vectors with $|\mathbf{x}^{\left(1\right)}|_{2}=1,$ $\ldots$ $,|\mathbf{x}^{\left(r\right)}|_{2}=1$ such that

[TABLE]

To finish the proof we have to show that $\mathbf{x}^{\left(r-1\right)}$ and $\mathbf{x}^{\left(r\right)}$ may be chosen equal. To this end, define a square $2$-matrix $B$ of order $n$ by

[TABLE]

and note that $B$ is symmetric since $A$ is $\left(r-1,r\right)$-symmetric.

Next, for convenience, set $\mathbf{x}=\mathbf{x}^{\left(r-1\right)}$ and $\mathbf{y}=\mathbf{x}^{\left(r\right)}.$ Obviously

[TABLE]

Clearly $\mathbf{x}$ and $\mathbf{y}$ maximize $L_{B}(\mathbf{x},\mathbf{y})$ subject to $x_{1}^{2}+\cdots+x_{n}^{2}=1$ and $y_{1}^{2}+\cdots+y_{n}^{2}=1.$ Therefore,

[TABLE]

Moreover, using Lagrange’s multipliers, it follows that there exists $\lambda$ such that

[TABLE]

and

[TABLE]

As in the proof of Theorem 1, we see that $\lambda=\left\|B\right\|_{2},$ and so $\lambda=\left\|A\right\|_{2}.$

Further, adding the $i$th equation (3) and the $i$th equation (4), we get a new system of equations

[TABLE]

which implies that $\lambda$ is an eigenvalue of $B,$ unless $\mathbf{x}+\mathbf{y}=0.$ In the latter case, we immediately see that

[TABLE]

completing the proof. Thus, we may assume that $\lambda$ is an eigenvalue of $B.$ The Rayleigh-Ritz theorem implies that

[TABLE]

Hence,

[TABLE]

completing the proof. $\Box$

Proof of Theorem 4 Our proof is a combination of the proofs of Theorems 1 and 3.

Let $A$ be a $\left(j,k\right)$-symmetric nonnegative $r$-matrix of order $n_{1}\times\cdots\times n_{r}$. For convenience, let us reindex the variables so that $j=r-1,$ $k=r,$ and let $n=n_{r-1}=n_{r}.$

Suppose that $\mathbf{x}^{\left(1\right)}\in\mathbb{R}^{n_{1}},$ $\ldots$ $,\mathbf{x}^{\left(r\right)}\in\mathbb{R}^{n_{r}}$ are vectors with $|\mathbf{x}^{\left(1\right)}|_{p}=1,$ $\ldots$ $,|\mathbf{x}^{\left(r\right)}|_{p}=1$ such that

[TABLE]

Note that $\mathbf{x}^{\left(1\right)},$ $\ldots$ $,\mathbf{x}^{\left(r\right)}$ can be taken nonnegative in view of

[TABLE]

To finish the proof we have to show that $\mathbf{x}^{\left(r-1\right)}$ and $\mathbf{x}^{\left(r\right)}$ can be chosen equal. To this end, define a square $2$-matrix $B$ of order $n$ by

[TABLE]

and note that $B$ is symmetric since $A$ is $\left(r-1,r\right)$-symmetric. Moreover, $B$ is nonnegative.

Next, for convenience, set $\mathbf{x}=\mathbf{x}^{\left(r-1\right)}$ and $\mathbf{y}=\mathbf{x}^{\left(r\right)}.$ Obviously

[TABLE]

Clearly $\mathbf{x}$ and $\mathbf{y}$ maximize $L_{B}(\mathbf{x},\mathbf{y})$ subject to $x_{1}^{p}+\cdots+x_{n}^{p}=1$ and $y_{1}^{p}+\cdots+y_{n}^{p}=1.$ Therefore,

[TABLE]

In view of Theorem 1, there exists $\mathbf{z}\in\mathbb{R}^{n}$ with $\left|\mathbf{z}\right|_{p}=1$ such that

[TABLE]

Hence,

[TABLE]

completing the proof. $\Box$

3 Two open problems

Corollary 5 could be very useful because $\left\|A\right\|_{p}$ is usually easier to evaluate or estimate than $\rho^{\left(p\right)}\left(A\right).$ Thus, it is desirable to extend Corollary 5 to matrices that are essentially distinct from nonnegative matrices. We naturally arrive at the following problems:

Problem 8

Characterize all symmetric $r$-matrices $A$ such that

[TABLE]

for all sufficiently large $p$.

Problem 9

Characterize all symmetric $r$-matrices $A$ such that

[TABLE]

for all $p\in\left[1,2\right).$

Unfortunately, the above problems seem hopeless at present. Probably there are some chances for solving either of them for $r=2.$