On the Equivariance Properties of Self-adjoint Matrices

TL;DR

This paper characterizes the equivariance properties of self-adjoint matrices, linking their symmetry groups to eigenvalue multiplicities, with implications for bifurcation theory, graph symmetries, and matrix approximation problems.

Contribution

It provides a complete characterization of the symmetry groups of self-adjoint matrices based on their eigenvalues and multiplicities, extending understanding of their equivariance properties.

Findings

Self-adjoint matrices are equivariant under a specific group isomorphic to a product of Z2 groups.

Multiple eigenvalues induce larger symmetry groups isomorphic to products of orthogonal groups.

Results have applications in bifurcation analysis, graph symmetry, and matrix approximation problems.

Abstract

We investigate self-adjoint matrices $A\in\mathbb{R}^{n,n}$ with respect to their equivariance properties. We show in particular that a matrix is self-adjoint if and only if it is equivariant with respect to the action of a group $\Gamma_2(A)\subset \mathbf{O}(n)$ which is isomorphic to $\otimes_{k=1}^n\mathbf{Z}_2$. If the self-adjoint matrix possesses multiple eigenvalues -- this may, for instance, be induced by symmetry properties of an underlying dynamical system -- then $A$ is even equivariant with respect to the action of a group $\Gamma(A) \simeq \prod_{i = 1}^k \mathbf{O}(m_i)$ where $m_1,\ldots,m_k$ are the multiplicities of the eigenvalues $\lambda_1,\ldots,\lambda_k$ of $A$. We discuss implications of this result for equivariant bifurcation problems, and we briefly address further applications for the Procrustes problem, graph symmetries and Taylor expansions.

Equations

$$\gamma f(x)=f(\gamma x)\quad\mbox{for all $\gamma\in\Gamma$,}$$

$$\dot{x}=A(\mu )x-{\Vert x\Vert }_2^{2}x,$$

$$A(\mu )=\left(\begin{array}{ccc}2& \sqrt{2}(2\mu -1)& \sqrt{2}(2\mu -1)\\ \sqrt{2}(2\mu -1)& 3-2\mu & 2\mu -1\\ \sqrt{2}(2\mu -1)& 2\mu -1& 3-2\mu \end{array}\right).$$

$$S=\left(\begin{array}{ccc}1& 0& 0\\ 0& 0& 1\\ 0& 1& 0\end{array}\right).$$

$$SA(\mu)=A(\mu)S\quad\mbox{for all $\mu\in\mathbb{R}$.}$$

$$f(Sx,\mu)=Sf(x,\mu)\quad\mbox{for all $x\in\mathbb{R}^{3}$ and $\mu\in\mathbb{R}$.}$$

$$v={\left(\frac{1}{\sqrt{2}},\frac{1}{2},\frac{1}{2}\right)}^T.$$

$$\left(\begin{array}{cccc}\pm 1& 0& \cdots & 0\\ 0& \pm 1& \ddots & ⋮\\ ⋮& \ddots & \ddots & 0\\ 0& \cdots & 0& \pm 1\end{array}\right).$$

$$D=\left(\begin{array}{cccc}{\lambda }_1& 0& \cdots & 0\\ 0& {\lambda }_2& \ddots & ⋮\\ ⋮& \ddots & \ddots & 0\\ 0& \cdots & 0& {\lambda }_n\end{array}\right),{\lambda }_i\in \mathbb{R},i=1,2,\dots ,n,$$

$$\sigma D=D\sigma \forall \sigma \in \Sigma .$$

$$\sigma B=B\sigma\quad\forall\sigma\in\Sigma\quad\Longleftrightarrow\quad\mbox{$B$ is a diagonal matrix.}$$

$$\gamma A=A\gamma \forall \gamma \in {\Gamma }_2(A),$$

$${\Gamma }_2(A)=\{V^T\sigma V:\sigma \in \Sigma \}.$$

$$D=VA{V}^T$$

$$\sigma VA{V}^T=VA{V}^T\sigma ⟺V^T\sigma VA=A{V}^T\sigma V.$$

$$A^T=(V^{T}DV{)}^T=A$$

$$\gamma A=A\gamma \forall \gamma \in {\Gamma }_2(A),$$

$${\Gamma }_2(A)=\{W^{\ast }\sigma W:\sigma \in \Sigma \}.$$

$$A(0)=\left(\begin{array}{ccc}2& -\sqrt{2}& -\sqrt{2}\\ -\sqrt{2}& 3& -1\\ -\sqrt{2}& -1& 3\end{array}\right).$$

$$V=\left(\begin{array}{ccc}-0.7071& -0.5& -0.5\\ 0.6969& -0.3732& -0.6124\\ 0.1196& -0.7815& 0.6124\end{array}\right)$$

$${\gamma }_1=\left(\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\right),$$

$${\gamma }_3=\left(\begin{array}{ccc}0& \frac{1}{\sqrt{2}}& \frac{1}{\sqrt{2}}\\ \frac{1}{\sqrt{2}}& -\frac{1}{2}& \frac{1}{2}\\ \frac{1}{\sqrt{2}}& \frac{1}{2}& -\frac{1}{2}\end{array}\right),$$

