The Jordan canonical form for a class of weighted directed graphs

Linear Algebra and its Applications 438 (2013) 261–268

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The Jordan canonical form for a class of weighted directed graphs < Hans Nina a , Ricardo L. Soto a,∗ , Domingos M. Cardoso b a b

Departamento de Matemáticas, Universidad Católica del Norte, Antofagasta, Chile Departamento de Matemática, Universidade de Aveiro, Portugal

ARTICLE INFO

ABSTRACT

Article history: Received 9 March 2012 Accepted 28 July 2012 Available online 30 August 2012

Recently, Cardon and Tuckfield (2011) [1] have described the Jordan canonical form for a class of zero-one matrices, in terms of its associated directed graph. In this paper, we generalize this result to describe the Jordan canonical form of a weighted adjacency matrix A in terms of its weighted directed graph. © 2012 Elsevier Inc. All rights reserved.

Submitted by R.A. Brualdi AMS classification: 15A18 15A21 Keywords: Jordan canonical form Weighted directed graph Weighted adjacency matrix

1. Introduction In this paper we show for instance, how to compute the Jordan canonical form of the matrix ⎡

A

=

0 1 0

⎢ ⎢ ⎢ 3i 0 0 ⎢ ⎢ ⎢0 0 0 ⎢ ⎢ ⎢ 0 0 −4 ⎣

0

0

⎤

⎥ ⎥ 1 − 2i ⎥ ⎥ ⎥ 0 0 ⎥ ⎥ ⎥ 0 0 ⎥ ⎦ 0 0 0 1+i 0 0

< Supported by Fondecyt 1120180 and Mecesup UCN0711, Chile. ∗ Corresponding author. E-mail addresses: [email protected] (H. Nina), [email protected] (R.L. Soto), [email protected] (D.M. Cardoso). 0024-3795/$ - see front matter © 2012 Elsevier Inc. All rights reserved. http://dx.doi.org/10.1016/j.laa.2012.07.038

(1)

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H. Nina et al. / Linear Algebra and its Applications 438 (2013) 261–268

in terms of the weighted directed graph  (see Fig. 1) for which A is the weighted adjacency matrix. A recent result of Cardon and Tuckfield [1] describes the Jordan canonical form, of a certain n × n zeroone matrix An , in terms of the directed graph for which An is the adjacency matrix. In this paper we generalized the result in [1], to describe the Jordan canonical form of a weighted adjacency matrix A in terms of its weighted directed graph (see Fig. 2). Although the weights pj are usually define as positive, we also show that the matrix A many contain complex (non-real) entries. We follow the paper [1] to give, for lack of completeness, the necessary definitions and basic results to understand the procedure to construct the Jordan canonical form: Let f : N → N be a function. For a given n ∈ N, let A = (aij ) be the n × n matrix defined by

aij

=

⎧ ⎨ p , if i = f (j) for some i, j j ⎩ 0, otherwise.

∈ {1, . . . , n}

(2)

Then A is the weighted adjacency matrix for the weighted directed graph  , with vertices labeled 1, 2, . . . , n having a weight pj ∈ C  {0} associated to the edge from vertex j to vertex i if and only if i = f (j). We consider a partition of the weighted directed graph  , into chains and cycles. The Jordan decomposition of the weighted adjacency matrix A will be related to the lengths of these chains and cycles. Definition 1.1. A chain in  is an ordered list of distinct vertices C = {c1 , c2 , . . . , cr } such that f (cj ) = cj+1 for 1  j < r , but f (cr )  = c1 . A cycle in  is an ordered list of distinct vertices Z = {z1 , z2 , . . . , zr } such that f (zj ) = zj+1 for 1  j < r , but f (zr ) = z1 . We call r the length of the chain or cycle and write len C or len Z . A single vertex {i} is a chain or a cycle. Since either f (i) cycle.

