Graph Equivalence Classes for Spectral Projector-Based Graph Fourier Transforms

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by  Joya A. Deri

1
Graph Equivalence Classes for Spectral
Projector-Based Graph Fourier Transforms
Joya A. Deri, Member, IEEE, and José M. F. Moura, Fellow, IEEE
Abstract—Wedefineanddiscusstheutilityoftwoequiv- Consider a graph 𝒢 = 𝐺(𝐴) with adjacency matrix
alence graphclasses over whicha spectral projector-based 𝐴∈C𝑁×𝑁 with 𝑘 ≤𝑁 distinct eigenvalues and Jordan
graph Fourier transform is equivalent: isomorphic equiv- decomposition 𝐴 = 𝑉𝐽𝑉−1. The associated Jordan
7 alence classes and Jordan equivalence classes. Isomorphic
1 equivalence classes show that the transform is equivalent subspaces of 𝐴 are J𝑖𝑗, 𝑖 = 1,...𝑘, 𝑗 = 1,...,𝑔𝑖,
0 uptoapermutationonthenodelabels.Jordanequivalence where 𝑔𝑖 is the geometric multiplicity of eigenvalue 𝜆𝑖,
2 classespermitidenticaltransformsovergraphsofnoniden- or the dimension of the kernel of 𝐴−𝜆𝑖𝐼. The signal
n ticaltopologiesandallowabasis-invariantcharacterization space 𝒮 can be uniquely decomposed by the Jordan
of total variation orderings of the spectral components.
a subspaces (see [13], [14] and Section II). For a graph
J Methodstoexploittheseclassestoreducecomputationtime signal 𝑠∈𝒮, the graph Fourier transform (GFT) of [12]
of the transform as well as limitations are discussed.
1 is defined as
1 Index Terms—Jordan decomposition, generalized
eigenspaces, directed graphs, graph equivalence classes, ⨁︁𝑘 ⨁︁𝑔𝑖
] graphisomorphism,signalprocessingongraphs,networks ℱ :𝒮 → J𝑖𝑗
I
S 𝑖=1 𝑗=1
s. 𝑠→(𝑠̂︀11,...,𝑠̂︀1𝑔1,...,𝑠̂︀𝑘1,...,𝑠̂︀𝑘𝑔𝑘), (1)
c
[ I. INTRODUCTION where𝑠𝑖𝑗 isthe(oblique)projectionof𝑠ontotheJordan
subspace 𝐽 parallel to 𝒮∖J . That is, the Fourier
1 Graph signal processing [1], [2] permits applications transform o𝑖f𝑗𝑠, is the unique de𝑖c𝑗omposition
v of digital signal processing concepts to increasingly
4 larger networks. It is based on defining a shift filter, ∑︁𝑘 ∑︁𝑔𝑖
6 𝑠= 𝑠 , 𝑠 ∈J . (2)
for example, the adjacency matrix in [1], [3], [4] to ̂︀𝑖𝑗 ̂︀𝑖𝑗 𝑖𝑗
8
analyze undirected and directed graphs, or the graph 𝑖=1𝑗=1
2
0 Laplacian [2] that applies to undirected graph struc- ThespectralcomponentsaretheJordansubspacesofthe
. tures. The graph Fourier transform is defined through adjacency matrix with this formulation.
1
the eigendecomposition of this shift operator, see these
0 This paper presents graph equivalence classes where
7 references. Further developments have been considered equal GFT projections by (1) are the equivalence re-
1 in [5], [6], [7]. In particular, filter design [1], [5], [8]
lation. First, the transform (1) is invariant to node
: andsampling[9],[10],[11]canbeappliedtoreducethe
v permutations, which we formalize with the concept of
i computational complexity of graph Fourier transforms.
X isomorphic equivalence classes. Furthermore, the GFT
With the objective of simplifying graph Fourier trans-
permits degrees of freedom in graph topologies, which
r formsforlargenetworkapplications,thispaperexplores
a we formalize by defining Jordan equivalence classes, a
methods based on graph equivalence classes to reduce
concept that allows graph Fourier transform computa-
the computation time of the subspace projector-based
tions over graphs of simpler topologies. A frequency-
graphFouriertransformproposedin[12].Thistransform
like ordering based on total variation of the spectral
extendsthegraphsignalprocessingframeworkproposed
componentsisalsopresentedtomotivatelow-pass,high-
by[1],[3],[4]toconsiderspectralanalysisoverdirected
pass, and pass-band graph signals.
graphs with potentially non-diagonalizable (defective)
adjacency matrices. The graph signal processing frame- Section II provides the graph signal processing and
work of [12] allows for a unique, unambiguous signal linear algebra background for the graph Fourier trans-
representation over defective adjacency matrices. form (1). Isomorphic equivalence classes are defined in
SectionIII,andJordanequivalenceclassesaredefinedin
ThisworkwaspartiallysupportedbyNSFgrantsCCF-1011903and SectionIV.TheJordanequivalenceclassesinfluencethe
CCF-1513936andanSYS-CMUgrant. definitionoftotalvariation-basedorderingsoftheJordan
The authors are with the Department of Electrical and Computer
subspaces, which is discussed in detail in Section V.
Engineering,CarnegieMellonUniversity,Pittsburgh,PA15213USA
(email:jderi,[email protected]) Section VI illustrates Jordan equivalence classes and
total variation orderings. Limitations of the method are Jordan decomposition. Let 𝑉 denote the 𝑁 ×𝑟
𝑖𝑗 𝑖𝑗
discussed in Section VII. matrix whose columns form a Jordan chain of eigen-
value 𝜆 that spans Jordan subspace J . Then the
𝑖 𝑖𝑗
II. BACKGROUND eigenvector matrix 𝑉 of 𝐴 is
This section reviews the concepts of graph signal 𝑉 =[︀𝑉 ···𝑉 ··· 𝑉 ···𝑉 ]︀, (9)
processing and the GFT (1). Background on graphs 11 1𝑔1 𝑘1 𝑘𝑔𝑘
signal processing, including definitions of graph signals where 𝑘 is the number of distinct eigenvalues. The
and the graph shift, is described in greater detail in [1], columns of 𝑉 are a Jordan basis of C𝑁. Then 𝐴
[3], [4], [12]. For background on eigendecompositions, has block-diagonal Jordan normal form 𝐽 consisting of
the reader is directed to in [13], [15], [16]. Jordan blocks
⎡ ⎤
𝜆 1
A. Eigendecomposition ⎢⎢ 𝜆 ... ⎥⎥
valCuoesnsi𝜆de,r..m.a,t𝜆rix, 𝐴𝑘 ∈≤C𝑁𝑁.×𝑁Thewiathlge𝑘brdaiisctinmctuletiipgleinc-- 𝐽(𝜆)=⎢⎢⎣ ... 1⎥⎥⎦. (10)
1 𝑘
𝜆
ity 𝑎 of 𝜆 represents the corresponding exponent of
𝑖 𝑖
the characteristic polynomial of 𝐴. Denote by Ker(𝐴) of size 𝑟 . The Jordan normal form 𝐽 of 𝐴 is unique
𝑖𝑗
the kernel or null space of matrix 𝐴. The geometric up to a permutation of the Jordan blocks. The Jordan
multiplicity 𝑔𝑖 of eigenvalue 𝜆𝑖 equals the dimension of decomposition of 𝐴 is 𝐴=𝑉𝐽𝑉−1.
Ker(𝐴−𝜆 𝐼),whichistheeigenspaceof𝜆 where𝐼 is
𝑖 𝑖
the 𝑁×𝑁 identity matrix. The generalized eigenspaces B. Spectral Components
G , 𝑖=1,...,𝑘, of 𝐴 are defined as
𝑖 The spectral components of the Fourier trans-
G =Ker(𝐴−𝜆 𝐼)𝑚𝑖, (3) form (1) are expressed in terms of the eigenvector
𝑖 𝑖
basis 𝑣 ,...,𝑣 and its dual basis 𝑤 ,...,𝑤 since
1 𝑁 1 𝑁
where𝑚𝑖 istheindexofeigenvalue 𝜆𝑖.Thegeneralized the Jordan basis may not be orthogonal. Denote the
eigenspaces uniquely decompose C𝑁 as the direct sum basis and dual basis matrices by 𝑉 = [𝑣 ···𝑣 ] and
1 𝑁
𝑊 =[𝑤 ··· ,𝑤 ]. The dual basis matrix is the inverse
𝑘 1 𝑁
C𝑁 =⨁︁G . (4) Hermitian 𝑊 =𝑉−𝐻 [14], [17].
