Graph Equivalence Classes for Spectral Projector-Based Graph Fourier Transforms Joya A

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Graph Equivalence Classes for Spectral Projector-Based Graph Fourier Transforms Joya A. Deri, Member, IEEE, and José M. F. Moura, Fellow, IEEE

Abstract—We deﬁne and discuss the utility of two equiv- Consider a graph 풢 = 퐺(퐴) with adjacency matrix alence graph classes over which a spectral projector-based 퐴 ∈ C푁×푁 with 푘 ≤ 푁 distinct eigenvalues and Jordan graph Fourier transform is equivalent: isomorphic equiv- decomposition 퐴 = 푉 퐽푉 −1. The associated Jordan alence classes and Jordan equivalence classes. Isomorphic equivalence classes show that the transform is equivalent subspaces of 퐴 are J푖푗, 푖 = 1, . . . 푘, 푗 = 1, . . . , 푔푖, up to a permutation on the node labels. Jordan equivalence where 푔푖 is the geometric multiplicity of eigenvalue 휆푖, classes permit identical transforms over graphs of noniden- or the dimension of the kernel of 퐴 − 휆푖퐼. The signal tical topologies and allow a basis-invariant characterization space 풮 can be uniquely decomposed by the Jordan of total variation orderings of the spectral components. subspaces (see [13], [14] and Section II). For a graph Methods to exploit these classes to reduce computation time of the transform as well as limitations are discussed. signal 푠 ∈ 풮, the graph Fourier transform (GFT) of [12] is deﬁned as Index Terms—Jordan decomposition, generalized 푘 푔푖 ؁ ؁ ,eigenspaces, directed graphs, graph equivalence classes graph isomorphism, signal processing on graphs, networks ℱ : 풮 → J푖푗 푖=1 푗=1 푠 → (푠 ,..., 푠 ,..., 푠 ,..., 푠 ) , (1) ̂︀11 ̂︀1푔1 ̂︀푘1 ̂︀푘푔푘

I.INTRODUCTION where 푠푖푗 is the (oblique) projection of 푠 onto the Jordan subspace 퐽푖푗 parallel to 풮∖J푖푗. That is, the Fourier Graph signal processing [1], [2] permits applications transform of 푠, is the unique decomposition of digital signal processing concepts to increasingly 푘 푔푖 larger networks. It is based on defining a shift filter, ∑︁ ∑︁ for example, the adjacency matrix in [1], [3], [4] to 푠 = 푠̂︀푖푗, 푠̂︀푖푗 ∈ J푖푗. (2) analyze undirected and directed graphs, or the graph 푖=1 푗=1 Laplacian [2] that applies to undirected graph struc- The spectral components are the Jordan subspaces of the tures. The graph Fourier transform is defined through adjacency matrix with this formulation. the eigendecomposition of this shift operator, see these This paper presents graph equivalence classes where references. Further developments have been considered equal GFT projections by (1) are the equivalence re- in [5], [6], [7]. In particular, filter design [1], [5], [8] lation. First, the transform (1) is invariant to node and sampling [9], [10], [11] can be applied to reduce the permutations, which we formalize with the concept of computational complexity of graph Fourier transforms. isomorphic equivalence classes. Furthermore, the GFT With the objective of simplifying graph Fourier trans- permits degrees of freedom in graph topologies, which forms for large network applications, this paper explores arXiv:1701.02864v1 [cs.SI] 11 Jan 2017 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- graph Fourier transform proposed in [12]. This transform like ordering based on total variation of the spectral extends the graph signal processing framework proposed components is also presented to motivate low-pass, high- by [1], [3], [4] to consider spectral analysis over directed 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 Section III, and Jordan equivalence classes are defined in This work was partially supported by NSF grants CCF-1011903 and Section IV. The Jordan equivalence classes influence the CCF-1513936 and an SYS-CMU grant. definition of total variation-based orderings of the Jordan The authors are with the Department of Electrical and Computer Engineering, Carnegie Mellon University, Pittsburgh, PA 15213 USA subspaces, which is discussed in detail in Section V. (email: jderi,[email protected]) Section VI illustrates Jordan equivalence classes and total variation orderings. Limitations of the method are Jordan decomposition. Let 푉푖푗 denote the 푁 × 푟푖푗 discussed in Section VII. matrix whose columns form a Jordan chain of eigenvalue 휆푖 that spans Jordan subspace J푖푗. Then the II.BACKGROUND eigenvector matrix 푉 of 퐴 is This section reviews the concepts of graph signal [︀ ]︀ 푉 = 푉11 ··· 푉1푔 ··· 푉푘1 ··· 푉푘푔 , (9) processing and the GFT (1). Background on graphs 1 푘 signal processing, including definitions of graph signals where 푘 is the number of distinct eigenvalues. The and the graph shift, is described in greater detail in [1], columns of 푉 are a Jordan basis of C푁 . Then 퐴 [3], [4], [12]. For background on eigendecompositions, has block-diagonal Jordan normal form 퐽 consisting of the reader is directed to in [13], [15], [16]. Jordan blocks ⎡휆 1 ⎤ A. Eigendecomposition ⎢ .. ⎥ ⎢ 휆 . ⎥ 푁×푁 퐽(휆) = . (10) Consider matrix 퐴 ∈ C with 푘 distinct eigen- ⎢ . ⎥ ⎢ .. 1⎥ values 휆1, . . . , 휆푘, 푘 ≤ 푁. The algebraic multiplic- ⎣ ⎦ 휆 ity 푎푖 of 휆푖 represents the corresponding exponent of the characteristic polynomial of 퐴. Denote by Ker(퐴) of size 푟푖푗. The Jordan normal form 퐽 of 퐴 is unique the kernel or null space of matrix 퐴. The geometric up to a permutation of the Jordan blocks. The Jordan multiplicity 푔푖 of eigenvalue 휆푖 equals the dimension of decomposition of 퐴 is 퐴 = 푉 퐽푉 −1. Ker (퐴 − 휆푖퐼), which is the eigenspace of 휆푖 where 퐼 is the 푁 × 푁 identity matrix. The generalized eigenspaces B. Spectral Components G푖, 푖 = 1, . . . , 푘, of 퐴 are defined as The spectral components of the Fourier trans-

푚푖 form (1) are expressed in terms of the eigenvector G푖 = Ker(퐴 − 휆푖퐼) , (3) basis 푣1, . . . , 푣푁 and its dual basis 푤1, . . . , 푤푁 since where 푚푖 is the index of eigenvalue 휆푖. The generalized the Jordan basis may not be orthogonal. Denote the 푁 eigenspaces uniquely decompose C as the direct sum basis and dual basis matrices by 푉 = [푣1 ··· 푣푁 ] and 푘 푊 = [푤1 ··· , 푤푁 ]. The dual basis matrix is the inverse 퐻− ؁ 푁 C = G푖. (4) Hermitian 푊 = 푉 [14], [17]. 푖=1 Consider the 푗th spectral component of 휆푖 Jordan chains. Let 푣 ∈ Ker(퐴 − 휆 퐼), 푣 ̸= 0, 1 푖 1 J푖푗 = span(푣1, ··· 푣푟푖푗 ). (11) be a proper eigenvector of 퐴 that generates generalized 푁 eigenvectors by the recursion The projection matrix onto J푖푗 parallel to C ∖J푖푗 is 퐻 푃푖푗 = 푉푖푗푊 , (12) 퐴푣푝 = 휆푖푣푝 + 푣푝−1, 푝 = 2, . . . , 푟 (5) 푖푗 where 푟 is the minimal positive integer such that where 푟 푟−1 푉푖푗 = [푣1 ··· 푣푟 ] (13) (퐴 − 휆푖퐼) 푣푟 = 0 and (퐴 − 휆푖퐼) 푣푟 ̸= 0. A se- 푖푗 quence of vectors (푣1, . . . , 푣푟) that satisfy (5) is a Jordan 퐻 푟푖푗 ×푁 is the corresponding submatrix of 푉 and 푊푖푗 ∈ C chain of length 푟 [13]. The vectors in a Jordan chain are is the corresponding submatrix of 푊 partitioned as linearly independent and generate the Jordan subspace 푊 = [··· 푊 퐻 ··· 푊 퐻 ··· ]푇 . 푖1 푖푔푖 (14) J = span (푣1, 푣2, . . . , 푣푟) . (6) As shown in [12], the projection of signal 푠 ∈ C푁 Denote by the 푗th Jordan subspace of 휆 with J푖푗 푖 onto Jordan subspace J푖푗 can be written as dimension 푟푖푗, 푖 = 1, . . . , 푘, 푗 = 1, . . . , 푔푖. The Jordan 푠 = 푠 푣 + ··· + 푠 푣 (15) spaces are disjoint and uniquely decompose the general- ̂︀푖푗 ̃︀1 1 ̃︀푟푖푗 푟푖푗 퐻 ized eigenspace G푖 (3) of 휆푖 as = 푉푖푗푊푖푗 푠. (16) 푔푖 The next sections show that invariance of the graph ؁ G푖 = J푖푗. (7) Fourier transform (1) is a useful equivalence relation on 푗=1 a set of graphs. Equivalence classes with respect to the The space C푁 can be expressed as the unique decom- GFT are explored in Sections III and IV. position of Jordan spaces III.ISOMORPHIC EQUIVALENCE CLASSES 푘 푔푖 -This section demonstrates that the graph Fourier trans ؁ ؁ 푁 C = J푖푗. (8) 푖=1 푗=1 form (1) is invariant up to a permutation of node labels