= i or f (i) = i, {i} is not both a chain and a

Definition 1.2. If C = {c1 , c2 , . . . , cs } is a chain of  , then cs is called the terminal point of the chain. A vertex k of  such that f (k) > n is a terminal point of  . If k is a vertex of  such that f (i) = f (j) = k for some i and j with i  = j, then k is said a merge point of  . Definition 1.3. A partition of  is a collection of disjoint cycles and chains whose union is  . A proper partition of  is a partition P

= {Z1 , . . . , Zr , C1 , . . . , Cs },

where Z1 , . . . , Zr are cycles and C1 , . . . , Cs are chains satisfying: 1. Each cycle in  is equal to Zi for some i. 2. If  (i) is the subgraph of  obtained by deleting the vertices in the cycles Z1 , . . . , Zr and in the chains C1 , . . . , Cs , then Ci+1 is a chain of maximal length in  (i) . It is proved in [1], that proper partitions of  exist, and that if C proper partition of  , then exactly one of the following occurs:

= {c1 , . . . , cm } is a chain in a

1. The terminal point cm of the chain is a terminal point of the graph  . 2. The point f (cm ) is a merge point of  . In Section 2 we show how the chains and cycles of the weighted directed graph are associated to the eigenvalues of the weighted adjacency matrix, and we prove our main result, Theorem 2.3, which describes the Jordan structure of A. We also give two examples to illustrate the results.

H. Nina et al. / Linear Algebra and its Applications 438 (2013) 261–268

263

2. The Jordan structure of A Let A be the matrix defined in (2). In this section we describe the Jordan canonical form of A, in terms of the weighted directed graph  for which A is the weighted adjacency matrix. Definition 2.1. Let A be an n × n matrix with complex entries. A nonzero vector v is a generalized eigenvector of A corresponding to the complex number λ if (A − λI )p v = 0 for some positive integer p. Definition 2.2. Let v be a generalized eigenvector of A for the eigenvalue λ and let p be the smallest positive integer such that (A − λI )p v = 0. Then the ordered set

{(A − λI )p−1 v , (A − λI )p−2 v , . . . , (A − λI )v , v }, is a chain of generalized eigenvectors of A corresponding to λ. Proofs of the following standard results can be found in [2,3]: Proposition 2.1. Let λ be an eigenvalue of A and let γ1 , . . . , γs be chains of generalized eigenvectors of A form a linearly independent set, then the chains are corresponding to λ. If the initial vectors of the chains  disjoint (γi ∩ γj = ∅ for i  = j) and the union si=1 γi is linearly independent. Proposition 2.2. Let A be an n × n complex matrix. Then there exists a basis β of Cn consisting of disjoint chains β1 , . . . , βr of generalized eigenvectors of lengths n1 , . . . , nr for the eigenvalues λ1 , . . . , λr with n = n1 + · · · + nr such that if Q is the matrix whose columns are the members of the basis β, then Q −1 AQ

= Jn1 (λ1 ) ⊕ · · · ⊕ Jnj (λj ),

where Jnk (λk ) denotes the nk

× nk Jordan block associated to λk .

Next, we show that the eigenvalues associated with the cycles of a proper partition P of the weighted directed graph  are multiples of a primitive root of unity, while the eigenvalues associated with the chains are all zeros. Lemma 2.1. Let Z = {z1 , . . . , z } be any cycle of a proper partition P of the weighted directed graph  , and let ω = exp(2π i/) be a primitive th root of unity. Let κ be any particular th root of the product  k i=1 pzi . Then λk = κω is an eigenvalue of A with

vk

=



j=1

⎛ ⎝κ −j ω−kj

j −1 i=1

⎞ pzi ⎠ezj ,

as its associated eigenvector for k span{v1 , . . . , v }

= 1, . . . , . We will say that vk is attached to the vertex zk . Also,

= span{ez1 , . . . , ez },

where ei is the vector with 1 in the ith position and zeros elsewhere.