𝑖
Consider the 𝑗th spectral component of 𝜆
𝑖=1 𝑖
Jordan chains. Let 𝑣1 ∈ Ker(𝐴 − 𝜆𝑖𝐼), 𝑣1 ̸= 0, J𝑖𝑗 =span(𝑣1,···𝑣𝑟𝑖𝑗). (11)
be a proper eigenvector of 𝐴 that generates generalized
The projection matrix onto J parallel to C𝑁∖J is
eigenvectors by the recursion 𝑖𝑗 𝑖𝑗
𝑃 =𝑉 𝑊𝐻, (12)
𝐴𝑣𝑝 =𝜆𝑖𝑣𝑝+𝑣𝑝−1, 𝑝=2,...,𝑟 (5) 𝑖𝑗 𝑖𝑗 𝑖𝑗
where
where 𝑟 is the minimal positive integer such that
(𝐴−𝜆𝑖𝐼)𝑟𝑣𝑟 = 0 and (𝐴−𝜆𝑖𝐼)𝑟−1𝑣𝑟 ̸= 0. A se- 𝑉𝑖𝑗 =[𝑣1···𝑣𝑟𝑖𝑗] (13)
quenceofvectors(𝑣1,...,𝑣𝑟)thatsatisfy(5)isaJordan isthecorrespondingsubmatrixof𝑉 and𝑊𝐻 ∈C𝑟𝑖𝑗×𝑁
𝑖𝑗
chainoflength𝑟 [13].ThevectorsinaJordanchainare is the corresponding submatrix of 𝑊 partitioned as
linearly independent and generate the Jordan subspace
𝑊 =[···𝑊𝐻···𝑊𝐻 ···]𝑇. (14)
J =span(𝑣 ,𝑣 ,...,𝑣 ). (6) 𝑖1 𝑖𝑔𝑖
1 2 𝑟
As shown in [12], the projection of signal 𝑠 ∈ C𝑁
Denote by J𝑖𝑗 the 𝑗th Jordan subspace of 𝜆𝑖 with onto Jordan subspace J can be written as
𝑖𝑗
dimension 𝑟 , 𝑖 = 1,...,𝑘, 𝑗 = 1,...,𝑔 . The Jordan
𝑖𝑗 𝑖
spaces are disjoint and uniquely decompose the general- 𝑠̂︀𝑖𝑗 =𝑠̃︀1𝑣1+···+𝑠̃︀𝑟𝑖𝑗𝑣𝑟𝑖𝑗 (15)
ized eigenspace G𝑖 (3) of 𝜆𝑖 as =𝑉𝑖𝑗𝑊𝑖𝐻𝑗𝑠. (16)
⨁︁𝑔𝑖 The next sections show that invariance of the graph
G = J . (7)
𝑖 𝑖𝑗 Fourier transform (1) is a useful equivalence relation on
𝑗=1
a set of graphs. Equivalence classes with respect to the
ThespaceC𝑁 canbeexpressedastheuniquedecom- GFT are explored in Sections III and IV.
position of Jordan spaces
III. ISOMORPHICEQUIVALENCECLASSES
C𝑁 =⨁︁𝑘 ⨁︁𝑔𝑖 J . (8) ThissectiondemonstratesthatthegraphFouriertrans-
𝑖𝑗
form (1) is invariant up to a permutation of node labels
𝑖=1 𝑗=1
2
andestablishessetsofisomorphicgraphsasequivalence with respect to the invariance of the GFT (1) up to a
classes with respect to invariance of the GFT (1). Two permutationofthegraphsignalandinversepermutation
graphs 𝒢(𝐴) and 𝒢(𝐵) are isomorphic if their adja- of the graph Fourier transform.
cency matrices are similar with respect to a permutation
Theorem 1 establishes an invariance of the GFT over
matrix 𝑇, or 𝐵 = 𝑇𝐴𝑇−1 [18]. The graphs have
graphs that only differ up to a node labeling, and
the same Jordan normal form and the same spectra.
Theorem 2 follows.
Also, if 𝑉 and 𝑉 are eigenvector matrices of 𝐴
𝐴 𝐵 The isomorphic equivalence of graphs is important
and 𝐵, respectively, then 𝑉 = 𝑇𝑉 . We prove that
𝐵 𝐴 since it signifies that the rows and columns of an adja-
the set G𝐼 of all graphs that are isomorphic to 𝒢(𝐴) is
𝐴 cencymatrix canbepermuted toacceleratethe eigende-
an equivalence class over which the GFT is preserved.
composition.Forexample,permutationsofhighlysparse
Thenexttheoremshowsthatanappropriatepermutation
adjacency matrices can convert an arbitrary matrix to
can be imposed on the graph signal and GFT to ensure
nearly diagonal forms, such as with the Cuthill-McKee
invariance of the GFT over all graphs 𝒢 ∈G𝐼.
𝐴 algorithm [19]. Optimizations for such matrices in this
Theorem 1. The graph Fourier transform of a signal form are discussed in [16] and [20], for example. In the
𝑠 is invariant to the choice of graph 𝒢 ∈ G𝐼 up to a next section, the degrees of freedom in graph topology
𝐴
permutationonthegraphsignalandinversepermutation are explored to define another GFT equivalence class.
on the graph Fourier transform.
IV. JORDANEQUIVALENCECLASSES
Proof:For𝒢(𝐴),𝒢(𝐵)∈G𝐼,thereexistsapermu-
𝐴
tation matrix 𝑇 such that 𝐵 =𝑇𝐴𝑇−1. For eigenvector Since the Jordan subspaces of defective adjacency
matrices 𝑉 and 𝑉 of 𝐴 and 𝐵, respectively, let 𝑉 matrices are nontrivial (i.e., they have dimension larger
𝐴 𝐵 𝐴,𝑖𝑗
and 𝑉 denote the 𝑁×𝑟 submatrices of 𝑉 and 𝑉 than one), a degree of freedom exists on the graph
𝐵,𝑖𝑗 𝑖𝑗 𝐴 𝐵
whose columns span the 𝑗th Jordan subspaces J structure so that the graph Fourier transform of a signal
𝐴,𝑖𝑗
andJ ofthe𝑖theigenvalueof𝐴and𝐵,respectively. is equal over multiple graphs of different topologies.
𝐵,𝑖𝑗
Let 𝑊 = 𝑉−𝐻 and 𝑊 = 𝑉−𝐻 denote the matrices ThissectiondefinesJordanequivalenceclassesofgraph
𝐴 𝐴 𝐵 𝐵
whose columns form dual bases of 𝑉 and 𝑉 . Since structures over which the GFT (1) is equal for a given
𝐴 𝐵
𝑉 =𝑇𝑉 , graph signal. The section proves important properties of
𝐵 𝐴
this equivalence class that are used to explore inexact
𝑊 =(𝑇𝑉 )−𝐻 (17)
𝐵 𝐴 methods and real-world applications in [21].
=(𝑉−1𝑇−1)𝐻 (18) The intuition behind Jordan equivalence is presented
𝐴
=𝑇−𝐻𝑉−𝐻 (19) in Section IV-A, and properties of Jordan equivalence
𝐴
are described in Section IV-B. Section IV-C com-
=𝑇𝑊 , (20)
𝐴 pares isomorphic and Jordan equivalent graphs. Sec-
where 𝑇−𝐻 =𝑇 since 𝑇 is a permutation matrix. Thus, tions IV-D, IV-E, IV-F, and IV-G prove properties for
Jordan equivalence classes when adjacency matrices
𝑊𝐻 =𝑊𝐻𝑇𝐻 =𝑊𝐻𝑇−1. (21)
𝐵 𝐴 𝐴 have particular Jordan block structures.
Considergraphsignal𝑠.By(16),thesignalprojection
onto J𝐴,𝑖𝑗 is A. Intuition
𝑠 =𝑉 𝑊𝐻 𝑠. (22) Consider Figure 1, which shows a basis {𝑉} =
̂︀𝐴,𝑖𝑗 𝐴,𝑖𝑗 𝐴,𝑖𝑗
{𝑣 ,𝑣 ,𝑣 } of R3 such that 𝑣 and 𝑣 span a two-
1 2 3 2 3
Permit a permutation 𝑠=𝑇𝑠 on the graph signal. Then dimensional Jordan space J of adjacency matrix 𝐴
the projection of 𝑠 onto J𝐵,𝑖𝑗 is with Jordan decomposition 𝐴=𝑉𝐽𝑉−1. The resulting
̂︀𝑠𝐵,𝑖𝑗 =𝑇𝑉𝐴,𝑖𝑗𝑊𝐴𝐻,𝑖𝑗𝑇−1𝑇𝑠 (23) projection of a signal 𝑠∈R𝑁 as in (16) is unique.