$${\gamma }_{7}v=-v.$$

$${\Sigma }_V(A)=\{\sigma \in \mathbf{O}(n)\mid \sigma VA{V}^T=VA{V}^T\sigma \}$$

$$\Gamma (A)=V^T{\Sigma }_V(A)V.$$

$$\gamma A=A\gamma ⟺\gamma \in \Gamma (A).$$

$$\gamma A=A\gamma$$

$$⟺\sigma VA{V}^T=VA{V}^T\sigma \text{ for }\sigma =V\gamma V^T$$

$$⟺\gamma \in \Gamma (A).\text{qed}$$

$$\vartheta (\gamma )A=\gamma A{\gamma }^T,$$

$${\Sigma }_U(A)=W^T{\Sigma }_V(A)W.$$

$$W^T{\Sigma }_V(A)W={\Sigma }_U(A).$$

Full text

Key words: self-adjoint matrix; equivariance; bifurcation theory; Procrustes problem; Taylor expansion

AMS subject classifications. 15B57, 15A24, 37G40, 41A58

1 Introduction

For more than 30 years the influence of symmetry properties of a dynamical system on its qualitative temporal behavior has been intensively studied. Such symmetry properties are typically induced by network structures or geometric properties of the underlying mathematical model. The related research focuses on a variety of topics, for instance, the classification of symmetry breaking bifurcations ([1]) or the explanation of the occurrence of stable heteroclinic cycles. For an overview of this area and their relevance in the sciences we refer to [2].

Formally, symmetry properties of a dynamical system $\dot{x}=f(x)$ manifest themselves by an equivariance property of the right-hand side. That is, $f\colon\mathbb{R}^{n}\to\mathbb{R}^{n}$ satisfies

[TABLE]

where $\Gamma\subset\mathbf{O}(n)$ is a compact Lie group. It is well known that equivariance properties are inherited by the linearization $Df(x^{*})$ of $f$ from the symmetry properties of the steady-state solutions $x^{*}$. In fact, this is the reason why generically $Df(x^{*})$ may possess multiple eigenvalues, which implies the occurrence of complex symmetry breaking bifurcations in dynamical systems. This happens, for instance, if $\Gamma=\mathbf{O}(n)$ ($n\geq 2$) or $\Gamma=\mathbf{D}_{\ell}$ ($\ell\geq 3$), where $\mathbf{D}_{\ell}$ is the dihedral group of order $\ell$, that is, the symmetry group of the $\ell$-sided regular polygon.

The investigations in this article are motivated by the analysis of equivariant dynamical systems where the linearization $A=Df(x)$ is additionally self-adjoint, that is, the matrix $A\in\mathbb{R}^{n,n}$ satisfies $A=A^{T}$. Recently, it has been observed that a matrix is self-adjoint if and only if it is equivariant with respect to the action of a group $\Gamma_{2}(A)\subset\mathbf{O}(n)$ which is isomorphic to $\prod_{i=1}^{n}\mathbf{Z}_{2}$ (see [3]). This underlying equivariance property is implicitly present in articles concerning the development of dynamical systems for the solution of certain optimization problems (e.g. [4, 5, 6]). But to the best of our knowledge it has not explicitly been stated elsewhere before – and definitely not in the dynamical systems context.

In this article, we extend this result from [3] significantly in the sense that we completely characterize the equivariance properties of self-adjoint matrices by their spectra. In fact, we will show in our main result on the equivariance properties of self-adjoint matrices (Corollary 4.5) that $\Gamma(A)$ is isomorphic to $\prod_{i=1}^{k}\mathbf{O}(m_{i})$ where $m_{1},\ldots,m_{k}$ are the multiplicities of the eigenvalues $\lambda_{1},\ldots,\lambda_{k}$ of $A$. In particular, if $A$ has only simple eigenvalues, then $\Gamma(A)=\Gamma_{2}(A)\simeq\prod_{i=1}^{n}\mathbf{Z}_{2}$.

One important consequence of this result is the following observation: Suppose that the underlying dynamical system is $\mathbf{D}_{\ell}$-equivariant for an $\ell\geq 3$. Then – as already mentioned above – the linearization $Df(x^{*})$ at a $\mathbf{D}_{\ell}$-symmetric steady-state solution $x^{*}$ generically possesses double eigenvalues. Our result implies that in this case the linearization will not just be $\mathbf{D}_{\ell}$-equivariant but (at least) even equivariant with respect to an action of $\Gamma(A)\simeq\mathbf{O}(2)\times\prod_{i=1}^{n-2}\mathbf{Z}_{2}$.