2 and establishes sets of isomorphic graphs as equivalence with respect to the invariance of the GFT (1) up to a classes with respect to invariance of the GFT (1). Two permutation of the graph signal and inverse permutation graphs 풢(퐴) and 풢(퐵) 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 푇 , or 퐵 = 푇 퐴푇 −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 푉 and 푉 are eigenvector matrices of 퐴 퐴 퐵 The isomorphic equivalence of graphs is important and 퐵, respectively, then 푉 = 푇 푉 . We prove that 퐵 퐴 since it signiﬁes that the rows and columns of an adja- the set G퐼 of all graphs that are isomorphic to 풢(퐴) is 퐴 cency matrix can be permuted to accelerate the eigende- an equivalence class over which the GFT is preserved. composition. For example, permutations of highly sparse The next theorem shows that an appropriate permutation 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 풢 ∈ G퐼 . 퐴 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 퐼 next section, the degrees of freedom in graph topology 푠 is invariant to the choice of graph 풢 ∈ G퐴 up to a permutation on the graph signal and inverse permutation are explored to deﬁne another GFT equivalence class. on the graph Fourier transform.

퐼 IV. JORDAN EQUIVALENCE CLASSES Proof: For 풢(퐴), 풢(퐵) ∈ G퐴, there exists a permutation matrix 푇 such that 퐵 = 푇 퐴푇 −1. For eigenvector Since the Jordan subspaces of defective adjacency matrices 푉퐴 and 푉퐵 of 퐴 and 퐵, respectively, let 푉퐴,푖푗 matrices are nontrivial (i.e., they have dimension larger and 푉퐵,푖푗 denote the 푁 × 푟푖푗 submatrices of 푉퐴 and 푉퐵 than one), a degree of freedom exists on the graph whose columns span the 푗th Jordan subspaces J퐴,푖푗 structure so that the graph Fourier transform of a signal and J퐵,푖푗 of the 푖th eigenvalue of 퐴 and 퐵, respectively. is equal over multiple graphs of different topologies. −퐻 −퐻 Jordan equivalence classes Let 푊퐴 = 푉퐴 and 푊퐵 = 푉퐵 denote the matrices This section defines of graph whose columns form dual bases of 푉퐴 and 푉퐵. Since structures over which the GFT (1) is equal for a given 푉퐵 = 푇 푉퐴, graph signal. The section proves important properties of −퐻 this equivalence class that are used to explore inexact 푊퐵 = (푇 푉퐴) (17) methods and real-world applications in [21]. −1 −1 퐻 = (푉퐴 푇 ) (18) The intuition behind Jordan equivalence is presented −퐻 −퐻 in Section IV-A, and properties of Jordan equivalence = 푇 푉퐴 (19) are described in Section IV-B. Section IV-C com- = 푇 푊 , (20) 퐴 pares isomorphic and Jordan equivalent graphs. Sec- where 푇 −퐻 = 푇 since 푇 is a permutation matrix. Thus, tions IV-D, IV-E, IV-F, and IV-G prove properties for 퐻 퐻 퐻 퐻 −1 Jordan equivalence classes when adjacency matrices 푊퐵 = 푊퐴 푇 = 푊퐴 푇 . (21) have particular Jordan block structures. Consider graph signal 푠. By (16), the signal projection onto J퐴,푖푗 is A. Intuition 퐻 푠퐴,푖푗 = 푉퐴,푖푗푊퐴,푖푗푠. (22) Consider Figure 1, which shows a basis {푉 } = ̂︀ 3 {푣1, 푣2, 푣3} of R such that 푣2 and 푣3 span a two- Permit a permutation 푠 = 푇 푠 on the graph signal. Then dimensional Jordan space J of adjacency matrix 퐴 the projection of 푠 onto J퐵,푖푗 is with Jordan decomposition 퐴 = 푉 퐽푉 −1. The resulting 퐻 −1 projection of a signal 푠 ∈ 푁 as in (16) is unique. ̂︀푠퐵,푖푗 = 푇 푉퐴,푖푗푊퐴,푖푗푇 푇 푠 (23) R Note that the definition of the two-dimensional Jordan = 푇 푉 푊 퐻 푠 (24) 퐴,푖푗 퐴,푖푗 subspace J in Figure 1 is not basis-dependent because = 푇 푠 (25) ̂︀퐴,푖푗 any spanning set {푤2, 푤3} could be chosen to define J . by (22). Therefore, the graph Fourier transform (1) is This can be visualized by rotating 푣2 and 푣3 on the two- invariant to a choice among isomorphic graphs up to a dimensional plane. Any choice {푤2, 푤3} corresponds to −1 permutation on the graph signal and inverse permutation a new basis 푉̃︀. Note that 퐴̃︀ = 푉̃︀ 퐽푉̃︀ does not equal −1 on the Fourier transform. 퐴 = 푉 퐽푉 for all choices of {푤2, 푤3}; the underlying graph topologies may be different, or the edge weights 푁×푁 퐼 Theorem 2. Consider 퐴 ∈ C . Then the set G퐴 may be different. Nevertheless, their spectral components of graphs isomorphic to 풢(퐴) is an equivalence class (the Jordan subspaces) are identical, and, consequently,

3 푁×푁 퐽 Theorem 4. For 퐴 ∈ C , the set G퐴 of all graphs that are Jordan equivalent to 풢(퐴) is an equivalence class with respect to invariance of the GFT (1). Jordan equivalent graphs have adjacency matrices with identical Jordan subspaces and identical Jordan normal forms. This implies equivalence of graph spectra, proven in Theorem 5 below.