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H. Nina et al. / Linear Algebra and its Applications 438 (2013) 261–268

= pzj ezj+1 , 1  j < , Aez = pz ez1 and ω = 1, we have

Proof. Since Aezj

Avk

= =

⎛



⎝κ −j ω−kj

j=1

−

1 j=1

⎛ ⎝κ −j ω−kj

= ez1 + = κω κ k

= κω

k

−

1 j=2

⎛

ω

j=1

i=1 j −1 i=1

⎞ pzi ⎠ Aezj ⎞ pzi ⎠ pzj ezj+1

⎝κ −j ω−kj

−1 −k



j −1

+

ez1

⎛ ⎝κ −j ω−kj

j  i=1



j=2 j −1 i=1

1 + ω−k p− z pz ez1

⎞ pzi ⎠ ezj+1

⎛ ⎝κ −(j−1) ω−k(j−1) ⎞

pzi ⎠ ezj

j −1 i=1

⎞ pzi ⎠ ezj

= κωk vk = λk vk .

Since v1 , . . . , v all belong to distinct eigenvalues they form a linearly independent set whose span has dimension . But span{v1 , . . . , v } is a subspace of span{ez1 , . . . , ez } whose dimension is also . Thus, the two spaces are equal.  Lemma 2.2 [1]. In a proper partition P = {Z1 , . . . , Zr , C1 , . . . , Cs } of  , the set of all eigenvectors attached to the vertices in the cycles Z1 , . . . , Zr is linearly independent. Lemma 2.3. Let C

= {c1 , . . . , cm } be any chain in a proper partition P of the weighted directed graph  .

(i) If cm is a terminal point of  , then 0 is an eigenvalue of A, and  ecm ,

ecm−1 pcm−1

,

ecm−2 pcm−2 pcm−1

,...,

ec2 pc2



ec1

,

. . . pcm−1 pc1 pc2 . . . pcm−1

is a chain of generalized eigenvectors of A corresponding to 0.

(ii) If f (cm ) is a merge point of  and z is the vertex in the cycle or chain containing f (cm ) in such a way that f m (z ) = f (cm ), then 0 is an eigenvalue of A and 

ecm pcm

−

ef m−1 (z)

,

ecm−1

pf m−1 (z) pcm−1 pcm ec2 pc2

· · · pcm

−

−

ef m−2 (z) pf m−2 (z) pf m−1 (z) ef (z)

,

pf (z) · · · pf m−1 (z) pc1

,..., ec1

· · · pcm

is a chain of generalized eigenvectors of  A corresponding to 0.  In the first case, we say that the vector that the vector ecj m

i=j pci

ef j−1 (z)

− m−1

i=j−1 pf i (z)

is attached to the vertex cj .

ecj

pcj ···pcm

−

ez



pf (z) · · · pf m−1 (z)

is attached to the vertex cj . In the second case, we say

H. Nina et al. / Linear Algebra and its Applications 438 (2013) 261–268

265

Proof. Let cm be a terminal point of  , then f (cm ) > n and the cm th column of A is zero. Then Aecm Besides, Aeci = pci eci+1 for 1  i < m. Therefore  A



eci

m

pci

= 0.

= 0.

· · · pcm

Thus, (i) is satisfied. Now, if f (cm ) is a merge  point of  , letz the vertex in the cycle or chain containing f (cm ) such that f m (z )  A

ecm pcm

−

ef m−1 (z) pf m−1 (z)

ecm−(j−1)

A

pm−(j−1)

· · · pcm

ef m−1 (z)

pf m−1 (z)

is nonzero since f m−1 (z )

= cm . Therefore,

= ef (cm ) − ef m (z) = 0. 



−



Thus, zero is an eigenvalue of A and

j

ecm pcm

= f (cm ). The vector

−

ecm pcm

−

ef m−1 (z) pf m−1 (z)

ef m−j (z) pf m−j (z) · · · pf m−1 (z)

 is its associated eigenvector. Since C is a chain,



= 0, 1  j < m.