Moreover, if in addition the entire function $f$ is $\Gamma(A)$-equivariant, then symmetry-related bifurcations of the system will be governed by $\Gamma(A)$ rather than $\mathbf{D}_{\ell}$, and this leads to phenomena which would generically be unexpected if only $\mathbf{D}_{\ell}$ is taken into account. This would apply, for instance, to numerical discretizations of the cubic or quintic Ginzburg–Landau equation on a $\mathbf{D}_{\ell}$-symmetric spatial domain ([7, 8]). Thus, from an abstract point of view our results are strongly related to the notion of hidden symmetries which has been introduced in connection with the occurrence of unexpected bifurcations in partial differential equations with Neumann boundary conditions ([9, 2]). We will illustrate this fact by several examples in the following sections.

A detailed outline of the structure of this article is as follows. In Section 2, we introduce a specific $\mathbf{Z}_{2}$-equivariant dynamical system as a guiding example. This system exhibits unexpected dynamical phenomena driven by the underlying $\Gamma(A)$-equivariance: A symmetry-preserving pitchfork bifurcation and the existence of an entire orbit of steady-state solutions. Then, in Section 3, we review briefly the main result from [3]. This will allow us to reveal the structure which leads to the symmetric pitchfork bifurcation in the guiding example. Our main results concerning the equivariance properties of self-adjoint matrices are stated in Section 4. In Section 5, we discuss the consequences for bifurcations in equivariant dynamical systems. Finally, in Section 6, we discuss a couple of further applications: First we characterize all solutions of the two-sided orthogonal Procrustes problem (Section 6.1). Then we briefly discuss consequences for the graph isomorphism problem for undirected graphs (Section 6.2). We conclude with the construction of simple approximations of derivatives of higher order for real valued functions (Section 6.3). Here we make use of the fact that each Hessian $H$ is symmetric and therefore also $\Gamma(H)$-equivariant.

2 Motivation – the Guiding Example

As a guiding example we consider the differential equation

[TABLE]

where $x\in\mathbb{R}^{3},\mu\in\mathbb{R}$ and

[TABLE]

Observe that this problem has an obvious $\mathbf{Z}_{2}$-symmetry: first the matrix $A(\mu)$ commutes for all $\mu$ with the permutation matrix

[TABLE]

That is,

[TABLE]

Moreover, $\left\lVert x\right\rVert_{2}^{2}$ is invariant under orthogonal transformations. Therefore, the right-hand side $f(x,\mu)=A(\mu)x-\left\lVert x\right\rVert_{2}^{2}x$ in (1) is $\mathbf{Z}_{2}$-equivariant and satisfies

[TABLE]

Thus, by genericity results from classical bifurcation or singularity theory (see [1]) we would particularly expect that

• $(i)$

the only steady-state bifurcations that occur in (1) are turning points or (symmetry-breaking) pitchfork bifurcations corresponding to the underlying symmetry given by $\mathbf{Z}_{2}=\{I,S\}$ (e.g. [1, 10]);

• $(ii)$

equilibria of (1) are isolated.

In contrast to this expectation we observe the following two phenomena for (1):

• $(i)^{*}$

Apparently the system undergoes a pitchfork bifurcation at $\mu=0$. The corresponding local bifurcation diagram is shown in Figure 1 (a). However, at the bifurcation point a normalized kernel vector of $A(0)$ is given by

[TABLE]

In particular, this eigenvector is $S$-symmetric ($Sv=v$) rather than antisymmetric ($Sv=-v$) as expected. Accordingly, also the equilibria on the bifurcating branches are $S$-symmetric, see Figure 1 (a).

• $(ii)^{*}$

For $\mu=-0.25$ we find not just $x^{*}=0$, but in addition an entire continuous orbit of equilibria for (1), see Figure 1 (b). In fact, we will see in Section 5 that such orbits exist for an entire range of parameter values.

It is the purpose of this work to explain such phenomena, and we will see that this is strongly related to the fact that $A$ is self-adjoint. In fact, our results will imply that the dynamical system (1) is $\mathbf{Z}_{2}\times\mathbf{O}(2)$-equivariant with $\{I,S\}\subset\mathbf{O}(2)$, and this will explain the phenomena described in $(i)^{*}$ and $(ii)^{*}$.

3 Self-adjoint Matrices are Equivariant – a Warm-up

In this section, we briefly summarize the main result from [3]. With this we illustrate the underlying structure, namely that equivariance properties of self-adjoint matrices are induced by the symmetry properties of diagonal matrices.

Let $\Sigma\subset\mathbf{O}(n)$ be the abelian group consisting of the $2^{n}$ matrices

[TABLE]

For any diagonal matrix

[TABLE]

we obviously have

[TABLE]

In fact, it is easy to verify for an arbitrary matrix $B\in\mathbb{R}^{n,n}$ that

[TABLE]

Proposition 3.1** ([3]).**

A matrix $A\in\mathbb{R}^{n,n}$ is self-adjoint (i.e. $A=A^{T}$) if and only if there is an orthogonal matrix $V\in\mathbf{O}(n)$ such that

[TABLE]

where the group $\Gamma_{2}(A)\subset\mathbf{O}(n)$ is defined by

[TABLE]

We state the proof for the sake of completeness.