Theorem 5. Denote by Λ퐴 and Λ퐵 the sets of eigen- 퐽 values of 퐴 and 퐵, respectively. Let 풢(퐴), 풢(퐵) ∈ G퐴. Fig. 1: Projections 푠̂︀1 and 푠̂︀2 (shown in red) of a signal Then Λ = Λ ; that is, 풢(퐴) and 풢(퐵) are cospectral. 푠 (shown in black) onto a nontrivial Jordan subspace (span 퐴 퐵 3 of 푣1 and 푣2) and the span of 푣3, respectively, in R . The Proof: Since 풢(퐴) and 풢(퐵) are Jordan equivalent, projection onto the nontrivial subspace is invariant to basis their Jordan forms are equal, so their spectra (the unique choices {푣1, 푣2} (in blue) or {푣1, 푣2} (in green). ̃︀ ̃︀ elements on the diagonal of the Jordan form) are equal. the spectral projections of a signal onto these compo- Once a Jordan decomposition for an adjacency matrix nents are identical; i.e., the GFT (1) is equivalent over is found, it is useful to characterize other graphs in graphs 풢(퐴) and 풢(퐴̃︀). This observation leads to the the same Jordan equivalence class. To this end, Theo- definition of Jordan equivalence classes which preserve rem 6 presents a transformation that preserves the Jordan the GFT (1) as well as the underlying structure captured equivalence class of a graph. by the Jordan normal form 퐽 of 퐴. These classes are Theorem 6. Consider 퐴, 퐵 ∈ C푁×푁 with Jordan formally defined in the next section. decompositions 퐴 = 푉 퐽푉 −1 and 퐵 = 푋퐽푋−1 and eigenvector matrices 푉 = [푉푖푗] and 푋 = [푋푖푗], respec- 퐽 B. Definition and Properties tively. Then, 풢(퐵) ∈ G퐴 if and only if 퐵 has eigenvector This section defines the Jordan equivalence class of matrix 푋 = 푉 푌 for block diagonal 푌 with invertible 푟푖푗 ×푟푖푗 graphs, over which the graph Fourier transform (1) is submatrices 푌푖푗 ∈ C , 푖 = 1, . . . , 푘, 푗 = 1, . . . , 푔푖. invariant. We will show that certain Jordan equivalence Proof: The Jordan normal forms of 퐴 and 퐵 are classes allow the GFT computation to be simplified. equal. By Definition 3, it remains to show J퐴 = J퐵 so Consider graph 풢(퐴) where 퐴 has a Jordan chain 퐽 that 풢(퐵) ∈ G퐴. The identity J퐴 = J퐵 must be true that spans Jordan subspace J푖푗 of dimension 푟푖푗 > 1. when span{푉푖푗} = span{푋푖푗} = J푖푗, which implies Then (15), and consequently, (16), would hold for a non- that 푋푖푗 represents an invertible linear transformation of Jordan basis of J푖푗; that is, a basis could be chosen to the columns of 푉푖푗. Thus, 푋푖푗 = 푉푖푗푌푖푗, where 푌푖푗 is find spectral component 푠푖푗 such that the basis vectors ̂︀ invertible. Defining 푌 = diag(푌11, . . . , 푌푖푗, . . . , 푌푘,푔푘 ) do not form a Jordan chain of 퐴. This highlights that yields 푋 = 푉 푌 . the Fourier transform (1) is characterized not by the Jordan basis of 퐴 but by the set J퐴 = {J푖푗}푖푗 of Jordan subspaces spanned by the Jordan chains of 퐴. 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 퐼 Lemma 7. Consider graphs 풢(퐴), 풢(퐵) ∈ G퐴 so that graphs 풢(퐴) and 풢(퐵) with adjacency matrices 퐴, 퐵 ∈ 퐵 = 푇 퐴푇 −1 for a permutation matrix 푇 . Denote by 푁×푁 . Then 풢(퐴) and 풢(퐵) are Jordan equivalent C J퐴 and J퐵 the sets of Jordan subspaces for 퐴 and 퐵, graphs if all of the following are true: respectively. If {푣1, . . . , 푣푟} is a basis of J퐴 ∈ J퐴, then 1) J퐴 = J퐵; and there exists J퐵 ∈ J퐵 with basis {푥1, . . . , 푥푟} such that 2) 퐽퐴 = 퐽퐵 (with respect to a fixed permutation of [푥1 ··· 푥푟] = 푇 [푣1 ··· 푣푟]; i.e., 퐴 and 퐵 have isomorphic Jordan blocks). Jordan subspaces. 퐽 Let G퐴 denote the set of graphs that are Jordan Proof: Consider 퐴 with Jordan decomposition 퐴 = equivalent to 풢(퐴). Definition 3 and (1) establish that 푉 퐽푉 −1. Since 퐵 = 푇 퐴푇 −1, it follows that 퐽 G퐴 is an equivalence class. 퐵 = 푇 푉 퐽푉 −1푇 −1 (26)

4 = 푋퐽푋−1 (27) where 푋 = 푇 푉 represents an eigenvector matrix of 퐵 that is a permutation of the rows of 푉 . (It is clear that the Jordan forms of 퐴 and 퐵 are equivalent.) Let columns 푣1, . . . , 푣푟 of 푉 denote a Jordan chain of 퐴 that spans Jordan subspace J퐴. The corresponding columns in 푋 are 푥1, . . . , 푥푟 and span(푥1, . . . , 푥푟) = J퐵. Fig. 2: Jordan equivalent graph structures with unicellular adjacency matrices. Since [푥1 ··· 푥푟] = 푇 [푣1 ··· 푣푟], J퐴 and J퐵 are isomorphic subspaces [13]. Theorem 8. A graph isomorphism does not imply Jordan 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 = C푁 and eigen- 퐼 Proof: Consider 풢(퐴), 풢(퐵) ∈ G퐴 and 퐵 = value 0; therefore, they are Jordan equivalent. On the −1 푇 퐴푇 for permutation matrix 푇 . By (27), 퐽퐴 = 퐽퐵. other hand, they are not isomorphic since the graph on 퐽 To show 풢(퐴), 풢(퐵) ∈ G퐴, it remains to check whether the right has more edges then the graph on the left. J퐴 = J퐵. Theorem 8 shows that changing the graph node labels By Lemma 7, for any J퐴 ∈ J퐴, there exists J퐵 ∈ may change the Jordan subspaces and the Jordan equiv- J퐵 that is isomorphic to J퐴. That is, if 푣1, . . . , 푣푟 alence class of the graph, while Theorem 9 shows that and 푥1, . . . , 푥푟 are bases of J퐴 and J퐵, respectively, a Jordan equivalence class may include graphs with dif- then [푥1 ··· 푥푟] = 푇 [푣1 ··· 푣푟]. Checking J퐴 = J퐵 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 훼 푣 + ··· + 훼 푣 = 훽 푥 + ··· + 훽 푥 (28) 1 1 푟 푟 1 1 푟 푟 invariance of the graph Fourier transform with respect = 훽1푇 푣1 + ··· + 훽푟푇 푣푟 (29) to equivalence relations as stated in Theorems 1 and 4. The next theorem establishes an isomorphism between for some coefficients 훼푖 and 훽푖, 푖 = 1, . . . , 푟. How- ever, (29) does not always hold. Consider matrices 퐴 Jordan equivalence classes. and 퐵 푁×푁 Theorem 10. If 퐴, 퐵 ∈ C and 풢(퐴) and 풢(퐵) [︃2 0 −1]︃ [︃ 1 0 0]︃ are isomorphic, then their respective Jordan equiva- 퐴 = 0 2 −1 , 퐵 = −1 2 0 . (30) 퐽 퐽 lence classes G퐴 and G퐵 are isomorphic; i.e., any 0 0 1 −1 0 2 ′ 퐽 ′ graph 풢(퐴 ) ∈ G퐴 is isomorphic to a graph 풢(퐵 ) ∈ 퐽 These matrices are similar with respect to a permutation G퐵. matrix and thus correspond to isomorphic graphs. Their Proof: Let 풢(퐴) and 풢(퐵) be isomorphic by per- Jordan normal forms are both mutation matrix 푇 such that 퐵 = 푇 퐴푇 −1. Consider [︃1 0 0]︃ ′ 퐽 풢(퐴 ) ∈ G퐴, which implies that Jordan normal forms 퐽 = 0 2 0 (31) 퐽퐴′ = 퐽퐴 and sets of Jordan subspaces J퐴′ = J퐴 by 0 0 2 ′ Definition 3. Denote by 퐴 = 푉퐴′ 퐽퐴′ 푉퐴′ the Jordan ′ ′ ′ −1 with possible eigenvector matrices 푉퐴 and 푉퐵 given by decomposition of 퐴 . Define 퐵 = 푇 퐴 푇 . It suffices ′ 퐽 [︃1 1 0]︃ [︃1 0 0]︃ to show 풢(퐵 ) ∈ G퐵. First simplify: 푉퐴 = 1 0 1 , 푉퐵 = 1 1 0 . (32) ′ ′ −1 1 0 0 1 0 1 퐵 = 푇 퐴 푇 (33) −1 −1 = 푇 푉 ′ 퐽 ′ 푉 ′ 푇 (34) Equation (32) shows that 퐴 and 퐵 both have Jordan 퐴 퐴 퐴 −1 −1 ′ 퐽 푇 = 푇 푉 ′ 퐽 푉 ′ 푇 (since 풢(퐴 ) ∈ G ) (35) subspaces J1 = span([1 1 1] ) for 휆1 = 1 and 퐴 퐴 퐴 퐴 푇 −1 −1 퐼 J21 = span([0 1 0] ) for one Jordan subspace of = 푇 푉퐴′ 퐽퐵푉퐴′ 푇 (since 풢(퐴) ∈ G퐵). (36) 휆2 = 2. However, the remaining Jordan subspace is span([1 0 0]푇 ) for 퐴 but span([0 0 1]푇 ) for 퐵, so (29) From (36), it follows that 퐽퐵′ = 퐽퐵. It remains to fails. Thus, 풢(퐴) and 풢(퐵) are not Jordan equivalent. show that J퐵′ = J퐵. Choose arbitrary Jordan subspace J퐴,푖푗 = span{푉퐴,푖푗} of 퐴. Then J퐴′,푖푗 = ′ 퐽 span{푉퐴′,푖푗} = J퐴,푖푗 since 풢(퐴 ) ∈ G . Then the The next theorem shows that Jordan equivalent graphs 퐴 푗th Jordan subspace of eigenvalue 휆 for 퐵 is may not be isomorphic. 푖 J퐵,푖푗 = span{푇 푉퐴,푖푗} (37) Theorem 9. Jordan equivalence does not imply the existence of a graph isomorphism. = 푇 span{푉퐴,푖푗}. (38)