Thus, (ii) is proved.  Corollary 2.1. Every eigenvalue of A is either 0 or a multiple of a primitive root of unity. Lemma 2.4. Let P = {Z1 , . . . , Zr , C1 , . . . , Cs } be a proper partition of  . The set of all generalized eigenvectors attached to vertices of the cycles Z1 , . . . , Zr and to the vertices of the chains C1 , . . . , Cs is a linearly independent set of n eigenvectors. Consequently, this set forms a Jordan basis of Cn for the matrix A. Proof. The proof is entirely similar to the proof of Lemma 11 in [1].  Theorem 2.3. Let f : N → N be a function and for a given n ∈ N, let  be its associated weighted directed graph. Let A be the weighted adjacency matrix defined in (2). Suppose that P

= {Z1 , . . . , Zr , C1 , . . . , Cs }

is a proper partition of  , where Z1 , . . . , Zr are the cycles and C1 , . . . , Cs are the chains. Let the lengths of the cycles and chains be denoted as len Zj = j , 1  j  r , len Cj = mj , 1  j  s. Let ωj = exp(2π i/j ) be a primitive j th root of unity and let κ be any particular th root of the product i=1 pzi . The Jordan decomposition of A contains the following 1 × 1 Jordan blocks for the eigenvalues which are multiples of a root of unity:  J1

κωjk

 for 1

 j  r and 1  k  j .

The Jordan decomposition contains the following blocks associated with the eigenvalue 0: Jm1 (0), Jm2 (0), . . . , Jms (0). Proof. In Lemma 2.1 it was shown that the eigenvalues of A associated with the cycles are multiple of a root of unity, and we attach an eigenvector of A, associated to an eigenvalue which is multiple of a root of unity, to each vertex of each cycle of  . In Lemma 2.2 (see [1]) it is shown that this set of eigenvectors is linearly independent. In Lemma 2.3, we show that the eigenvalues associated with the chains are all zeros, and we attach chains of generalized eigenvectors of A for the eigenvalue 0 to

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H. Nina et al. / Linear Algebra and its Applications 438 (2013) 261–268

Fig. 1. Directed graph with complex weights.

chains of the graph  . In Lemma 2.4 it is shown that these generalized eigenvectors also form a linearly independent set and the set of all generalized eigenvectors attached to the vertices of  via Lemmas 2.1 and 2.3 is a Jordan basis of Cn for the matrix A.  We finish this section with the following Example 2.1. We apply Theorem 2.3 to find the Jordan canonical form J (A) of the matrix A in (1): ⎡

A

=

0 1 0

⎢ ⎢ ⎢ 3i 0 0 ⎢ ⎢ ⎢0 0 0 ⎢ ⎢ ⎢ 0 0 −4 ⎣

0

0

⎤

⎥ ⎥ 1 − 2i ⎥ ⎥ ⎥ 0 0 ⎥. ⎥ ⎥ 0 0 ⎥ ⎦ 0 0 0 1+i 0 0

The associated weighted directed graph  is shown in Fig. 1. A proper partition of  is Z1 = {1, 2}, C1 = {3, 4, 5}. The weights are p1 = 3i, p2 = 1, p3 = −4, p4 = 1 + i, p5 = 1 − 2i. Then ω = exp(2π i/2) is a primitive root of unity and ω2 = 1. Besides, f (5) = f (1) = 2 is a merge point. From Lemmas 2.1, 2.3 and 2.4 we compute the matrix ⎡

Q

=

1√ −i

⎢ 6 ⎢ ⎢ −1 ⎢ ⎢ ⎢ 0 ⎢ ⎢ ⎢ ⎢ 0 ⎣ 0

−√ 1+i 6

i 3

−1

0

0

0

0

0

0

1+2i 5

0

1 9

i 3

0

6 (1 2

+ i)

⎤

⎥ ⎥ ⎥ ⎥ ⎥ −3−i ⎥ 0 40 ⎥ ⎥ ⎥ 3+i 0 ⎥ 10 ⎦ 0 0

and ⎡

J (A)

= Q −1 AQ =

⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣

−

√

√

0

0

6 (1 2

0

0

0

0

0

0

0 0 0

⎤ ⎥

+ i) 0 0 0 ⎥ ⎥

⎥ ⎥ 0 1 0⎥. ⎥ ⎥ 0 0 1⎥ ⎦ 0 0 0

Example 2.2. Consider the weighted directed graph  (see Fig. 2): whose weighted adjacency matrix is

H. Nina et al. / Linear Algebra and its Applications 438 (2013) 261–268

267

Fig. 2. Directed graph with real weights.