Proof.

Suppose that $A=A^{T}$. Then there is $V\in\mathbf{O}(n)$ such that

[TABLE]

is a diagonal matrix. By (4) we have for all $\sigma\in\Sigma$

[TABLE]

Therefore, $A$ satisfies the equivariance condition (5).

Now suppose that (5) is satisfied for some $V\in\mathbf{O}(n)$. Then the matrix $VAV^{T}$ commutes with every $\sigma\in\Sigma$, and by (4) it follows that $D=VAV^{T}$ is a diagonal matrix. Therefore,

[TABLE]

as desired. ∎

Remark 3.2**.**

• (a)

Observe that the implication ‘$\Longrightarrow$’ could also be proved by using the well-known fact that two matrices $A$ and $B$ commute if there is an orthogonal transformation $V$ such that both $V^{T}AV$ and $V^{T}BV$ are diagonal.

• (b)

By construction all the eigenvalues of every $\gamma\in\Gamma_{2}(A)$ are $1$ or $-1$. In particular $\gamma^{2}=I$ for all $\gamma\in\Gamma_{2}(A)$, and $\Gamma_{2}(A)\simeq\prod_{i=1}^{n}\mathbf{Z}_{2}$. Moreover, by (a) the matrix $A$ and all $\gamma\in\Gamma_{2}(A)$ possess the same set of eigenvectors.

• (c)

Obviously, analogous results can be obtained for Hermitian or normal matrices: Using essentially the same proof as in Proposition 3.1 one can show that a matrix $A\in\mathbb{C}^{n,n}$ is normal (i.e. $AA^{*}=A^{*}A$) if and only if there is a unitary matrix $W\in\mathbf{U}(n)$ such that

[TABLE]

where the group $\Gamma_{2}(A)\subset\mathbf{U}(n)$ is defined by

[TABLE]

Example 3.3**.**

Let us consider the matrix $A(0)$ from our guiding example in Section 2, i.e.

[TABLE]

The matrix

[TABLE]

transforms $A(0)$ into a diagonal matrix $D$ with the eigenvalues $0,4,4$ of $A(0)$ on the diagonal. With Proposition 3.1 we can compute eight matrices which commute with $A(0)$ – the elements of $\Gamma_{2}(A)$ –, and these matrices are given by

[TABLE]

and $\gamma_{j+4}=-\gamma_{j}$ for $j=1,2,3,4$. (The entries in the matrices are exact up to four decimal places.) For the eigenvector $v$ in (3) we compute

[TABLE]

Thus, a symmetry breaking pitchfork bifurcation at $\mu=0$ is induced by the group $\mathbf{Z}_{2}=\{I,\gamma_{7}\}$ rather than $\{I,S\}$ (see (2)), and this explains the phenomenon $(i)^{*}$ discussed in Section 2.

Observe that $S$ is not among the matrices $\gamma_{j}$ $(j=1,\ldots,8)$, i.e. $S\not\in\Gamma_{2}(A)$, so that there is still another structure to be revealed. We will see in the following section that this is related to the fact that $4$ is a double eigenvalue of $A(0)$.

4 Self-adjoint Matrices are Equivariant – the General Case

In this section, we generalize Proposition 3.1 significantly. In fact, we will show that in general the group $\Sigma$ may be much more complex – even in the case where $A$ is not equivariant in the classical sense where e.g. underlying geometric symmetries lead to equvariance properties of a dynamical system.

4.1 Orthogonal Isotropy Subgroups for Matrices

The following observation forms the theoretical basis for our analytical investigations. With this result we state a useful characterization of the group $\Gamma(A)$ containing all $\gamma\in\mathbf{O}(n)$ which commute with a given matrix $A$.

Proposition 4.1**.**

Let $A\in\mathbb{R}^{n,n}$ and $V\in\mathbf{O}(n)$. Define the compact group

[TABLE]

and let

[TABLE]

Then for every $\gamma\in\mathbf{O}(n)$

[TABLE]

In particular, $\Gamma(A)$ does not depend on $V$, and we refer to $\Gamma(A)$ as the orthogonal isotropy subgroup of $A$.

Proof.

Let $\gamma\in\mathbf{O}(n)$ such that $\gamma A=A\gamma$. Then we have for each $V\in\mathbf{O}(n)$

[TABLE]

Remark 4.2**.**

• (a)

Recall that the isotropy subgroup for a point $x$ in some space $X$ characterizes the symmetry of $x$ with respect to a certain group action. More precisely consider a group action $\vartheta$ of a group $G$ on a linear space $X$. Then the isotropy subgroup of $x\in X$ is given by $\{g\in G:\vartheta(g)x=x\}$.

If we let $G=\mathbf{O}(n)$ act on matrices $A\in X=\mathbb{R}^{n,n}$ by

[TABLE]

then $\Gamma(A)\subset\mathbf{O}(n)$ in Proposition 4.1 is the isotropy subgroup of $A$ with respect to this action.