5 ′ For the 푗th Jordan subspace of eigenvalue 휆푖 for 퐵 , it Proof: Since the Jordan subspaces of a diagonal- follows from (36) that izable matrix are one-dimensional, the possible choices of Jordan basis are limited to nonzero scalar multiples ′ = span{푇 푉 ′ } (39) J퐵 ,푖푗 퐴 ,푖푗 of the eigenvectors. Then, given eigenvector matrix 푉 = 푇 span{푉퐴′,푖푗} (40) of 퐴, all possible eigenvector matrices of 퐴 are given ′ 퐽 = 푇 span{푉퐴,푖푗} (since 풢(퐴 ) ∈ G퐴) by 푋 = 푉 푈, where 푈 is a diagonal matrix with nonzero (41) diagonal entries. Let 퐵 = 푋퐽푋−1, where 퐽 is the diagonal canonical Jordan form of 퐴. Since 푈 and 퐽 = J퐵,푖푗. (by (38)) (42) are both diagonal, they commute, yielding Since (42) holds for all 푖 and 푗, the sets of Jordan −1 ′ 퐵 = 푋퐽푋 (43) subspaces J퐵′ = J퐵. Therefore, 풢(퐵 ) and 풢(퐵) are Jordan equivalent, which proves the theorem. = 푉 푈퐽푈 −1푉 −1 (44) Theorem 10 shows that the Jordan equivalence classes = 푉 퐽푈푈 −1푉 −1 (45) of two isomorphic graphs are also isomorphic. This = 푉 퐽푉 −1 (46) result permits an frequency ordering on the spectral components of a matrix 퐴 that is invariant to both the = 퐴. (47) 퐽 choice of graph in G퐴 and the choice of node labels, as Thus, a graph with a diagonalizable adjacency matrix is demonstrated in Section V. the one and only element in its Jordan equivalence class. Relation to matrices with the same set of invariant Inv subspaces. Let G퐴 denote the set of all matrices When a matrix has nondefective but repeated eigen- with the same set of invariant subspaces of 퐴; i.e., values, there are infinitely many choices of eigenvec- Inv 풢(퐵) ∈ G퐴 if and only if Inv(퐴) = Inv(퐵). The tors [16]. An illustrative example is the identity matrix, Inv next theorem shows that G퐴 is a proper subset of the which has a single eigenvalue but is diagonalizable. 퐽 Jordan equivalence class G퐴 of 퐴. Since it has infinitely many choices of eigenvectors, the 푁×푁 Inv 퐽 identity matrix corresponds to infinitely many Jordan Theorem 11. For 퐴 ∈ C , G퐴 ⊂ G퐴. Inv equivalence classes. By Theorem 12, each of these equiv- Proof: If 풢(퐵) ∈ G퐴 , then the set of Jordan alence classes have size one. This observation highlights subspaces are equal, or J퐴 = J퐵. 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 퐴 with Jordan decomposition 퐴 = decomposition of the discrete Fourier transform matrix 푉 퐽푉 −1 where 퐽 is a single Jordan block and 푉 = in terms of projections onto invariant spaces. The Jor- [푣1 ··· 푣푁 ] is an eigenvector matrix. Then 퐴 is a repre- dan equivalence class relaxes the assumption that all sentation of a unicellular transformation 푇 : C푁 → C푁 invariant subspaces of two adjacency matrices must be with respect to Jordan basis 푣1, . . . 푣푁 (see [15, Sec- 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 = C푁 . 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. Let 풢(퐴) be an element of the unicellular 퐽 D. Diagonalizable Matrices Jordan equivalence class G퐴. Then all graph filters 퐻 ∈ 퐽 If the canonical Jordan form 퐽 of 퐴 is diagonal (퐴 G퐴 are all-pass. is diagonalizable), then there are no Jordan chains and Proof: 퐴 푁 Since is unicellular, it has a single Jordan the set of Jordan subspaces J퐴 = {J푝} where 푝=1 chain 푣1, . . . , 푣푁 of length 푁. Consider a graph signal 푠 J푝 = span(푣푝) and 푣푝 is the 푝th eigenvector of 퐴. over graph 풢(퐴), and let 푠̃︀ represent the coordinate Graphs with diagonalizable adjacency matrices include 푁 vector of 푠 in terms of the basis {푣푖}푖=1. Then the undirected graphs, directed cycles, and other digraphs spectral decomposition of signal 푠 is given by with normal adjacency matrices such as normally regular digraphs [23]. A graph with a diagonalizable adjacency 푠 = 푠̃︀1푣1 + ··· 푠̃︀푁 푣푁 = 푠̂︀; (48) matrix is Jordan equivalent only to itself, as proven next. that is, the unique projection of 푠 onto the spectral Theorem 12. A graph 풢(퐴) with diagonalizable adja- component J = C푁 is itself. Therefore, 풢(퐴) acts as cency matrix 퐴 ∈ C푁×푁 belongs to a Jordan equiva- an all-pass filter. Moreover, (48) holds for all graphs in 퐽 lence class of size one. Jordan equivalence class G퐴.