Notethatthedefinitionofthetwo-dimensionalJordan
=𝑇𝑉 𝑊𝐻 𝑠 (24)
𝐴,𝑖𝑗 𝐴,𝑖𝑗 subspace J in Figure 1 is not basis-dependent because
=𝑇𝑠̂︀𝐴,𝑖𝑗 (25) anyspanningset{𝑤2,𝑤3}couldbechosentodefineJ.
Thiscanbevisualizedbyrotating𝑣 and𝑣 onthetwo-
by (22). Therefore, the graph Fourier transform (1) is 2 3
dimensional plane. Any choice {𝑤 ,𝑤 } corresponds to
invariant to a choice among isomorphic graphs up to a 2 3
permutationonthegraphsignalandinversepermutation a new basis 𝑉̃︀. Note that 𝐴̃︀ = 𝑉̃︀𝐽𝑉̃︀−1 does not equal
𝐴=𝑉𝐽𝑉−1 forallchoicesof{𝑤 ,𝑤 };theunderlying
on the Fourier transform. 2 3
graph topologies may be different, or the edge weights
Theorem 2. Consider 𝐴 ∈ C𝑁×𝑁. Then the set G𝐼 maybedifferent.Nevertheless,theirspectralcomponents
𝐴
of graphs isomorphic to 𝒢(𝐴) is an equivalence class (the Jordan subspaces) are identical, and, consequently,
3
Theorem 4. For 𝐴∈C𝑁×𝑁, the set G𝐽 of all graphs
𝐴
that are Jordan equivalent to 𝒢(𝐴) is an equivalence
class with respect to invariance of the GFT (1).
Jordanequivalentgraphshaveadjacencymatriceswith
identical Jordan subspaces and identical Jordan normal
forms.Thisimpliesequivalenceofgraphspectra,proven
in Theorem 5 below.
Theorem 5. Denote by Λ and Λ the sets of eigen-
𝐴 𝐵
values of 𝐴 and 𝐵, respectively. Let 𝒢(𝐴),𝒢(𝐵)∈G𝐽.
Fig. 1: Projections 𝑠 and 𝑠 (shown in red) of a signal 𝐴
̂︀1 ̂︀2 ThenΛ =Λ ;thatis,𝒢(𝐴)and𝒢(𝐵)arecospectral.
𝑠 (shown in black) onto a nontrivial Jordan subspace (span 𝐴 𝐵
of 𝑣1 and 𝑣2) and the span of 𝑣3, respectively, in R3. The Proof:Since𝒢(𝐴)and𝒢(𝐵)areJordanequivalent,
projection onto the nontrivial subspace is invariant to basis
theirJordanformsareequal,sotheirspectra(theunique
choices {𝑣 ,𝑣 } (in blue) or {𝑣 ,𝑣 } (in green).
1 2 ̃︀1 ̃︀2
elements on the diagonal of the Jordan form) are equal.
the spectral projections of a signal onto these compo- OnceaJordandecompositionforanadjacencymatrix
nents are identical; i.e., the GFT (1) is equivalent over is found, it is useful to characterize other graphs in
graphs 𝒢(𝐴) and 𝒢(𝐴̃︀). This observation leads to the the same Jordan equivalence class. To this end, Theo-
definition of Jordan equivalence classes which preserve rem6presentsatransformationthatpreservestheJordan
the GFT (1) as well as the underlying structure captured equivalence class of a graph.
by the Jordan normal form 𝐽 of 𝐴. These classes are Theorem 6. Consider 𝐴,𝐵 ∈ C𝑁×𝑁 with Jordan
formally defined in the next section.
decompositions 𝐴 = 𝑉𝐽𝑉−1 and 𝐵 = 𝑋𝐽𝑋−1 and
eigenvector matrices 𝑉 =[𝑉 ] and 𝑋 =[𝑋 ], respec-
𝑖𝑗 𝑖𝑗
B. Definition and Properties tively.Then,𝒢(𝐵)∈G𝐽 ifandonlyif𝐵haseigenvector
𝐴
This section defines the Jordan equivalence class of matrix 𝑋 = 𝑉𝑌 for block diagonal 𝑌 with invertible
graphs, over which the graph Fourier transform (1) is submatrices 𝑌𝑖𝑗 ∈C𝑟𝑖𝑗×𝑟𝑖𝑗, 𝑖=1,...,𝑘, 𝑗 =1,...,𝑔𝑖.
invariant. We will show that certain Jordan equivalence
Proof: The Jordan normal forms of 𝐴 and 𝐵 are
classes allow the GFT computation to be simplified.
equal. By Definition 3, it remains to show J =J so
𝐴 𝐵
Consider graph 𝒢(𝐴) where 𝐴 has a Jordan chain that 𝒢(𝐵) ∈ G𝐽. The identity J = J must be true
that spans Jordan subspace J𝑖𝑗 of dimension 𝑟𝑖𝑗 > 1. when span{𝑉 }𝐴= span{𝑋 } =𝐴J ,𝐵which implies
𝑖𝑗 𝑖𝑗 𝑖𝑗
Then(15),andconsequently,(16),wouldholdforanon-
that𝑋 representsaninvertiblelineartransformationof
Jordan basis of J ; that is, a basis could be chosen to 𝑖𝑗
𝑖𝑗 the columns of 𝑉 . Thus, 𝑋 = 𝑉 𝑌 , where 𝑌 is
𝑖𝑗 𝑖𝑗 𝑖𝑗 𝑖𝑗 𝑖𝑗
find spectral component 𝑠 such that the basis vectors
̂︀𝑖𝑗 invertible. Defining 𝑌 = diag(𝑌 ,...,𝑌 ,...,𝑌 )
do not form a Jordan chain of 𝐴. This highlights that 11 𝑖𝑗 𝑘,𝑔𝑘
yields 𝑋 =𝑉𝑌.
the Fourier transform (1) is characterized not by the
Jordan basis of 𝐴 but by the set J = {J } of
𝐴 𝑖𝑗 𝑖𝑗
Jordan subspaces spanned by the Jordan chains of 𝐴. C. Jordan Equivalent Graphs vs. Isomorphic Graphs
Thus, graphs with topologies yielding the same Jordan
This section shows that isomorphic graphs do not
subspace decomposition of the signal space have the
imply Jordan equivalence, and vice versa. First it is
same spectral components. Such graphs are termed Jor-
shown that isomorphic graphs have isomorphic Jordan
dan equivalent with the following formal definition.
subspaces.
Definition 3 (Jordan Equivalent Graphs). Consider Lemma 7. Consider graphs 𝒢(𝐴),𝒢(𝐵)∈G𝐼 so that
𝐴
graphs𝒢(𝐴)and𝒢(𝐵)withadjacencymatrices𝐴,𝐵 ∈ 𝐵 = 𝑇𝐴𝑇−1 for a permutation matrix 𝑇. Denote by
C𝑁×𝑁. Then 𝒢(𝐴) and 𝒢(𝐵) are Jordan equivalent
J and J the sets of Jordan subspaces for 𝐴 and 𝐵,
𝐴 𝐵
graphs if all of the following are true: respectively.If{𝑣 ,...,𝑣 }isabasisofJ ∈J ,then
1 𝑟 𝐴 𝐴
1) J𝐴 =J𝐵; and thereexistsJ𝐵 ∈J𝐵 withbasis{𝑥1,...,𝑥𝑟}suchthat
2) 𝐽𝐴 = 𝐽𝐵 (with respect to a fixed permutation of [𝑥1···𝑥𝑟]=𝑇[𝑣1···𝑣𝑟];i.e.,𝐴and𝐵 haveisomorphic
Jordan blocks). Jordan subspaces.
Let G𝐽𝐴 denote the set of graphs that are Jordan Proof: Consider 𝐴 with Jordan decomposition 𝐴=
equivalent to 𝒢(𝐴). Definition 3 and (1) establish that 𝑉𝐽𝑉−1. Since 𝐵 =𝑇𝐴𝑇−1, it follows that
G𝐽 is an equivalence class.
𝐴 𝐵 =𝑇𝑉𝐽𝑉−1𝑇−1 (26)
4
=𝑋𝐽𝑋−1 (27)
where 𝑋 = 𝑇𝑉 represents an eigenvector matrix of
𝐵 that is a permutation of the rows of 𝑉. (It is clear
that the Jordan forms of 𝐴 and 𝐵 are equivalent.) Let
columns𝑣 ,...,𝑣 of𝑉 denoteaJordanchainof𝐴that
1 𝑟
spansJordansubspaceJ .Thecorrespondingcolumns
𝐴
in 𝑋 are 𝑥 ,...,𝑥 and span(𝑥 ,...,𝑥 ) = J . Fig. 2: Jordan equivalent graph structures with unicellular
1 𝑟 1 𝑟 𝐵
Since [𝑥 ··· 𝑥 ] = 𝑇[𝑣 ··· 𝑣 ], J and J are adjacency matrices.