• (b)

If we replace $\mathbf{O}(n)$ by $\mathbf{U}(n)$ (unitary matrices) or $\mathbf{GL}(n,\mathbb{R})$ (invertible matrices) and the matrix $V^{T}$ by $V^{*}$ or $V^{-1}$, respectively, then we obtain an analogous result for the unitary and invertible isotropy subgroup. It is also possible to formulate Proposition 4.1 for general orthogonal, unitary or invertible operators.

We now show that $\Sigma_{V}(A)$ in (7) is unique up to orthogonal transformations.

Corollary 4.3**.**

Let $V\in\mathbf{O}(n)$.

• (a)

For each $U\in\mathbf{O}(n)$ there exists $W\in\mathbf{O}(n)$ such that

[TABLE]

• (b)

For each $W\in\mathbf{O}(n)$ there exists $U\in\mathbf{O}(n)$ such that

[TABLE]

Thus,

[TABLE]

Proof.

Let $V\in\mathbf{O}(n)$ be given. Then, by Proposition 4.1, we have for $U\in\mathbf{O}(n)$

[TABLE]

and therefore

[TABLE]

With $W=VU^{T}$ we obtain (a) and (b) follows by setting $U=W^{T}V$. ∎

4.2 Equivariance Properties of Self-adjoint Matrices

We now return to the case where $A$ is self-adjoint. Our aim is to extend significantly Proposition 3.1. This leads to the surprising fact that self-adjoint matrices may possess hidden symmetries due to repeated eigenvalues. Denote by $\lambda_{1}<\dots<\lambda_{k}$ the (real) sorted eigenvalues of $A$ with multiplicities $m=(m_{1},...,m_{k})$ and let $V\in\mathbf{O}(n)$ so that $VAV^{T}=D$, where $D\in\mathbb{R}^{n,n}$ is a diagonal matrix containing the sorted eigenvalues $\lambda_{i}\in\mathbb{R}$ of $A$ on its diagonal.

Definition 4.4**.**

Let $k\leq n$ and $m\in\mathbb{N}^{k}$ so that $\sum_{i=1}^{k}m_{i}=n$. Define $\mathbf{O_{B}}(m)$ to be the set of block-diagonal matrices where the $i$-th block is in $\mathbf{O}(m_{i})$, i.e.

[TABLE]

With this useful definition we are able to completely characterize the symmetries of a self-adjoint matrix $A\in\mathbb{R}^{n,n}$.

Corollary 4.5**.**

Let $A\in\mathbb{R}^{n,n}$ be self-adjoint and $V\in\mathbf{O}(n)$ so that $V$ diagonalizes $A$ (and the eigenvalues on the diagonal are sorted). Then

[TABLE]

where $m$ is the vector that contains the multiplicities of the eigenvalues of $A$. In particular, by Proposition 4.1 we have

[TABLE]

Proof.

Using the fact that $V$ diagonalizes $A$ we can write $\Sigma_{V}(A)$ as

[TABLE]

Thus, we only need to show that $\sigma D=D\sigma$ is equivalent to the fact that $\sigma\in\Sigma_{V}$ is a block-diagonal matrix. Let $\sigma\in\Sigma_{V}(A)$ and write $\sigma=(\sigma_{i,j})_{i,j=1,...,k}$ with rectangular blocks $\sigma_{i,j}\in\mathbb{R}^{m_{i},m_{j}}$. Then we have $\sigma D=(\lambda_{j}\sigma_{i,j})_{i,j}$ and $D\sigma=(\lambda_{i}\sigma_{i,j})_{i,j}$. Therefore, $\sigma\in\Sigma_{V}$ translates into

[TABLE]

which is equivalent to $\sigma_{i,j}=0$ for $i\neq j$ and $\sigma_{i,i}\in\mathbf{O}(m_{i})$ for $i=1,...,k$. ∎

Remark 4.6**.**

1. (a)

The order of the eigenvalues $\lambda_{i}\in\mathbb{R}$ is not relevant as long as $V$ is chosen in such a way that all instances of the same eigenvalue on the diagonal of $VAV^{T}$ are next to each other. Otherwise, the elements in $\Sigma_{V}$ are not block-diagonal. 2. (b)

Analogous results for normal matrices in the unitary case and diagonalizable matrices in the invertible case (cf. Remark 4.2 (b)) follow in the same way. 3. (c)

Since $\mathbf{O}(1)=\{+1,-1\}=\mathbf{Z}_{2}$ and $\mathbf{Z}_{2}^{\ell}=\prod_{i=1}^{\ell}\mathbf{Z}_{2}\subseteq\mathbf{O_{B}}(\ell)=\mathbf{O}(\ell)$ for every $\ell\in\mathbb{N}$, we have $\mathbf{Z}_{2}^{n}\subseteq\Sigma_{V}$ for each self-adjoint matrix $A$ (independently of the multiplicities of the eigenvalues of $A$). That is, we always have $\Gamma_{2}(A)\subset\Gamma(A)$ (cf. Proposition 3.1 and Remark 3.2 (b)), and equality holds if and only if $A$ has only simple eigenvalues. In particular, $\Gamma(A)$ is finite if and only if $A$ has only simple eigenvalues.