6 In addition to the all-pass property of unicellular graph where Inv(·) represents the set of invariant spaces of a filters, unicellular isomorphic graphs are also Jordan matrix). If 푎 = 휆, Definition 3 can be applied, which 퐽 equivalent, as proven next. yields 풢(퐴) ∈ G퐽 . On the other hand, consider a unicellular matrix 퐵 Theorem 14. Let 풢(퐴), 풢(퐵) ∈ G퐼 where 퐴 is a 퐴 such that its eigenvector is not in the span of a canonical unicellular matrix. Then 풢(퐴), 풢(퐵) ∈ G퐽 . 퐴 vector, e.g., Proof: Since 풢(퐴) and 풢(퐵) are isomorphic, Jor- ⎡ 1 − 1 1 1 ⎤ dan normal forms 퐽 = 퐽 . Therefore, 퐵 is also 2 2 2 2 퐴 퐵 1 1 1 1 푁 ⎢ − − − ⎥ unicellular, so J퐴 = J퐵 = {C }. By Definition 3, 퐵 = ⎢ 2 2 2 2 ⎥ (49) 퐽 ⎢ 1 1 ⎥ 풢(퐴), 풢(퐵) ∈ G퐴. ⎣0 0 2 − 2 ⎦ The dual basis of 푉 can also be used to construct 1 1 0 0 2 − 2 graphs in the Jordan equivalence class of unicellular 퐴. with Jordan normal form 퐽(0). Since the span of Theorem 15. Denote by 푉 an eigenvector matrix of the eigenvectors of 퐽(0) and 퐵 are not identical, unicellular 퐴 ∈ C푁×푁 and 푊 = 푉 −퐻 is the dual basis. Inv(퐽(0)) ̸= Inv(퐵). However, by Definition 3, 풢(퐵) is −1 Consider decompositions 퐴 = 푉 퐽푉 and 퐴푊 = in the same class of unicellular Jordan equivalent graphs 푊 퐽푊 −1. Then 풢(퐴 ) ∈ G퐽 . 퐽 푊 퐴 as those of Figure 2, i.e., 풢(퐵) ∈ G퐽 . In other words, for matrices 퐴 and 퐵 with the same Jordan normal forms Proof: Matrices 퐴 and 퐴푊 have the same Jordan normal form by definition. Since there is only one Jordan (퐽퐴 = 퐽퐵), Jordan equivalence, i.e., J퐴 = J퐵, is a more general condition than Inv(퐴) = Inv(퐵). This illustrates block, both matrices have a single Jordan subspace C푁 . that graphs having adjacency matrices with equal Jordan By Definition 3, 풢(퐴푊 ) and 풢(퐴) are Jordan equivalent. normal forms and the same sets of invariant spaces form The next theorem characterizes the special case of a proper subset of a Jordan equivalence class, as shown graphs in the Jordan equivalence class that contains 풢(퐽) above in Theorem 11. Remark on topology. Note that replacing each with adjacency matrix equal to Jordan block 퐽 = 퐽(휆). nonzero element of (49) with a unit entry results in a Theorem 16. Denote by 퐽 = 퐽(휆) is the 푁 ×푁 Jordan matrix that is not unicellular. Therefore, its correspond- 퐽 block (10) for eigenvalue 휆. Then 풢(퐴) ∈ G퐽 if 퐴 ∈ ing graph is not in a unicellular Jordan equivalence class. C푁×푁 is upper triangular with diagonal entries 휆 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 퐴 = F. Two Jordan Blocks [푎푖푗] with diagonal entries 푎11 = ··· = 푎푁푁 and nonzero elements on the first off-diagonal. By [15, Consider 푁 × 푁 matrix 퐴 with Jordan normal Example 10.2.1], 퐴 has the same invariant subspaces form consisting of two Jordan subspaces J1 = 푁 span(푣 , . . . , 푣 ) and = span(푣 , . . . , 푣 ) of as 퐽 = 퐽(휆), which implies J퐽 = J퐴 = {C }. 1 푟1 J2 푟1+1 푟2 Therefore, the Jordan normal form of 퐴 is the Jordan dimensions 푟1 > 1 and 푟2 = 푁 − 푟1 and corresponding block 퐽퐴 = 퐽(푎11). Restrict the diagonal entries of 퐴 to eigenvalues 휆1 and 휆2, respectively. The spectral decom- 퐽 position of signal 푠 over 풢(퐴) yields 휆 so 퐽퐴 = 퐽. Then, 풢(퐽), 풢(퐴) ∈ G퐽 by Definition (3). 푠 = 푠 푣 + ··· + 푠 푣 + 푠 푣 + ··· + 푠 푣 ̃︀1 1 ̃︀푟1 푟1 ̃︀푟1+1 푟1+1 ̃︀푁 푁 Figure 2 shows graph structures that are in the same ⏟ ⏞ ⏟ ⏞ unicellular Jordan equivalence class by Theorem 16. In 푠̂︀1 푠̂︀2 (50) addition, the theorem implies that it is sufficient to deter- 퐽 = 푠1 + 푠2. (51) mine the GFT of unicellular 퐴 by replacing 풢(퐴) ∈ G퐽 ̂︀ ̂︀ with 풢(퐽), where 퐽 is a single 푁 ×푁 Jordan block. That Spectral components 푠̂︀1 and 푠̂︀2 are the unique pro- is, without loss of generality, 풢(퐴) can be replaced with jections of 푠 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 푋 can be eigenvector matrix 푉 = 퐼 chosen to compute the GFT chosen for 퐴 = 푉 퐽푉 −1 such that 푋 = 푉 푈, where 푈 of a graph signal. commutes with 퐽 and has a particular form as follows. Remark on invariant spaces. Example 10.2.1 of [15] If 휆1 ̸= 휆2, then 푈 = diag(푈1, 푈2), where 푈푖, shows that a matrix 퐴 ∈ 푁×푁 having upper triangular C 푖 = 1, 2, is an 푟푖 × 푟푖 upper triangular Toeplitz matrix; entries with constant diagonal entries 푎 and nonzero otherwise, 푈 has form entries on the first off-diagonal is both necessary and [︃ ]︃ sufficient for 퐴 to have the same invariant subspaces as 0 푈12 푈 = diag(푈1, 푈2) + (52) 푁 × 푁 Jordan block 퐽 = 퐽(휆) (i.e., Inv(퐽) = Inv(퐴), 푈21 0

7 where 푈푖 is an 푟푖 × 푟푖 upper triangular Toeplitz matrix 퐽푟푖 (휆) is the 푟푖 × 푟푖 Jordan block for eigenvalue 휆. and 푈 and 푈 are extended upper triangular Toeplitz 12 21 Proof: By Lemma 10.3.3 in [15], 퐴 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 퐴 can be obtained by transforming eigenvector subspaces as 퐽 = diag(퐽 (휆), 퐽 (휆)). Therefore, 퐴 matrix 푉 as 푋 = 푉 푈. 푟1 푟2 퐽 A corresponding theorem to Theorem 16 is presented and have the same Jordan normal form and Jordan to characterize Jordan equivalent classes when the Jordan subspaces and so are Jordan equivalent. form consists of two Jordan blocks. The reader is di- Theorems 17 and 18 demonstrate two types of Jordan rected to Sections 10.2 and 10.3 in [15] for more details. equivalences that arise from block diagonal matrices The following definitions are needed. Denote 푝×푝 upper with submatrices of form (53) and (54). These theorems imply that computing the GFT (1) over the block triangular Toeplitz matrices 푇푟 (푏1, . . . , 푏푟 ) of form 2 2 diagonal matrices can be simplified to computing the ⎡ ⎤ 푏1 푏2 ··· 푏푝−1 푏푝 transform over the adjacency matrix of a union of ⎢ . ⎥ directed chain graphs. That is, the canonical basis can ⎢ 0 푏 .. 푏 푏 ⎥ ⎢ 1 푝−2 푝−1⎥ be chosen for 푉 without loss of generality. 푇 (푏 , . . . , 푏 ) = ⎢ . . . . . ⎥ , (53) 푝 1 푝 ⎢ ...... ⎥ As for the case of unicellular transformations, it is ⎢ ⎥ ⎢ ⎥ possible to pick bases of J1 and J2 that do not form ⎣ 0 0 ··· 푏1 푏2 ⎦ a Jordan basis of 퐴. Any two such choices of bases are 0 0 ··· 0 푏 1 related by Theorem 6. Concretely, if 푉 is the eigenvector and define 푞×푞 upper triangular matrix for some 푞 > 푝 matrix of 퐴 and 푋 is the matrix corresponding to another choice of basis, then Theorem 6 states that a transfor- 푅 (푏 , . . . , 푏 ; 퐹 ) = 푞 1 푝 mation matrix 푌 can be found such that 푋 = 푉 푌 , ⎡푏1 ··· 푏푝 푓11 푓12 ··· 푓1,푞−푝−1 푓1,푞−푝 ⎤ where 푌 is partitioned as 푌 = diag(푌1, 푌2) with full- 0 푏1 ··· 푏푝 푓22 ··· 푓2,푞−푝−1 푓2,푞−푝 ⎢ ⎥ rank submatrices 푌 ∈ 푟푖×푟푖 , 푖 = 1, 2. ⎢ ...... ⎥ 푖 C ⎢ ...... ⎥ ⎢ ⎥ ⎢ 0 0 ··· 푏푝 푓푞−푝,푞−푝⎥ ⎢ 0 0 ··· 푏 푏 ⎥ (54) ⎢ 푝−1 푝 ⎥ G. Multiple Jordan Blocks ⎢ . . . . . ⎥ ⎢ . . . . . ⎥ This section briefly describes a special case of Jordan ⎣ 0 0 ··· 0 푏 푏 ⎦ 1 2 equivalence classes whose graphs have adjacency matri- 0 0 ··· 0 0 푏1 ces 퐴 ∈ C푁×푁 with 푝 Jordan blocks, 1 < 푝 < 푁. where 퐹 = [푓 ] is a (푞 − 푝) × (푞 − 푝) upper triangular 푖푗 Consider matrix 퐴 with Jordan normal form 퐽 com- matrix and 푏 ∈ , 푖 = 1, . . . , 푝. The theorems are 푖 C prised of 푝 Jordan blocks and eigenvalues 휆 , . . . , 휆 . presented below. 1 푘 By Theorem 10.2.1 in [15], there exists an upper tri- −1 Theorem 17. Consider 퐴 = diag(퐴1, 퐴2) where each angular 퐴 with Jordan decomposition 퐴 = 푉 퐽푉 퐽 matrix 퐴푖, 푖 = 1, 2, is upper triangular with diag- such that 풢(퐴) ∈ G퐽 . Note that the elements in the 퐽 onal elements 휆푖 and nonzero elements on the first Jordan equivalence class G퐽 of 풢(퐽) are useful since off-diagonal. Let 휆1 ̸= 휆2. Then 풢(퐴) is Jordan signals over a graph in this class can be computed equivalent to the graph with adjacency matrix 퐽 = with respect to the canonical basis with eigenvector