1 𝑟 1 𝑟 𝐴 𝐵
isomorphic subspaces [13].
Theorem8. AgraphisomorphismdoesnotimplyJordan Proof: A counterexample is provided. The top two
equivalence. graphs in Figure 2 correspond to 0/1 adjacency matrices
with a single Jordan subspace J = C𝑁 and eigen-
Proof: Consider 𝒢(𝐴),𝒢(𝐵) ∈ G𝐼 and 𝐵 = value 0; therefore, they are Jordan equivalent. On the
𝐴
𝑇𝐴𝑇−1 for permutation matrix 𝑇. By (27), 𝐽𝐴 = 𝐽𝐵. other hand, they are not isomorphic since the graph on
Toshow𝒢(𝐴),𝒢(𝐵)∈G𝐽,itremainstocheckwhether the right has more edges then the graph on the left.
𝐴
J𝐴 =J𝐵. Theorem8showsthatchangingthegraphnodelabels
By Lemma 7, for any J𝐴 ∈J𝐴, there exists J𝐵 ∈ may change the Jordan subspaces and the Jordan equiv-
J𝐵 that is isomorphic to J𝐴. That is, if 𝑣1,...,𝑣𝑟 alence class of the graph, while Theorem 9 shows that
and 𝑥1,...,𝑥𝑟 are bases of J𝐴 and J𝐵, respectively, a Jordan equivalence class may include graphs with dif-
then [𝑥1···𝑥𝑟] = 𝑇[𝑣1···𝑣𝑟]. Checking J𝐴 = J𝐵 is ferent topologies. Thus, graph isomorphism and Jordan
equivalent to checking equivalence are not identical concepts. Nevertheless, the
isomorphic and Jordan equivalence classes both imply
𝛼 𝑣 +···+𝛼 𝑣 =𝛽 𝑥 +···+𝛽 𝑥 (28)
1 1 𝑟 𝑟 1 1 𝑟 𝑟 invariance of the graph Fourier transform with respect
=𝛽 𝑇𝑣 +···+𝛽 𝑇𝑣 (29)
1 1 𝑟 𝑟 to equivalence relations as stated in Theorems 1 and 4.
Thenexttheoremestablishesanisomorphismbetween
for some coefficients 𝛼 and 𝛽 , 𝑖 = 1,...,𝑟. How-
𝑖 𝑖
Jordan equivalence classes.
ever, (29) does not always hold. Consider matrices 𝐴
and 𝐵 Theorem 10. If 𝐴,𝐵 ∈ C𝑁×𝑁 and 𝒢(𝐴) and 𝒢(𝐵)
[︃2 0 −1]︃ [︃ 1 0 0]︃ are isomorphic, then their respective Jordan equiva-
𝐴= 0 2 −1 ,𝐵 = −1 2 0 . (30) lence classes G𝐽 and G𝐽 are isomorphic; i.e., any
𝐴 𝐵
0 0 1 −1 0 2 graph 𝒢(𝐴′) ∈ G𝐽 is isomorphic to a graph 𝒢(𝐵′) ∈
𝐴
These matrices are similar with respect to a permutation G𝐽.
𝐵
matrix and thus correspond to isomorphic graphs. Their
Proof: Let 𝒢(𝐴) and 𝒢(𝐵) be isomorphic by per-
Jordan normal forms are both
mutation matrix 𝑇 such that 𝐵 = 𝑇𝐴𝑇−1. Consider
[︃1 0 0]︃ 𝒢(𝐴′) ∈ G𝐽, which implies that Jordan normal forms
𝐴
𝐽 = 0 2 0 (31)
𝐽 = 𝐽 and sets of Jordan subspaces J = J by
𝐴′ 𝐴 𝐴′ 𝐴
0 0 2
Definition 3. Denote by 𝐴′ = 𝑉 𝐽 𝑉 the Jordan
𝐴′ 𝐴′ 𝐴′
with possible eigenvector matrices 𝑉 and 𝑉 given by decomposition of 𝐴′. Define 𝐵′ = 𝑇𝐴′𝑇−1. It suffices
𝐴 𝐵
[︃1 1 0]︃ [︃1 0 0]︃ to show 𝒢(𝐵′)∈G𝐽𝐵. First simplify:
𝑉𝐴 = 1 0 1 ,𝑉𝐵 = 1 1 0 . (32) 𝐵′ =𝑇𝐴′𝑇−1 (33)
1 0 0 1 0 1
=𝑇𝑉 𝐽 𝑉−1𝑇−1 (34)
Equation (32) shows that 𝐴 and 𝐵 both have Jordan 𝐴′ 𝐴′ 𝐴′
subspaces J1 = span([1 1 1]𝑇) for 𝜆1 = 1 and =𝑇𝑉𝐴′𝐽𝐴𝑉𝐴−′1𝑇−1 (since 𝒢(𝐴′)∈G𝐽𝐴) (35)
J21 = span([0 1 0]𝑇) for one Jordan subspace of =𝑇𝑉𝐴′𝐽𝐵𝑉𝐴−′1𝑇−1 (since 𝒢(𝐴)∈G𝐼𝐵). (36)
𝜆 = 2. However, the remaining Jordan subspace is
2
From (36), it follows that 𝐽 = 𝐽 . It remains to
span([1 0 0]𝑇) for 𝐴 but span([0 0 1]𝑇) for 𝐵, so (29) 𝐵′ 𝐵
show that J = J . Choose arbitrary Jordan sub-
fails. Thus, 𝒢(𝐴) and 𝒢(𝐵) are not Jordan equivalent. 𝐵′ 𝐵
space J = span{𝑉 } of 𝐴. Then J =
𝐴,𝑖𝑗 𝐴,𝑖𝑗 𝐴′,𝑖𝑗
span{𝑉 } = J since 𝒢(𝐴′) ∈ G𝐽. Then the
ThenexttheoremshowsthatJordanequivalentgraphs 𝐴′,𝑖𝑗 𝐴,𝑖𝑗 𝐴
𝑗th Jordan subspace of eigenvalue 𝜆 for 𝐵 is
may not be isomorphic. 𝑖
J =span{𝑇𝑉 } (37)
Theorem 9. Jordan equivalence does not imply the 𝐵,𝑖𝑗 𝐴,𝑖𝑗
=𝑇span{𝑉 }. (38)
existence of a graph isomorphism. 𝐴,𝑖𝑗
5
For the 𝑗th Jordan subspace of eigenvalue 𝜆 for 𝐵′, it Proof: Since the Jordan subspaces of a diagonal-
𝑖
follows from (36) that izable matrix are one-dimensional, the possible choices
of Jordan basis are limited to nonzero scalar multiples
J =span{𝑇𝑉 } (39)
𝐵′,𝑖𝑗 𝐴′,𝑖𝑗 of the eigenvectors. Then, given eigenvector matrix 𝑉
=𝑇span{𝑉𝐴′,𝑖𝑗} (40) of 𝐴, all possible eigenvector matrices of 𝐴 are given
=𝑇span{𝑉 } (since 𝒢(𝐴′)∈G𝐽) by𝑋 =𝑉𝑈,where𝑈 isadiagonalmatrixwithnonzero
𝐴,𝑖𝑗 𝐴
(41) diagonal entries. Let 𝐵 = 𝑋𝐽𝑋−1, where 𝐽 is the
=J . (by (38)) (42) diagonal canonical Jordan form of 𝐴. Since 𝑈 and 𝐽
𝐵,𝑖𝑗
are both diagonal, they commute, yielding
Since (42) holds for all 𝑖 and 𝑗, the sets of Jordan
subspaces J = J . Therefore, 𝒢(𝐵′) and 𝒢(𝐵) are 𝐵 =𝑋𝐽𝑋−1 (43)
𝐵′ 𝐵
Jordan equivalent, which proves the theorem. =𝑉𝑈𝐽𝑈−1𝑉−1 (44)
Theorem10showsthattheJordanequivalenceclasses =𝑉𝐽𝑈𝑈−1𝑉−1 (45)
of two isomorphic graphs are also isomorphic. This
=𝑉𝐽𝑉−1 (46)
result permits an frequency ordering on the spectral
components of a matrix 𝐴 that is invariant to both the =𝐴. (47)
choice of graph in G𝐽 and the choice of node labels, as
𝐴 Thus, a graph with a diagonalizable adjacency matrix is
demonstrated in Section V.
theoneandonlyelementinitsJordanequivalenceclass.