Example 4.7**.**

• (a)

Let us return to our guiding example from Section 2 and consider the matrix

[TABLE]

The eigenvalues of $A(-0.25)$ are a simple eigenvalue $\lambda_{1}=-1$ and a double eigenvalue $\lambda_{2}=5$. Thus, Corollary 4.5 yields $\Sigma_{V}=\mathbf{O_{B}}(1,2)$ and it turns out that $A(-0.25)$ is in fact $\mathbf{Z}_{2}\times\mathbf{O}(2)$-equivariant. Here $\mathbf{Z}_{2}=\{I,\gamma_{7}\}$ (cf. Example 3.3), and a reflection $S$ and a rotation $R$ by $\pi/2$ (exact up to four decimal places) within $\mathbf{O}(2)$ are given by

[TABLE]

It follows that if we have an equilibrium which is not $\mathbf{O}(2)$-symmetric, then we obtain an entire nontrivial $\mathbf{O}(2)$-orbit of equilibria. This explains the phenomenon described in $(ii)^{*}$ for the guiding example in Section 2, see also Figure 1 (b).

Finally, observe that the bifurcating equilibria in Figure 1 (a) are $\mathbf{O}(2)$-symmetric and therefore isolated for each fixed value of $\mu$. Note that generically, it is not expected for a one-parameter family of self-adjoint matrices to possess multiple eigenvalues (see Appendix 10 in **[11]**; or also **[12]** where results of **[13]** on general matrices have been extended to self-adjoint matrices in the equivariant context). Rather we have constructed this family for the purpose of illustration.

• (b)

Consider the parameter-dependent family of matrices (see **[12]**)

[TABLE]

with

[TABLE]

and

[TABLE]

for $\mu\in[-3,3]$. Then it is easy to verify that $A(\mu)$ is $\mathbf{D}_{4}$-equivariant. Here the action of the dihedral group $\mathbf{D}_{4}$ is generated by a rotation $R$ and a reflection $S$, where $R,S\in\mathbb{R}^{16,16}$. Written in a $4\times 4$-block structure these are given by

[TABLE]

However, $A(\mu)$ is self-adjoint and therefore this matrix family is not just $\mathbf{D}_{4}$- but even $\Gamma(A)$-equivariant where $\Gamma(A)$ is given in (9). Now $A(\mu)$ has $8$ simple and $4$ double eigenvalues, where the multiple eigenvalues are induced by the two-dimensional irreducible representations of $\mathbf{D}_{4}$ (see **[1]**). Hence, for each parameter value $A(\mu)$ is indeed $\mathbf{Z}_{2}^{8}\otimes\mathbf{O}(2)^{4}$-equivariant by Corollary 4.5 and (9).

Finally, we explicitly list for $\mu=0$ a couple of elements of $\Gamma(A)$ and $\Sigma_{V}(A)$ for illustration purposes. For instance, the matrices $R$ and $S$ above are given by

[TABLE]

where $\sigma_{R},\sigma_{S}\in\mathbf{O_{B}}(1,1,1,2,1,2,2,2,1,1,1,1)$ and

[TABLE]

and

[TABLE]

A couple of ‘hidden symmetries’ – i.e. elements of $\Gamma(A)$ which are not contained in $\mathbf{D}_{4}$ – are given by

[TABLE]

[TABLE]

or

[TABLE]

[TABLE]

5 Implications for Equivariant Dynamical Systems

In this section, we discuss by an example the implications for dynamical systems of the form

[TABLE]

where $A(\mu)\in\mathbb{R}^{n,n}$ is self-adjoint for all $\mu\in\mathbb{R}$ and $f\colon\mathbb{R}^{n}\times\mathbb{R}\to\mathbb{R}^{n}$ is $\mathbf{O}(n)$-equivariant, i.e.

[TABLE]

It follows that the right-hand side in (10) is $\Gamma(A(\mu))$-equivariant. In particular, the symmetry group varies with the parameter $\mu$, and a detailed bifurcation analysis in this context should be developed elsewhere. Here, we rather focus on the description of qualitative dynamical phenomena induced by the hidden symmetries.

Remark 5.1**.**

• (a)

Observe that the requirement on $f$ is satisfied if, for instance,

[TABLE]

where $g\colon\mathbb{R}^{n}\times\mathbb{R}\to\mathbb{R}$ is $\mathbf{O}(n)$-invariant, that is $g(\gamma x,\mu)=g(x,\mu)$ for all $\gamma\in\mathbf{O}(n)$. In particular, the equivariance condition (11) would hold for $g(x,\mu)=h(x)$ or $g(x,\mu)=h(A(\mu)x)$ where $h$ is $\mathbf{O}(n)$-invariant.