diag(퐽푟1 (휆1), 퐽푟2 (휆2)) where 퐽푟푖 (휆푖) is the 푟푖 × 푟푖 matrix 푉 = 퐼. Theorem 19 characterizes the possible −1 Jordan block for eigenvalue 휆푖. eigenvector matrices 푉 such that 퐴 = 푉 퐽푉 allows 퐽 풢(퐴) ∈ G퐽 . Proof: By Theorem 16, 풢(퐴푖) and 풢(퐽푟푖 (휆푖)) are −1 Jordan equivalent for 푖 = 1, 2 and 퐴푖 upper triangular Theorem 19. Let 퐴 = 푉 퐽푉 be the Jordan decom- 푁×푁 퐽 with nonzero elements on the ﬁrst off-diagonal. There- position of 퐴 ∈ C and 풢(퐴) ∈ G퐽 . Then 푉 must fore, the Jordan normal forms of 퐽 and 퐴 are the same. be an invertible block diagonal matrix. Moreover, the set of irreducible subspaces of 퐽 is the union of the irreducible subspaces of [퐽 0]푇 and [0 퐽 ]푇 , Proof: Consider 풢(퐽) with eigenvector matrix 퐼. 1 2 풢(퐴) ∈ G퐽 which are the same as the irreducible subspaces of By Theorem 6, 퐽 implies 푇 푇 [퐴1 0] and [0 퐴2] , respectively. Therefore, J퐴 = J퐽 , 푉 = 퐼푌 = 푌 (55) so 풢(퐴) and 풢(퐽) are Jordan equivalent. where 푌 is an invertible block diagonal matrix. Theorem 18. Consider 퐴 = diag(퐴1, 퐴2) where 퐴1 = The structure of 푉 given in Theorem 19 allows a char- 푈푟 (휆, 푏1, . . . , 푏푟 , 퐹 ) and 퐴2 = 푇푟 (휆, 푏1, . . . , 푏푟 ), 퐽 1 2 2 2 acterization of graphs in the Jordan equivalence class G퐽 푟1 ≥ 푟2. Then 풢(퐴) is Jordan equivalent to the graph with the dual basis of 푉 as proved in Theorem 20. with adjacency matrix 퐽 = diag(퐽푟1 (휆), 퐽푟2 (휆)) where

8 퐽 Theorem 20. Let 풢(퐴) ∈ G퐽 , where 퐴 has Jordan can be chosen without loss of generality. decomposition 퐴 = 푉 퐽푉 −1 and 푊 = 푉 −퐻 is the dual Jordan equivalence classes show that the GFT (1) −1 퐽 basis of 푉 . If 퐴푊 = 푊 퐽푊 , then 풢(퐴푊 ) ∈ G퐽 . permits degrees of freedom in graph topologies. This has ramifications for total variation-based orderings of Proof: By Theorem 19, 푉 is block diagonal with the spectral components, as discussed in the next section. −퐻 invertible submatrices 푉푖. Thus, 푊 = 푉 is block diagonal with submatrices 푊 = 푉 −퐻 . By Theorem 6, 푖 푖 V. FREQUENCY ORDERINGOF SPECTRAL 푊 is an appropriate eigenvector matrix such that, for −1 퐽 COMPONENTS 퐴푊 = 푊 퐽푊 , 풢(퐴푊 ) ∈ G . 퐽 This section defines a mapping of spectral components 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 generalized low and high frequencies as in [4]. An upper 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. in use would allow a practitioner to determine the Jordan The graph total variation of a graph signal 푠 ∈ 푁 equivalence classes (assuming the eigenvalues can be C is defined as [4] computed) and potentially choose a different matrix in

the class for which the GFT can be computed more TV퐺 (푠) = ‖푠 − 퐴푠‖1 . (56) easily. Sections IV-E and IV-G show that working with Matrix 퐴 can be replaced by 퐴norm = 1 퐴 when unicellular matrices and matrices in Jordan normal form |휆max| permits the choice of the canonical basis. In this way, the maximum eigenvalue satisfies |휆max| > 0. for matrices with Jordan blocks of size greater than Equation (56) can be generalized to define the total one, finding a spanning set for each Jordan subspace variation of the Jordan subspaces of the graph shift 퐴 as described in [12]. Choose a Jordan basis of 퐴 so may be more efficient than attempting to compute the −1 Jordan chains. Nevertheless, relying on graph topology that 푉 is the eigenvector matrix of 퐴, i.e., 퐴 = 푉 퐽푉 , is not always possible. Such an example was presented where 퐽 is the Jordan form of 퐴. Partition 푉 into in Section IV-E with adjacency matrix (49). 푁 × 푟푖푗 submatrices 푉푖푗 whose columns are a Jordan chain of (and thus span) the 푗th Jordan subspace J푖푗 Relation to algebraic signal processing. The emer- of eigenvalue 휆푖, 푖 = 1, . . . , 푘 ≤ 푁, 푗 = 1, . . . , 푔푖. Then gence of Jordan equivalence from the graph Fourier the (graph) total variation of 푉푖푗 is defined as [12] transform (1) is related to algebraic signal processing and the signal model (풜, ℳ, Φ), where 풜 is a signal algebra TV퐺 (푉푖푗) = ‖푉푖푗 − 퐴푉푖푗‖1 , (57) corresponding to the filter space, ℳ is a module of 풜 where ‖·‖ represents the induced L1 matrix norm (equal corresponding to the signal space, and Φ: 푉 → ℳ 1 to the maximum absolute column sum). is a bijective linear mapping that generalizes the 푧- 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 diagonal adjacency matrix 퐴 = 휆퐼, where any basis that spectral components for graphs in the same isomorphic spans 푁 defines the eigenvectors (the Jordan subspaces C equivalence class as defined in Section III are equal. and spectral components) of a graph signal; that is, 푁×푁 퐼 a matrix, even a diagonalizable matrix, may not have Theorem 21. Let 퐴, 퐵 ∈ C and 풢(퐵) ∈ G퐴, i.e., 푁×푟 distinct spectral components. Similarly, the signal model 풢(퐵) is isomorphic to 풢(퐴). Let 푉퐴,푖푗 ∈ C 푖푗 be 푁×푟 (풜, ℳ, Φ) requires a choice of basis for module (signal a Jordan chain of matrix 퐴 and 푉퐵,푖푗 ∈ C 푖푗 the space) ℳ in order to define the frequency response corresponding Jordan chain of 퐵. Then (irreducible representation) of a signal [26]. On the other hand, this section demonstrated the equivalence of TV퐺(푉퐴,푖푗) = TV퐺(푉퐵,푖푗). (58) the GFT (1) over graphs in Jordan equivalence classes, Proof: Since 풢(퐴) and 풢(퐵) are isomorphic, there which implies an equivalence of certain bases. This exists a permutation matrix 푇 such that 퐵 = 푇 퐴푇 −1 observation suggests the concept of equivalent signal and the eigenvector matrices 푉퐴 and 푉퐵 of 퐴 and 퐵, models in the algebraic signal processing framework. respectively, are related by 푉퐵 = 푇 푉퐴. Thus, the Jordan Just as working with graphs that are Jordan equivalent chains are related by 푉퐵,푖푗 = 푇 푉퐴,푖푗. By (57), to those with adjacency matrices in Jordan normal form simplifies GFT computation, we expect similar classes TV (푉퐵) = ‖푉퐵,푖푗 − 퐵푉퐵,푖푗‖1 (59) ⃦ (︀ −1)︀ ⃦ of equivalent signal models for which the canonical basis = ⃦푇 푉퐴,푖푗 − 푇 퐴푇 푇 푉퐴,푖푗⃦1 (60)