Relation to matrices with the same set of invariant
subspaces. Let GInv denote the set of all matrices
𝐴 When a matrix has nondefective but repeated eigen-
with the same set of invariant subspaces of 𝐴; i.e.,
values, there are infinitely many choices of eigenvec-
𝒢(𝐵) ∈ GInv if and only if Inv(𝐴) = Inv(𝐵). The
𝐴 tors [16]. An illustrative example is the identity matrix,
next theorem shows that GInv is a proper subset of the
𝐴 which has a single eigenvalue but is diagonalizable.
Jordan equivalence class G𝐽 of 𝐴.
𝐴 Since it has infinitely many choices of eigenvectors, the
Theorem 11. For 𝐴∈C𝑁×𝑁, GInv ⊂G𝐽. identity matrix corresponds to infinitely many Jordan
𝐴 𝐴
equivalenceclasses.ByTheorem12,eachoftheseequiv-
Proof: If 𝒢(𝐵) ∈ GInv, then the set of Jordan
𝐴 alenceclasseshavesizeone.Thisobservationhighlights
subspaces are equal, or J =J .
𝐴 𝐵 that the definition of a Jordan equivalence class requires
Theorem 11 sets the results of this chapter apart
a choice of basis.
from analyses such as those in Chapter 10 of [15],
which describes structures for matrices with the same E. One Jordan Block
invariant spaces, and [22], which describes the eigen-
Consider matrix 𝐴 with Jordan decomposition 𝐴 =
decomposition of the discrete Fourier transform matrix
𝑉𝐽𝑉−1 where 𝐽 is a single Jordan block and 𝑉 =
in terms of projections onto invariant spaces. The Jor-
[𝑣 ···𝑣 ] is an eigenvector matrix. Then 𝐴 is a repre-
1 𝑁
dan equivalence class relaxes the assumption that all sentation of a unicellular transformation 𝑇 :C𝑁 →C𝑁
invariant subspaces of two adjacency matrices must be
with respect to Jordan basis 𝑣 ,...𝑣 (see [15, Sec-
1 𝑁
equal. This translates to more degrees of freedom in the
tion 2.5]). In this case the set of Jordan subspaces has
graph topology. The following sections present results one element J =C𝑁. Properties of unicellular Jordan
for adjacency matrices with diagonal Jordan forms, one
equivalence classes are demonstrated next.
Jordan block, and multiple Jordan blocks.
Theorem 13. Let𝒢(𝐴)beanelementoftheunicellular
D. Diagonalizable Matrices JordanequivalenceclassG𝐽.Thenallgraphfilters𝐻 ∈
𝐴
If the canonical Jordan form 𝐽 of 𝐴 is diagonal (𝐴 G𝐽 are all-pass.
𝐴
is diagonalizable), then there are no Jordan chains and
Proof: Since 𝐴 is unicellular, it has a single Jordan
the set of Jordan subspaces J = {J }𝑁 where
𝐴 𝑝 𝑝=1 chain𝑣 ,...,𝑣 oflength𝑁.Consideragraphsignal𝑠
J = span(𝑣 ) and 𝑣 is the 𝑝th eigenvector of 𝐴. 1 𝑁
𝑝 𝑝 𝑝 over graph 𝒢(𝐴), and let 𝑠 represent the coordinate
̃︀
Graphs with diagonalizable adjacency matrices include vector of 𝑠 in terms of the basis {𝑣 }𝑁 . Then the
𝑖 𝑖=1
undirected graphs, directed cycles, and other digraphs
spectral decomposition of signal 𝑠 is given by
withnormaladjacencymatricessuchasnormallyregular
digraphs [23]. A graph with a diagonalizable adjacency 𝑠=𝑠̃︀1𝑣1+···𝑠̃︀𝑁𝑣𝑁 =𝑠̂︀; (48)
matrixisJordanequivalentonlytoitself,asprovennext.
that is, the unique projection of 𝑠 onto the spectral
Theorem 12. A graph 𝒢(𝐴) with diagonalizable adja- component J = C𝑁 is itself. Therefore, 𝒢(𝐴) acts as
cency matrix 𝐴 ∈ C𝑁×𝑁 belongs to a Jordan equiva- an all-pass filter. Moreover, (48) holds for all graphs in
lence class of size one. Jordan equivalence class G𝐽.
𝐴
6
Inadditiontotheall-passpropertyofunicellulargraph where Inv(·) represents the set of invariant spaces of a
filters, unicellular isomorphic graphs are also Jordan matrix). If 𝑎 = 𝜆, Definition 3 can be applied, which
equivalent, as proven next. yields 𝒢(𝐴)∈G𝐽.
𝐽
On the other hand, consider a unicellular matrix 𝐵
Theorem 14. Let 𝒢(𝐴),𝒢(𝐵) ∈ G𝐼 where 𝐴 is a
𝐴 suchthatitseigenvectorisnotinthespanofacanonical
unicellular matrix. Then 𝒢(𝐴),𝒢(𝐵)∈G𝐽.
𝐴 vector, e.g.,
Proof: Since 𝒢(𝐴) and 𝒢(𝐵) are isomorphic, Jor- ⎡1 −1 1 1 ⎤
dan normal forms 𝐽 = 𝐽 . Therefore, 𝐵 is also 2 2 2 2
𝐴 𝐵
unicellular, so J𝐴 = J𝐵 = {C𝑁}. By Definition 3, 𝐵 =⎢⎢12 −12 −12 −21⎥⎥ (49)
𝒢(𝐴),𝒢(𝐵)∈G𝐽𝐴. ⎢⎣0 0 12 −21⎥⎦
The dual basis of 𝑉 can also be used to construct 0 0 1 −1
2 2
graphs in the Jordan equivalence class of unicellular 𝐴.
with Jordan normal form 𝐽(0). Since the span of
Theorem 15. Denote by 𝑉 an eigenvector matrix of the eigenvectors of 𝐽(0) and 𝐵 are not identical,
unicellular𝐴∈C𝑁×𝑁 and𝑊 =𝑉−𝐻 isthedualbasis. Inv(𝐽(0))̸=Inv(𝐵).However,byDefinition3,𝒢(𝐵)is
Consider decompositions 𝐴 = 𝑉𝐽𝑉−1 and 𝐴𝑊 = inthesameclassofunicellularJordanequivalentgraphs
𝑊𝐽𝑊−1. Then 𝒢(𝐴𝑊)∈G𝐽𝐴. as those of Figure 2, i.e., 𝒢(𝐵) ∈ G𝐽𝐽. In other words,
formatrices𝐴and𝐵withthesameJordannormalforms
Proof: Matrices 𝐴 and 𝐴 have the same Jordan
𝑊
(𝐽 =𝐽 ),Jordanequivalence,i.e.,J =J ,isamore
normalformbydefinition.SincethereisonlyoneJordan 𝐴 𝐵 𝐴 𝐵
block, both matrices have a single Jordan subspace C𝑁. generalconditionthanInv(𝐴)=Inv(𝐵).Thisillustrates
thatgraphshavingadjacencymatriceswithequalJordan
ByDefinition3,𝒢(𝐴 )and𝒢(𝐴)areJordanequivalent.
𝑊
normalformsandthesamesetsofinvariantspacesform
a proper subset of a Jordan equivalence class, as shown
The next theorem characterizes the special case of
above in Theorem 11.
graphsintheJordanequivalenceclassthatcontains𝒢(𝐽)
Remark on topology. Note that replacing each
with adjacency matrix equal to Jordan block 𝐽 =𝐽(𝜆).
nonzero element of (49) with a unit entry results in a
Theorem16. Denoteby𝐽 =𝐽(𝜆)isthe𝑁×𝑁 Jordan matrix that is not unicellular. Therefore, its correspond-
block (10) for eigenvalue 𝜆. Then 𝒢(𝐴) ∈ G𝐽 if 𝐴 ∈ inggraphisnotinaunicellularJordanequivalenceclass.
𝐽
C𝑁×𝑁 is upper triangular with diagonal entries 𝜆 and This observation demonstrates that topology may not
nonzero entries on the first off-diagonal. determine the Jordan equivalence class of a graph.