• (b)

If we consider e.g. the nonlinear Schrödinger/Gross–Pitaevskii equation **[14]** or the cubic Ginzburg–Landau equation **[8]** in two dimensions then a numerical discretization by the method of lines yields a dynamical system of the form (10), where $A$ does not explicitly depend on $\mu$. Moreover, if the underlying spatial domain is $\mathbf{D}_{\ell}$-symmetric ($\ell\geq 3$) then symmetry related bifurcations of the system will be governed by $\Gamma(A)$ rather than just $\mathbf{D}_{\ell}$. In fact, in this case we expect to observe phenomena driven by the hidden symmetries as already described in Example 4.7 (b).

As a concrete example, we consider our guiding example introduced in Section 2 and let

[TABLE]

and $g(x,\mu)=\left\lVert x\right\rVert_{2}^{2}$ (see (12)). The eigenvalues of $A(\mu)$ are $\lambda_{1}(\mu)=4\mu$ with multiplicity $1$ and $\lambda_{2}(\mu)=4(1-\mu)$ with multiplicity $2$.

In the following analytic considerations, we will use the fact that a point $x^{*}\not=0$ is an equilibrium of (10) if and only if $g(x^{*},\mu^{*})$ is an eigenvalue of $A(\mu)$ and $x^{*}$ is a corresponding eigenvector. This follows immediately from the structure of (10). For $g(x)=\left\lVert x\right\rVert_{2}^{2}$ this means that every appropriately scaled eigenvector of a positive eigenvalue of $A(\mu)$ is an equilibrium and vice versa (i.e. every equilibrium is an appropriately scaled eigenvector of $A(\mu)$).

Since [math] is always an equilibrium it will be omitted in the following considerations. By the results in this work, we immediately know how the set of equilibria of (10) changes with respect to $\mu$. Observe that if $x^{*}$ is an equilibrium, then the $\Gamma(A(\mu))$-equivariance implies that $\gamma x^{*}\in\mathbb{R}^{n}$ is an equilibrium for all $\gamma\in\Gamma(A(\mu))$.

• •

$\mu<0$: There is a circle of equilibria induced by the $\mathbf{O}(2)$ equivariance of $f$ (see Example 4.7 (a)).

• •

$\mu=0$: There still is a circle of equilibria and a pitchfork bifurcation occurs (cf. Section 2).

• •

$\mu\in(0,0.5)$: There is a circle of equilibria and two isolated equilibrium points.

• •

$\mu=0.5$: Since $A(0.5)$ possesses the threefold eigenvalue $\lambda_{1}(0.5)=\lambda_{2}(0.5)=2$ the set of equilibria becomes a sphere.

• •

$\mu\in(0.5,1)$: The sphere breaks up and there is again a circle of equilibria and two isolated equilibrium points.

• •

$\mu=1$: A subcritical pitchfork bifurcation occurs, and the circle of equilibria disappears.

• •

$\mu>1$: There are only two equilibria left.

Figure 2 shows the set of equilibria for different values of $\mu$.

6 Other Applications

In addition to the implications for symmetry breaking bifurcation phenomena as illustrated in Section 5, our results have further applications. We illustrate this briefly by the following three examples.

6.1 Two-sided orthogonal Procrustes problem

Given two symmetric matrices $A,B\in\mathbb{R}^{n,n}$, the two-sided orthogonal Procrustes problem can be defined as follows: Find an orthogonal matrix $P\in\mathbf{O}(n)$ such that the cost function $\left\lVert PA-BP\right\rVert_{F}$ is minimized. It is well known – see, for instance, [4, 15, 16] – that an optimal solution is given by $P=V_{B}^{T}V_{A}$, where $D_{A}=V_{A}AV_{A}^{T}$ and $D_{B}=V_{B}BV_{B}^{T}$ are the eigendecompositions of $A$ and $B$, respectively. Note that the eigenvalues in $D_{A}$ and $D_{B}$ both have to be sorted in nonincreasing (or, alternatively, nondecreasing) order. If the cost function is to be maximized, the eigenvalues need to be ordered in opposite order. With the aid of the results from Section 4, we can now characterize all solutions of this form, i.e.

[TABLE]

since for such $P$ we obtain

[TABLE]

which is indeed the optimal solution. Here, we used the invariance of the Frobenius norm under unitary transformations and the equivariance properties.

Remark 6.1**.**

1. (a)

If $D_{A}=D_{B}$, then $\sigma_{B}^{T}\sigma_{A}\in\Sigma_{V_{A}}(A)$ and it suffices to consider matrices of the form $P=V_{B}^{T}\sigma V_{A}$ for $\sigma\in\Sigma_{V_{A}}(A)$ (or, equivalently, $\sigma\in\Sigma_{V_{B}}(B)$). 2. (b)

If, furthermore, all eigenvalues are distinct, then the eigenvectors are determined up to the sign and we obtain the group $\Sigma$ and thus the special case derived in **[4]**. 3. (c)

Since minimizing the Procrustes cost function corresponds to maximizing the cost function $\operatorname{tr}(A^{T}P^{T}BP)$ and vice versa, the results can be extended to the orthogonal relaxation of the quadratic assignment problem (QAP) **[15, 16]**. 4. (d)

Given two undirected graphs $G_{A}$ and $G_{B}$ with adjacency matrices $A$ and $B$, the graphs are isomorphic if they are isospectral and $\Sigma_{A,B}$ contains a permutation matrix, see also **[17]**.