9 −1 = ‖푇 푉퐴,푖푗 − 푇 퐴푉퐴,푖푗‖1 (61) composition 퐵 = 푉 퐽푉 . Let the columns of eigenvector submatrix 푉 span the Jordan subspace of = ‖푇 (푉퐴,푖푗 − 퐴푉퐴,푖푗)‖ (62) 푖푗 J푖푗 1 퐴. Then the class total variation of spectral component = ‖푉퐴,푖푗 − 퐴푉퐴,푖푗‖1 (63) J푖푗 is defined as the supremum of the graph total = TV (푉 ) , (64) 퐴 variation of 푉푖푗 over the Jordan equivalence class (for 퐽 where (63) holds because the maximum absolute column all 풢(퐵) ∈ G퐴): sum of a matrix is invariant to a permutation on its rows. TVG퐽 (J푖푗) = sup TV퐺 (푉푖푗) . (68) 퐴 퐽 풢(퐵)∈G퐴 Theorem 21 shows that the graph total variation is 퐵=푉 퐽푉 −1 span{푉푖푗 }=J푖푗 invariant to a node relabeling, which implies that an ‖푉푖푗 ‖1=1 ordering of the total variations of the frequency com- 푁×푁 퐼 ponents is also invariant. Theorem 23. Let 퐴, 퐵 ∈ C and 풢(퐵) ∈ G퐴. Let Reference [12] demonstrates that each eigenvector 푉퐴 and 푉퐵 be the respective eigenvector matrices with submatrix corresponding to a Jordan chain can be nor- Jordan subspaces J퐴,푖푗 = span{푉퐴,푖푗} and J퐵,푖푗 = malized. This is stated as a property below: span{푉퐵,푖푗} spanned by the 푗th Jordan chain of eigenvalue 휆푖. Then TVG퐽 (J퐴,푖푗) = TVG퐽 (J퐵,푖푗). Property 22. The eigenvector matrix 푉 of adjacency 퐴 퐵 Proof: Let 푉 * denote the eigenvector matrix corre- matrix 퐴 ∈ C푁×푁 can be chosen so that each Jordan 퐴 sponding to 풢(퐴*) ∈ G퐽 that maximizes the class total chain represented by the eigenvector submatrix 푉푖푗 ∈ 퐴 푁×푟푖푗 variation of Jordan subspace J퐴,푖푗; i.e., C satisfies ‖푉푖푗‖1 = 1; i.e., ‖푉 ‖1 = 1 without (︀ * )︀ loss of generality. TV 퐽 (J퐴,푖푗) = TV퐺 푉 . (69) G퐴 퐴,푖푗 It is assumed that the eigenvector matrices are normal- * Similarly, let 푉퐵 denote the eigenvector matrix corre- ized as in Property 22 for the remainder of the section. * 퐽 sponding to 풢(퐵 ) ∈ G퐵 that maximizes the class total Furthermore, [12] shows that (57) can be written as variation of Jordan subspace J퐵,푖푗, or ⃦ ⃦ TV (푉 ) = 푉 (︀퐼 − 퐽 )︀ (65) (︀ * )︀ 퐺 푖푗 ⃦ 푖푗 푟푖푗 푖푗 ⃦1 TV 퐽 (J퐵,푖푗) = TV퐺 푉 . (70) G퐵 퐵,푖푗 = max {|1 − 휆| ‖푣1‖1 , ‖(1 − 휆) 푣푖 − 푣푖−1‖1} . 푖=2,...,푟푖푗 Since 풢(퐴) and 풢(퐵) are isomorphic, Theorem 10 ′ 퐽 ′ (66) implies that there exists 풢(퐵 ) ∈ G퐵 such that 퐵 = * −1 * and establishes the upper bound for the total variation 푇 퐴 푇 ; i.e., 푉퐵′ = 푇 푉퐴 where 푉퐵′ is an eigenvector matrix of 퐵′. By the class total variation definition (68), of spectral components as * TV퐺(푉퐵′,푖푗) ≤ TV퐺(푉퐵,푖푗). Applying Theorem 21 to * ′ TV퐺(푉푖푗) ≤ |1 − 휆푖| + 1. (67) isomorphic graphs 풢(퐴 ) and 풢(퐵 ) yields (︀ * )︀ (︀ * )︀ Equations (65), (66), and (67) characterize the (graph) TV퐺 푉퐴,푖푗 = TV퐺 (푉퐵′,푖푗) ≤ TV퐺 푉퐵,푖푗 . (71) total variation of a Jordan chain by quantifying the ′ 퐽 change in a set of vectors that spans the Jordan sub- Similarly, by Theorem 10, there exists 풢(퐴 ) ∈ G퐴 * ′ −1 * space when they are transformed by the graph such that 퐵 = 푇 퐴 푇 , or 푉퐵 = 푇 푉퐴′ where 푉퐴′ is J푖푗 ′ shift 퐴. These equations, however, are dependent on an eigenvector matrix of 퐴 . Apply (68) and Theorem 21 a particular choice of Jordan basis. As seen in Sec- again to obtain tions IV-E, IV-F, and IV-G, defective graph shift matrices (︀ * )︀ (︀ * )︀ TV퐺 푉퐴,푖푗 ≥ TV퐺 (푉퐴′,푖푗) = TV퐺 푉퐵,푖푗 . (72) 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 TV퐺 푉퐴,푖푗 = (︀ * )︀ over any graph in a given Jordan equivalence class. TV퐺 푉퐵,푖푗 , or 퐽 Furthermore, for any two graphs 풢(퐴), 풢(퐵) ∈ G퐴, 퐴 TVG퐽 (J퐴,푖푗) = TVG퐽 (J퐵,푖푗) . (73) and 퐵 have Jordan bases for the same Jordan subspaces, 퐴 퐵 but the respective total variations of the spanning Jordan 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 manner that is invariant to the choice of Jordan basis, we the nodes. This is significant because it means that an derive here a definition of the total variation of a spectral ordering of the spectral components by their class total component of 퐴 in relation to the Jordan equivalence variations is invariant to node labels. 퐽 class G퐴. Next, the significance of the class total variation (68) Class total variation. Let 풢(퐵) be an element in is illustrated for adjacency matrices with diagonal Jordan 퐽 Jordan equivalence class G퐴 where 퐵 has Jordan de- form, one Jordan block, and multiple Jordan blocks.

10 Diagonal Jordan Form. Section IV-D shows that a variation of J = C푁 satisﬁes graph shift 퐴 with diagonal Jordan form is the single 퐽 TVG퐽 (J푖) = sup TV퐺 (푉 ) (83) element of its Jordan equivalence class G퐴. This yields 퐴 퐽 풢(퐵)∈G퐴 the following result. 퐵=푉 퐽푉 −1 푁 span{푉 }=J =C ‖푉 ‖ =1 Theorem 24. Let 풢(퐴) have diagonalizable adjacency 1 matrix 퐴 with eigenvectors 푣1, . . . , 푣푁 . Then the class = TV퐺 (퐼) (84) total variation of the spectral component J푖, 푖 = = |1 − 휆| + 1. (85) 1, . . . , 푁, of 퐴 satisﬁes (for ‖푣푖‖ = 1)

TV 퐽 (J푖) = |1 − 휆푖| . (74) G퐴 Multiple Jordan blocks. Theorem 26 proves that 퐽 graphs in the Jordan equivalence class G퐽 where 퐽 is Proof: Each spectral component J푖 of 퐴 is the in Jordan normal form attains the bound (67). span of eigenvector 푣푖 corresponding to eigenvalue 휆푖. 퐽 The class total variation of J푖 is then Theorem 26. Let 풢(퐴) ∈ G퐽 where 퐽 is the Jordan normal form of 퐴 and J퐴 = {J푖푗}푖푗 for 푖 = 1, . . . , 푘, TVG퐽 (J푖) = sup TV퐺 (푣푖) (75) 퐴 퐽 푗 = 1, . . . , 푔푖. Then the class total variation of J푖푗 is 풢(퐵)∈G퐴 퐵=푉 퐽푉 −1 |1 − 휆푖| + 1. span{푣푖}=J푖 ‖푣 ‖ =1 퐽 푖 1 Proof: Since 풢(퐴) ∈ G퐽 , the GFT can be com- = TV퐺 (푣푖) (76) puted over 풢(퐽) with eigenvector matrix 푉 = 퐼. Then

each 푉푖푗 = 퐼푟푖푗 that spans J푖푗 has total variation = ‖푣푖 − 퐵푣푖‖1 (by (57)) (77) ⃦ ⃦ = ‖푣푖 − 휆푖푣푖‖1 (78) TV퐺 (퐼푖푗) = ⃦퐼푟푖푗 − 퐽푖푗⃦1 (86)