Proof: Consider upper triangular matrix 𝐴 =
F. Two Jordan Blocks
[𝑎 ] with diagonal entries 𝑎 = ··· = 𝑎 and
𝑖𝑗 11 𝑁𝑁
nonzero elements on the first off-diagonal. By [15, Consider 𝑁 × 𝑁 matrix 𝐴 with Jordan normal
Example 10.2.1], 𝐴 has the same invariant subspaces form consisting of two Jordan subspaces J1 =
as 𝐽 = 𝐽(𝜆), which implies J𝐽 = J𝐴 = {C𝑁}. span(𝑣1,...,𝑣𝑟1) and J2 = span(𝑣𝑟1+1,...,𝑣𝑟2) of
Therefore, the Jordan normal form of 𝐴 is the Jordan dimensions 𝑟1 >1 and 𝑟2 =𝑁 −𝑟1 and corresponding
block𝐽𝐴 =𝐽(𝑎11).Restrictthediagonalentriesof𝐴to eigenvalues𝜆1 and𝜆2,respectively.Thespectraldecom-
𝜆so𝐽 =𝐽.Then,𝒢(𝐽),𝒢(𝐴)∈G𝐽 byDefinition(3). position of signal 𝑠 over 𝒢(𝐴) yields
𝐴 𝐽
𝑠=𝑠 𝑣 +···+𝑠 𝑣 +𝑠 𝑣 +···+𝑠 𝑣
Figure 2 shows graph structures that are in the same ⏟̃︀1 1 ⏞ ̃︀𝑟1 𝑟 1 ⏟̃︀𝑟1+1 𝑟1+1 ⏞ ̃︀𝑁 𝑁
unicellular Jordan equivalence class by Theorem 16. In 𝑠̂︀1 𝑠̂︀2
(50)
addition,thetheoremimpliesthatitissufficienttodeter-
minetheGFTofunicellular𝐴byreplacing𝒢(𝐴)∈G𝐽 =𝑠̂︀1+𝑠̂︀2. (51)
𝐽
with𝒢(𝐽),where𝐽 isasingle𝑁×𝑁 Jordanblock.That
Spectral components 𝑠 and 𝑠 are the unique pro-
̂︀1 ̂︀2
is,withoutlossofgenerality,𝒢(𝐴)canbereplacedwith
jections of 𝑠 onto the respective Jordan subspaces. By
a directed chain graph with possible self-edges and the
Example 6.5.4 in [13], a Jordan basis matrix 𝑋 can be
eigenvector matrix 𝑉 = 𝐼 chosen to compute the GFT chosen for 𝐴 = 𝑉𝐽𝑉−1 such that 𝑋 = 𝑉𝑈, where 𝑈
of a graph signal.
commutes with 𝐽 and has a particular form as follows.
Remarkoninvariantspaces.Example10.2.1of[15]
If 𝜆 ̸= 𝜆 , then 𝑈 = diag(𝑈 ,𝑈 ), where 𝑈 ,
shows that a matrix 𝐴∈C𝑁×𝑁 having upper triangular 1 2 1 2 𝑖
𝑖 = 1,2, is an 𝑟 ×𝑟 upper triangular Toeplitz matrix;
𝑖 𝑖
entries with constant diagonal entries 𝑎 and nonzero
otherwise, 𝑈 has form
entries on the first off-diagonal is both necessary and
[︃ ]︃
sufficient for 𝐴 to have the same invariant subspaces as 𝑈 =diag(𝑈 ,𝑈 )+ 0 𝑈12 (52)
𝑁 ×𝑁 Jordan block 𝐽 =𝐽(𝜆) (i.e., Inv(𝐽)=Inv(𝐴), 1 2 𝑈 0
21
7
where 𝑈 is an 𝑟 ×𝑟 upper triangular Toeplitz matrix 𝐽 (𝜆) is the 𝑟 ×𝑟 Jordan block for eigenvalue 𝜆.
𝑖 𝑖 𝑖 𝑟𝑖 𝑖 𝑖
and 𝑈 and 𝑈 are extended upper triangular Toeplitz
12 21 Proof: By Lemma 10.3.3 in [15], 𝐴 with structure
matrices as in Theorem 12.4.1 in [13]. Thus, all Jordan
as described in the theorem have the same invariant
bases of 𝐴 can be obtained by transforming eigenvector
subspaces as 𝐽 = diag(𝐽 (𝜆),𝐽 (𝜆)). Therefore, 𝐴
matrix 𝑉 as 𝑋 =𝑉𝑈. 𝑟1 𝑟2
and 𝐽 have the same Jordan normal form and Jordan
A corresponding theorem to Theorem 16 is presented
subspaces and so are Jordan equivalent.
tocharacterizeJordanequivalentclasseswhentheJordan
Theorems 17 and 18 demonstrate two types of Jordan
form consists of two Jordan blocks. The reader is di-
equivalences that arise from block diagonal matrices
rectedtoSections10.2and10.3in[15]formoredetails.
with submatrices of form (53) and (54). These theo-
Thefollowingdefinitionsareneeded.Denote𝑝×𝑝upper
rems imply that computing the GFT (1) over the block
triangular Toeplitz matrices 𝑇 (𝑏 ,...,𝑏 ) of form
𝑟2 1 𝑟2 diagonal matrices can be simplified to computing the
⎡ ⎤
𝑏 𝑏 ··· 𝑏 𝑏 transform over the adjacency matrix of a union of
1 2 𝑝−1 𝑝
⎢⎢0 𝑏 ... 𝑏 𝑏 ⎥⎥ directed chain graphs. That is, the canonical basis can
⎢ 1 𝑝−2 𝑝−1⎥ be chosen for 𝑉 without loss of generality.
𝑇𝑝(𝑏1,...,𝑏𝑝)=⎢⎢ ... ... ... ... ... ⎥⎥, (53) As for the case of unicellular transformations, it is
⎢ ⎥
⎢ ⎥ possible to pick bases of J and J that do not form
⎣0 0 ··· 𝑏1 𝑏2 ⎦ 1 2
a Jordan basis of 𝐴. Any two such choices of bases are
0 0 ··· 0 𝑏
1 relatedbyTheorem6.Concretely,if𝑉 istheeigenvector
anddefine𝑞×𝑞 uppertriangularmatrixforsome𝑞 >𝑝 matrixof𝐴and𝑋 isthematrixcorrespondingtoanother
choice of basis, then Theorem 6 states that a transfor-
𝑅 (𝑏 ,...,𝑏 ;𝐹)=
𝑞 1 𝑝 mation matrix 𝑌 can be found such that 𝑋 = 𝑉𝑌,
⎡𝑏 ··· 𝑏 𝑓 𝑓 ···𝑓 𝑓 ⎤
1 𝑝 11 12 1,𝑞−𝑝−1 1,𝑞−𝑝 where 𝑌 is partitioned as 𝑌 = diag(𝑌 ,𝑌 ) with full-
0 𝑏 ··· 𝑏 𝑓 ···𝑓 𝑓 1 2
⎢⎢⎢ ... ...1 ... 𝑝 .2.2. 2,...𝑞−𝑝−1 2,𝑞...−𝑝 ⎥⎥⎥ rank submatrices 𝑌𝑖 ∈C𝑟𝑖×𝑟𝑖, 𝑖=1,2.
⎢0 0 ··· 𝑏 𝑓 ⎥
⎢ 𝑝 𝑞−𝑝,𝑞−𝑝⎥ (54)
⎢0 0 ··· 𝑏 𝑏 ⎥
⎢ 𝑝−1 𝑝 ⎥ G. Multiple Jordan Blocks
⎢ .. .. .. .. .. ⎥
⎢ . . . . . ⎥
⎣ ⎦ This section briefly describes a special case of Jordan
0 0 ··· 0 𝑏 𝑏
0 0 ··· 0 01 𝑏2 equivalence classes whose graphs have adjacency matri-
1
ces 𝐴∈C𝑁×𝑁 with 𝑝 Jordan blocks, 1<𝑝<𝑁.
where 𝐹 =[𝑓 ] is a (𝑞−𝑝)×(𝑞−𝑝) upper triangular
𝑖𝑗 Consider matrix 𝐴 with Jordan normal form 𝐽 com-
matrix and 𝑏 ∈ C, 𝑖 = 1,...,𝑝. The theorems are
𝑖 prised of 𝑝 Jordan blocks and eigenvalues 𝜆 ,...,𝜆 .
1 𝑘
presented below.
By Theorem 10.2.1 in [15], there exists an upper tri-
Theorem 17. Consider 𝐴 = diag(𝐴 ,𝐴 ) where each angular 𝐴 with Jordan decomposition 𝐴 = 𝑉𝐽𝑉−1
1 2
matrix 𝐴 , 𝑖 = 1,2, is upper triangular with diag- such that 𝒢(𝐴) ∈ G𝐽. Note that the elements in the
𝑖 𝐽
onal elements 𝜆 and nonzero elements on the first Jordan equivalence class G𝐽 of 𝒢(𝐽) are useful since
𝑖 𝐽
off-diagonal. Let 𝜆 ̸= 𝜆 . Then 𝒢(𝐴) is Jordan signals over a graph in this class can be computed
1 2
equivalent to the graph with adjacency matrix 𝐽 = with respect to the canonical basis with eigenvector
diag(𝐽 (𝜆 ),𝐽 (𝜆 )) where 𝐽 (𝜆 ) is the 𝑟 × 𝑟 matrix 𝑉 = 𝐼. Theorem 19 characterizes the possible
𝑟1 1 𝑟2 2 𝑟𝑖 𝑖 𝑖 𝑖
Jordan block for eigenvalue 𝜆 . eigenvector matrices 𝑉 such that 𝐴 = 𝑉𝐽𝑉−1 allows
𝑖
𝒢(𝐴)∈G𝐽.