6.2 Graph Symmetries

An isomorphism from a graph to itself is called an automorphism. Let $A$ be the adjacency matrix of an undirected graph $G_{A}$, then the automorphism group (or symmetry group) of $G_{A}$ is defined as

[TABLE]

A graph $G_{A}$ is called asymmetric if $\operatorname{Aut}(G_{A})$ is trivial, i.e. $\operatorname{Aut}(G_{A})=\{\mathrm{Id}\}$. Since $A$ is self-adjoint, we can use Corollary 4.5 to identify the orthogonal commutator $\Gamma(A)$ of $A$. Permutation matrices are orthogonal, hence $\operatorname{Aut}(G_{A})\subseteq\Gamma(A)$.

Our results show that even in the case where the graph $G_{A}$ is asymmetric it typically possesses additional symmetries – namely the elements of the group $\Gamma(A)$. We illustrate this with the following example (cf. [18], Figure 5):

Example 6.2**.**

Consider the graph $G_{A}$ with adjacency matrix $A$ shown in Figure 3. The graph is asymmetric (due to the edge between the vertices $2$ and $6$). The eigenvalues are $(\lambda_{1},\ldots,\lambda_{7})=(-2.24,-1.66,-0.83,0,0.74,1.29,2.70)$ with corresponding multiplicities $m=(1,1,1,2,1,1,1)$. (Here we use the notation from Section 4.2.) By Corollary 4.5, we know that if $V$ is the orthogonal matrix that diagonalizes $A$, then all $\gamma\in\Gamma(A)$ are of the form

[TABLE]

with an arbitrary $Q\in\mathbf{O}(2)$. For instance, we have $\gamma A=A\gamma$ for

[TABLE]

or for

[TABLE]

6.3 Taylor Expansions

Finally, let us briefly mention one implication involving Taylor expansions. In fact, in this context our main result Corollary 4.5 can be used to develop a novel general technique for the construction of higher order stencils for real valued functions of several variables.

Suppose that $f\colon\mathbb{R}^{n}\to\mathbb{R}$ is smooth in a neighborhood of $\bar{x}\in\mathbb{R}^{n}$. In what follows, we use Corollary 4.5 to construct a four-point stencil which provides a second order approximation of evaluations of the fourth-order derivative in $\bar{x}$. For convenience, we write the Taylor expansion of $f$ in $\bar{x}$ as

[TABLE]

where $g_{j}(\bar{x},h)=O(\|h\|^{j})$, $j=3,4,\ldots$, and $H(\bar{x})$ is the Hessian matrix of $f$ at $\bar{x}$.

Corollary 6.3**.**

Denote by $\Gamma(\bar{x})$ the group in (9) corresponding to the Hessian matrix $H(\bar{x})$. Then for all $\gamma\in\Gamma(\bar{x})$ we have

[TABLE]

and therefore for all $\gamma_{1},\gamma_{2}\in\Gamma(\bar{x})$

[TABLE]

In particular, $f(\bar{x}+\gamma_{1}h)+f(\bar{x}-\gamma_{1}h)-f(\bar{x}+\gamma_{2}h)-f(\bar{x}-\gamma_{2}h)=O(\|h\|^{4})$.

Proof.

For $h\in\mathbb{R}^{n}$ and $\gamma_{j}\in\Gamma(\bar{x})$ $(j=1,2)$ we compute

[TABLE]

Therefore, using the fact that $\Gamma(\bar{x})\subset\mathbf{O}(n)$

[TABLE]

and (13), (14) immediately follow. ∎

Obviously, if $\gamma_{1}=\pm\gamma_{2}$ then this result is not useful. However, for all other choices of $\gamma_{j}$ this leads to interesting approximations of the fourth-order derivative as long as $h$ is not an eigenvector of $\gamma_{j}$ ($j=1,2$).

Example 6.4**.**

Let $f\colon\mathbb{R}^{3}\to\mathbb{R}$ be defined by

[TABLE]

We choose $\bar{x}=(1,1,1)^{T}$ and compute

[TABLE]

The choice of

[TABLE]

where $\sigma_{1},\sigma_{2}\in\Sigma_{V}(H(\bar{x}))$ leads to

[TABLE]

For $h=(0.2,0.05,0.1)^{T}$ we obtain

[TABLE]

and for $h=\frac{1}{10}(0.2,0.05,0.1)^{T}$ one computes

[TABLE]

as expected.

Acknowledgements

This work is supported by the Priority Programme SPP 1881 Turbulent Superstructures of the Deutsche Forschungsgemeinschaft. We also thank an anonymous reviewer for important comments on the contents of this article.