= |1 − 휆푖| ‖푣푖‖1 (79) = |1 − 휆푖| + 1 (by (67)). (87) = |1 − 휆푖| . (80) Therefore,

TV 퐽 (J푖) = sup TV퐺 (푉푖푗) (88) G퐽 풢(퐵)∈G퐽 Theorem 24 is consistent with the total variation result 퐴 퐵=푉 퐽푉 −1 for diagonalizable graph shifts in [4]. Next, the class total span{푉푖푗 }=J푖푗 ‖푉 ‖ =1 variation for defective graph shifts is characterized. 푖푗 1 = TV (︀퐼 )︀ (89) One Jordan block. Consider the graph shift 퐴 with 퐺 푟푖푗 푁 a single spectral component J = C and Jordan = |1 − 휆푖| + 1. (90) form 퐽 = 퐽(휆). The next theorem proves that the total variation of J attains the upper bound (67). Although the total variation upper bound may not be 푁×푁 attained for a general graph shift 퐴, choosing this bound Theorem 25. Consider unicellular 퐴 ∈ C with Jordan normal form 퐽 = 퐽(휆). Then the class total as the ordering function provides a useful standard for variation of G퐽 is |1 − 휆| + 1. comparing spectral components for all graphs in a Jordan 퐴 equivalence class. The ordering proceeds as follows:

Proof: Graph 풢(퐴) is Jordan equivalent to 풢(퐽) 1) Order the eigenvalues 휆1, . . . , 휆푘 of 퐴 by increasing since 퐴 is unicellular. Therefore, the GFT of a graph |1 − 휆푖| + 1 (from low to high total variation). signal can be computed over 풢(퐽) by choosing the 2) Permute submatrices 푉푖푗 of eigenvector matrix 푉 the canonical vectors (푉 = 퐼) as the Jordan basis, as to respect the total variation ordering. shown in (48). By (66), the maximum of |1 − 휆| ‖푣 ‖ 1 1 Since the ordering is based on the class total vari- and ‖|1 − 휆| 푣 − 푣 ‖ for 푖 = 2, . . . , 푁 needs to be 푖 푖−1 1 ation (68), it is invariant to the particular choice of computed. The former term equals |1 − 휆| since 푣 is 1 Jordan basis for each nontrivial Jordan subspace. Such the ﬁrst canonical vector. The latter term has form an ordering can be used to study low frequency and high ⃦⎡ ⎤⃦ ⃦ 0 ⃦ ⃦ −1 ⃦ frequency behaviors of graph signals; see also [4]. ‖ |1 − 휆| 푣푖 − 푣푖−1‖ = ⃦⎣ ⎦⃦ (81) 1 ⃦ |1 − 휆| ⃦ ⃦ 0 ⃦1 = 1 + |1 − 휆| , (82) VI.EXAMPLE

Since |1 − 휆| + 1 > |1 − 휆|, TV퐺(퐼) = 1 + |1 − 휆|. This section illustrates the Jordan equivalence classes Therefore, (67) holds with equality, so the class total of Section IV and total variation ordering of Section V

11 2 These results show that the degrees of freedom in the X: 3.933 Y: 2 Jordan chain recurrence (5) can lead to fluctuating total 1.8 variations of the spectral components. We compare these results to the upper bound (67), 1.6 which is |휆 − 1| + 1 = 2 for 휆 = 0. Our results show that this upper bound is achieved with 푉3 (95). Total Variation 1.4 X: 0 In this way, the class total variation (68) of the Jordan Y: 1.181 1.2 subspace J2(0) = span{푉3} corresponding to Jordan block 퐽 (0) is 0 2 4 6 8 10 12 14 16 18 20 2 v 6 TV 퐽 (J2(0)) = 2. (99) G퐴 Fig. 3: Total variation of the spectral component of 퐽2(0) for the example in Section VI with respect to generalized This example shows that using the class total variation eigenvector component 푣6. The data points (gray squares) show or the upper bound (67) as a method of ranking the 59 total variation 1.181 when 푣6 = 0 and 2 when 푣6 = 15 . spectral components by (57) removes the dependency on the choice of generalized eigenvector. We modify 푉 (95) by varying the sixth component 푣 on the 10 × 10 matrix example 3 6 (and fourth component 푣4 as 푣4 = 1 − 1.5푣6) of the ⎡0 0 0 −2 0 −3 0 0 0 0⎤ generalized eigenvector in the second column. It can be 0 0 0 0 0 0 0 1 0 0 ⎢5 0 0 0 0 0 2 0 0 0⎥ verified by (5) that such vectors are valid generalized ⎢ ⎥ ⎢0 0 0 0 6 0 0 0 0 0⎥ eigenvectors. The results are shown in Figure 3 with ⎢0 0 0 0 0 0 0 1 0 0⎥ 퐴 = ⎢ ⎥ . (91) the total variation plotted versus the value of 푣6. The ⎢0 0 0 0 0 0 0 0 −2 0⎥ ⎢0 0 0 0 0 0 0 0 0 3⎥ data point at 푣6 = 0 corresponds to the total variation ⎢0 0 0 0 0 0 0 4 0 0⎥ ⎣ ⎦ of 푉1 (93). The figure illustrates that the total variation 0 1 0 0 0 0 0 0 0 0 59 0 0 −1 0 0 0 0 0 0 0 has a global maximum at 푣6 = 15 . Jordan equivalence. It can be shown that the images The Jordan normal form of 퐴 is √ √ √ of the projection matrices (12) corresponding to 푉1(93), (︀ 3 3 2 3 )︀ 퐽 = diag 4, −6휔, −6휔 , −6, 퐽4(0), 퐽2(0) , 푉2 (94), and 푉3 (95) are nonidentical; that is, each (92) choice of Jordan basis corresponds to a different Jordan where 휔 = exp(2휋푗/3) and 퐽4(0) and 퐽2(0) are 4 × 4 equivalence class. and 2 × 2 Jordan blocks corresponding to eigenvalue Consider an alternate basis for J2(0) = span{푉3} zero, respectively. provided by the columns of matrix Total variation. Possible Jordan chains for the Jordan [︂1 0 0 3 0 1 2 0 0 0]︂푇 block 퐽 (0) and their respective total variations (57) 2 푉̃︀1 = 0 0 0 −1 1 1 0 0 1 5 . (100) are computed. By applying the recurrence equation (5) 2 3 with 휆 = 0, the following eigenvector submatrices with If 푉̃︀ is defined as the matrix consisting of the columns columns that span potential Jordan subspaces corre- of 푉 that do not correspond to J2(0) in addition to the sponding to 퐽2(0) in (92) are obtained: −1 columns of 푉̃︀1 (100), it can be shown that 퐴̃︀ = 푉̃︀ 퐽푉̃︀ [︃ ]︃푇 does not equal 퐴. Nevertheless, the oblique projection −2 0 0 3 0 −2 5 0 0 0 matrices (12) corresponding to 푉1 (100) and 푉3 (95) 푉1 = 1 5 , (93) ̃︀ 0 0 0 1 2 0 0 0 1 3 onto the Jordan subspaces are identical; that is, the [︃ ]︃푇 GFT (1) is equivalent for both eigenvector matrices, −2 0 0 3 0 −2 5 0 0 0 푉2 = , (94) and graphs 풢(퐴) and 풢(퐴̃︀) are in the same Jordan 0 0 0 − 1 1 1 0 0 1 5 2 2 3 equivalence class corresponding to J2(0) = span{푉3}. [︃ ]︃푇 −2 0 0 3 0 −2 5 0 0 0 The total variation of 푉̃︀1 with respect to 퐴̃︀ is 푉3 = 49 1 59 5 . (95) ⃦ ⃦ 0 0 0 − 10 2 15 0 0 1 3 TV퐺(푉̃︀1) = ⃦푉̃︀1 − 퐴̃︀푉̃︀1⃦ = 1.452. (101) ⃦ ⃦1 Normalizing these matrices by their L1 norm as spec- Thus, 푉 does not achieve the class total variation (99). ified by Property 22 in Section V, the resulting total ̃︀1 variations (57) are VII.LIMITATIONS TV퐺(푉1) = 1.181 (96) The Jordan equivalence classes discussed in Sec- TV (푉 ) = 1.389 (97) 퐺 2 tion IV show that there are degrees of freedom over TV퐺(푉3) = 2. (98) graph topologies with defective adjacency matrices that

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