Proof: By Theorem 16, 𝒢(𝐴 ) and 𝒢(𝐽 (𝜆 )) are 𝐽
𝑖 𝑟𝑖 𝑖
Jordan equivalent for 𝑖 = 1,2 and 𝐴 upper triangular Theorem 19. Let 𝐴 = 𝑉𝐽𝑉−1 be the Jordan decom-
𝑖
with nonzero elements on the first off-diagonal. There- position of 𝐴 ∈ C𝑁×𝑁 and 𝒢(𝐴) ∈ G𝐽. Then 𝑉 must
𝐽
fore, the Jordan normal forms of 𝐽 and 𝐴 are the same. be an invertible block diagonal matrix.
Moreover, the set of irreducible subspaces of 𝐽 is the
Proof: Consider 𝒢(𝐽) with eigenvector matrix 𝐼.
unionoftheirreduciblesubspacesof[𝐽 0]𝑇 and[0𝐽 ]𝑇,
1 2 By Theorem 6, 𝒢(𝐴)∈G𝐽 implies
which are the same as the irreducible subspaces of 𝐽
[𝐴1 0]𝑇 and [0 𝐴2]𝑇, respectively. Therefore, J𝐴 =J𝐽, 𝑉 =𝐼𝑌 =𝑌 (55)
so 𝒢(𝐴) and 𝒢(𝐽) are Jordan equivalent.
where 𝑌 is an invertible block diagonal matrix.
Theorem 18. Consider 𝐴=diag(𝐴 ,𝐴 ) where 𝐴 =
1 2 1 Thestructureof𝑉 giveninTheorem19allowsachar-
𝑈𝑟𝑟1≥(𝜆𝑟,𝑏.1,T.h.e.n,𝑏𝒢𝑟2(,𝐴𝐹))isanJodrd𝐴an2 e=quiv𝑇a𝑟l2e(n𝜆t,t𝑏o1,th.e..g,𝑏ra𝑟2p)h, acterizationofgraphsintheJordanequivalenceclassG𝐽𝐽
1 2 with the dual basis of 𝑉 as proved in Theorem 20.
with adjacency matrix 𝐽 = diag(𝐽 (𝜆),𝐽 (𝜆)) where
𝑟1 𝑟2
8
Theorem 20. Let 𝒢(𝐴) ∈ G𝐽, where 𝐴 has Jordan can be chosen without loss of generality.
𝐽
decomposition 𝐴=𝑉𝐽𝑉−1 and 𝑊 =𝑉−𝐻 is the dual Jordan equivalence classes show that the GFT (1)
basis of 𝑉. If 𝐴 =𝑊𝐽𝑊−1, then 𝒢(𝐴 )∈G𝐽. permits degrees of freedom in graph topologies. This
𝑊 𝑊 𝐽
has ramifications for total variation-based orderings of
Proof: By Theorem 19, 𝑉 is block diagonal with
thespectralcomponents,asdiscussedinthenextsection.
invertible submatrices 𝑉 . Thus, 𝑊 = 𝑉−𝐻 is block
𝑖
diagonal with submatrices 𝑊 = 𝑉−𝐻. By Theorem 6,
𝑖 𝑖 V. FREQUENCYORDERINGOFSPECTRAL
𝑊 is an appropriate eigenvector matrix such that, for
COMPONENTS
𝐴 =𝑊𝐽𝑊−1, 𝒢(𝐴 )∈G𝐽.
𝑊 𝑊 𝐽 Thissectiondefinesamappingofspectralcomponents
Relation to graph topology. Certain types of ma-
to the real line to achieve an ordering of the spectral
trices have Jordan forms that can be deduced from
components. This ordering can be used to distinguish
their graph structure. For example, [24] and [25] relate
generalizedlowandhighfrequenciesasin[4].Anupper
the Jordan blocks of certain adjacency matrices to a
bound for a total-variation based mapping of a spectral
decomposition of their graph structures into unions of
component (Jordan subspace) is derived and generalized
cycles and chains. Applications where such graphs are
to Jordan equivalence classes.
inusewouldallowapractitionertodeterminetheJordan The graph total variation of a graph signal 𝑠 ∈ C𝑁
equivalence classes (assuming the eigenvalues can be
is defined as [4]
computed) and potentially choose a different matrix in
the class for which the GFT can be computed more TV (𝑠)=‖𝑠−𝐴𝑠‖ . (56)
𝐺 1
easily. Sections IV-E and IV-G show that working with
Matrix 𝐴 can be replaced by 𝐴norm = 1 𝐴 when
unicellular matrices and matrices in Jordan normal form |𝜆max|
the maximum eigenvalue satisfies |𝜆 |>0.
permits the choice of the canonical basis. In this way, max
Equation (56) can be generalized to define the total
for matrices with Jordan blocks of size greater than
variation of the Jordan subspaces of the graph shift 𝐴
one, finding a spanning set for each Jordan subspace
as described in [12]. Choose a Jordan basis of 𝐴 so
may be more efficient than attempting to compute the
that𝑉 istheeigenvectormatrixof𝐴,i.e.,𝐴=𝑉𝐽𝑉−1,
Jordan chains. Nevertheless, relying on graph topology
where 𝐽 is the Jordan form of 𝐴. Partition 𝑉 into
is not always possible. Such an example was presented
𝑁 ×𝑟 submatrices 𝑉 whose columns are a Jordan
in Section IV-E with adjacency matrix (49). 𝑖𝑗 𝑖𝑗
chain of (and thus span) the 𝑗th Jordan subspace J
𝑖𝑗
Relation to algebraic signal processing. The emer-
of eigenvalue 𝜆 , 𝑖=1,...,𝑘 ≤𝑁, 𝑗 =1,...,𝑔 . Then
𝑖 𝑖
gence of Jordan equivalence from the graph Fourier
the (graph) total variation of 𝑉 is defined as [12]
𝑖𝑗
transform(1)isrelatedtoalgebraicsignalprocessingand
thesignalmodel(𝒜,ℳ,Φ),where𝒜isasignalalgebra TV𝐺(𝑉𝑖𝑗)=‖𝑉𝑖𝑗 −𝐴𝑉𝑖𝑗‖1, (57)
corresponding to the filter space, ℳ is a module of 𝒜
where‖·‖ representstheinducedL1matrixnorm(equal
corresponding to the signal space, and Φ : 𝑉 → ℳ 1
to the maximum absolute column sum).
is a bijective linear mapping that generalizes the 𝑧-
Theorem 21 shows that the graph total variation of
transform [26], [27]. We emphasize that the GFT (1) is
a spectral component is invariant to a relabeling of
tied to a basis. This is most readily seen by considering
the graph nodes; that is, the total variations of the
diagonaladjacencymatrix𝐴=𝜆𝐼,whereanybasisthat
spectral components for graphs in the same isomorphic
spansC𝑁 definestheeigenvectors(theJordansubspaces
equivalence class as defined in Section III are equal.
and spectral components) of a graph signal; that is,
a matrix, even a diagonalizable matrix, may not have Theorem 21. Let 𝐴,𝐵 ∈C𝑁×𝑁 and 𝒢(𝐵)∈G𝐼, i.e.,
𝐴
distinctspectralcomponents.Similarly,thesignalmodel 𝒢(𝐵) is isomorphic to 𝒢(𝐴). Let 𝑉𝐴,𝑖𝑗 ∈ C𝑁×𝑟𝑖𝑗 be
(𝒜,ℳ,Φ) requires a choice of basis for module (signal a Jordan chain of matrix 𝐴 and 𝑉𝐵,𝑖𝑗 ∈ C𝑁×𝑟𝑖𝑗 the
space) ℳ in order to define the frequency response corresponding Jordan chain of 𝐵. Then
(irreducible representation) of a signal [26]. On the
TV (𝑉 )=TV (𝑉 ). (58)
other hand, this section demonstrated the equivalence of 𝐺 𝐴,𝑖𝑗 𝐺 𝐵,𝑖𝑗
the GFT (1) over graphs in Jordan equivalence classes, Proof: Since 𝒢(𝐴) and 𝒢(𝐵) are isomorphic, there
which implies an equivalence of certain bases. This exists a permutation matrix 𝑇 such that 𝐵 = 𝑇𝐴𝑇−1
observation suggests the concept of equivalent signal and the eigenvector matrices 𝑉 and 𝑉 of 𝐴 and 𝐵,
𝐴 𝐵
models in the algebraic signal processing framework. respectively,arerelatedby𝑉 =𝑇𝑉 .Thus,theJordan
𝐵 𝐴
Just as working with graphs that are Jordan equivalent chains are related by 𝑉 =𝑇𝑉 . By (57),
𝐵,𝑖𝑗 𝐴,𝑖𝑗
to those with adjacency matrices in Jordan normal form
TV(𝑉 )=‖𝑉 −𝐵𝑉 ‖ (59)
simplifies GFT computation, we expect similar classes 𝐵 𝐵,𝑖𝑗 𝐵,𝑖𝑗 1
ofequivalentsignalmodelsforwhichthecanonicalbasis =⃦⃦𝑇𝑉𝐴,𝑖𝑗 −(︀𝑇𝐴𝑇−1)︀𝑇𝑉𝐴,𝑖𝑗⃦⃦1 (60)
9
=‖𝑇𝑉 −𝑇𝐴𝑉 ‖ (61) composition 𝐵 = 𝑉𝐽𝑉−1. Let the columns of eigen-
𝐴,𝑖𝑗 𝐴,𝑖𝑗 1
=‖𝑇 (𝑉𝐴,𝑖𝑗 −𝐴𝑉𝐴,𝑖𝑗)‖1 (62) vector submatrix 𝑉𝑖𝑗 span the Jordan subspace J𝑖𝑗 of
𝐴. Then the class total variation of spectral component
=‖𝑉 −𝐴𝑉 ‖ (63)
𝐴,𝑖𝑗 𝐴,𝑖𝑗 1 J is defined as the supremum of the graph total
𝑖𝑗
=TV(𝑉𝐴), (64) variation of 𝑉𝑖𝑗 over the Jordan equivalence class (for
all 𝒢(𝐵)∈G𝐽):
where(63)holdsbecausethemaximumabsolutecolumn 𝐴
sumofamatrixisinvarianttoapermutationonitsrows. TV (J )= sup TV (𝑉 ). (68)
G𝐽 𝑖𝑗 𝐺 𝑖𝑗
𝐴 𝒢(𝐵)∈G𝐽
𝐴
Theorem 21 shows that the graph total variation is 𝐵=𝑉𝐽𝑉−1
invariant to a node relabeling, which implies that an span{𝑉𝑖𝑗}=J𝑖𝑗
‖𝑉𝑖𝑗‖1=1
ordering of the total variations of the frequency com-
Theorem 23. Let 𝐴,𝐵 ∈C𝑁×𝑁 and 𝒢(𝐵)∈G𝐼. Let
ponents is also invariant. 𝐴
𝑉 and 𝑉 be the respective eigenvector matrices with
Reference [12] demonstrates that each eigenvector 𝐴 𝐵
Jordan subspaces J = span{𝑉 } and J =
submatrix corresponding to a Jordan chain can be nor- 𝐴,𝑖𝑗 𝐴,𝑖𝑗 𝐵,𝑖𝑗
span{𝑉 } spanned by the 𝑗th Jordan chain of eigen-
malized. This is stated as a property below: 𝐵,𝑖𝑗
value 𝜆 . Then TV (J )=TV (J ).
𝑖 G𝐽 𝐴,𝑖𝑗 G𝐽 𝐵,𝑖𝑗
Property 22. The eigenvector matrix 𝑉 of adjacency 𝐴 𝐵
matrix 𝐴 ∈ C𝑁×𝑁 can be chosen so that each Jordan Proof: Let 𝑉𝐴* denote the eigenvector matrix corre-
sponding to 𝒢(𝐴*)∈G𝐽 that maximizes the class total
chain represented by the eigenvector submatrix 𝑉 ∈ 𝐴
C𝑁×𝑟𝑖𝑗 satisfies ‖𝑉𝑖𝑗‖1 = 1; i.e., ‖𝑉‖1 = 1 wit𝑖h𝑗out variation of Jordan subspace J𝐴,𝑖𝑗; i.e.,
loss of generality. TV (J )=TV (︀𝑉* )︀. (69)
G𝐽 𝐴,𝑖𝑗 𝐺 𝐴,𝑖𝑗
𝐴
Itisassumedthattheeigenvectormatricesarenormal- Similarly, let 𝑉* denote the eigenvector matrix corre-
𝐵
ized as in Property 22 for the remainder of the section. sponding to 𝒢(𝐵*)∈G𝐽 that maximizes the class total
𝐵
Furthermore, [12] shows that (57) can be written as variation of Jordan subspace J , or
𝐵,𝑖𝑗
TV𝐺(𝑉𝑖𝑗)=⃦⃦𝑉𝑖𝑗(︀𝐼𝑟𝑖𝑗 −𝐽𝑖𝑗)︀⃦⃦1 (65) TVG𝐽 (J𝐵,𝑖𝑗)=TV𝐺(︀𝑉𝐵*,𝑖𝑗)︀. (70)
𝐵
= max {|1−𝜆|‖𝑣 ‖ ,‖(1−𝜆)𝑣 −𝑣 ‖ }.
𝑖=2,...,𝑟𝑖𝑗 1 1 𝑖 𝑖−1 1 Since 𝒢(𝐴) and 𝒢(𝐵) are isomorphic, Theorem 10
(66) implies that there exists 𝒢(𝐵′) ∈ G𝐽 such that 𝐵′ =
𝐵
𝑇𝐴*𝑇−1; i.e., 𝑉 =𝑇𝑉* where 𝑉 is an eigenvector
and establishes the upper bound for the total variation 𝐵′ 𝐴 𝐵′
matrixof 𝐵′. Bythe classtotal variationdefinition (68),
of spectral components as
TV (𝑉 ) ≤ TV (𝑉* ). Applying Theorem 21 to
𝐺 𝐵′,𝑖𝑗 𝐺 𝐵,𝑖𝑗
TV𝐺(𝑉𝑖𝑗)≤|1−𝜆𝑖|+1. (67) isomorphic graphs 𝒢(𝐴*) and 𝒢(𝐵′) yields
Equations(65),(66),and(67)characterizethe(graph) TV (︀𝑉* )︀=TV (𝑉 )≤TV (︀𝑉* )︀. (71)
𝐺 𝐴,𝑖𝑗 𝐺 𝐵′,𝑖𝑗 𝐺 𝐵,𝑖𝑗
total variation of a Jordan chain by quantifying the
Similarly, by Theorem 10, there exists 𝒢(𝐴′) ∈ G𝐽
change in a set of vectors that spans the Jordan sub- 𝐴
space J𝑖𝑗 when they are transformed by the graph asuncehigtehnavte𝐵ct*or=m𝑇at𝐴rix′𝑇o−f1𝐴,′o.rA𝑉p𝐵p*ly=(6𝑇8𝑉)𝐴an′dwThheereor𝑉em𝐴′2i1s
shift 𝐴. These equations, however, are dependent on
again to obtain
a particular choice of Jordan basis. As seen in Sec-
tionsIV-E,IV-F,andIV-G,defectivegraphshiftmatrices TV (︀𝑉* )︀≥TV (𝑉 )=TV (︀𝑉* )︀. (72)
𝐺 𝐴,𝑖𝑗 𝐺 𝐴′,𝑖𝑗 𝐺 𝐵,𝑖𝑗
belong to Jordan equivalence classes that contain more
than one element, and the GFT of a signal is the same Equations (71) and (72) imply that TV𝐺(︀𝑉𝐴*,𝑖𝑗)︀ =
over any graph in a given Jordan equivalence class. TV𝐺(︀𝑉𝐵*,𝑖𝑗)︀, or
Furthermore, for any two graphs 𝒢(𝐴),𝒢(𝐵) ∈ G𝐽, 𝐴
𝐴 TV (J )=TV (J ). (73)
and𝐵 haveJordanbasesforthesameJordansubspaces, G𝐽𝐴 𝐴,𝑖𝑗 G𝐽𝐵 𝐵,𝑖𝑗
buttherespectivetotalvariationsofthespanningJordan
chains as computed by (65) may be different. Since it Theorem 23 shows that the class total variation of
is desirable to be able to order spectral components in a a spectral component is invariant to a relabeling of
mannerthatisinvarianttothechoiceofJordanbasis,we the nodes. This is significant because it means that an
derivehereadefinitionofthetotalvariationofaspectral ordering of the spectral components by their class total
component of 𝐴 in relation to the Jordan equivalence variations is invariant to node labels.
class G𝐽. Next, the significance of the class total variation (68)
𝐴
Class total variation. Let 𝒢(𝐵) be an element in isillustratedforadjacencymatriceswithdiagonalJordan
Jordan equivalence class G𝐽 where 𝐵 has Jordan de- form, one Jordan block, and multiple Jordan blocks.
𝐴
10