⎡

A

=

0 0 0

⎢ ⎢ ⎢0 ⎢ ⎢ ⎢5 ⎢ ⎢ ⎢0 ⎢ ⎢ ⎢0 ⎢ ⎢ ⎢0 ⎢ ⎢ ⎢ ⎢0 ⎢ ⎢ ⎢0 ⎢ ⎢ ⎢0 ⎣ 0

−2 0 −3 0 0 0 0

⎤

⎥ ⎥ 0 0 0 0 0 0 0⎥ ⎥ ⎥ 0 0 0 0 2 0 0 0⎥ ⎥ ⎥ 0 0 6 0 0 0 0 0⎥ ⎥ ⎥ 0 0 0 0 0 0 0 0⎥ ⎥ ⎥. 0 0 0 0 0 0 −2 0 ⎥ ⎥ ⎥ ⎥ 0 0 0 0 0 0 0 3⎥ ⎥ ⎥ 0 0 0 0 0 4 0 0⎥ ⎥ ⎥ 0 0 0 0 0 0 0 0⎥ ⎦ −1 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0

A proper partition of  is Z1 = {8}, Z2 = {7, 3, 10}, C1 two cycles with corresponding eigenvalues

λ1 =

= {5, 4, 1}, C2 = {9, 6}, C3 = {2}. We have

√ √ √ 3 −6ω, λ2 = 3 −6ω2 , λ3 = 3 −6 and μ1 = 4.

Then from Theorem 2.3 we have that J (A) = diag {J1 (μ1 ), J1 (λ1 ), J1 (λ2 ), J1 (λ3 ), J3 (0), J2 (0), J1 (0)} ⎤ ⎡ 4 0 0 0 0 0 0 0 0 0 ⎥ ⎢ √ ⎥ ⎢ ⎢ 0 3 −6ω 0 0 0 0 0 0 0 0⎥ ⎥ ⎢ √ ⎥ ⎢ 3 2 ⎥ ⎢0 0 − 6 ω 0 0 0 0 0 0 0 ⎥ ⎢ √ ⎥ ⎢ 3 ⎥ ⎢0 0 0 − 6 0 0 0 0 0 0 ⎥ ⎢ ⎥ ⎢ ⎥ ⎢0 0 0 0 0 1 0 0 0 0 ⎥ ⎢ ⎥ =⎢ ⎢0 0 0 0 0 0 1 0 0 0⎥ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ 0 0 0 0 0 0 0 0 0⎥ ⎢0 ⎥ ⎢ ⎥ ⎢ ⎢0 0 0 0 0 0 0 0 1 0⎥ ⎥ ⎢ ⎥ ⎢ ⎢0 0 0 0 0 0 0 0 0 0⎥ ⎦ ⎣ 0 0 0 0 0 0 0 0 0 0

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H. Nina et al. / Linear Algebra and its Applications 438 (2013) 261–268

is the Jordan canonical form of A. Besides, we may compute from Theorem 2.3, the matrix ⎡

Q

=

0

⎢ ⎢ ⎢0 ⎢ ⎢ ⎢0 ⎢ ⎢ ⎢1 ⎢ ⎢ ⎢0 ⎢ ⎢ ⎢0 ⎢ ⎢ ⎢ ⎢0 ⎢ ⎢ ⎢0 ⎢ ⎢ ⎢0 ⎣ 0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

1 6

0

0

1 2

√ 3

6ω

√ 3

such that Q −1 AQ

√ 3

6

0

0

0

0

1 − 10

0

0

0

1 5

0

0

0

0

− 13

− 12

0

0

0

0

−(

6ω2

√ 3

0

6) 2

2

ω2 − (

√ 3

0

6) 2

2

ω −(

√ 3

6) 2

2

1 − 60 0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

− 16

0

0

1

1

1

0 0

⎤

⎥ ⎥ 0 1⎥ ⎥ ⎥ 0 0⎥ ⎥ ⎥ 0 0⎥ ⎥ ⎥ 1 ⎥ 0 ⎥ 2 ⎥ 0 0⎥ ⎥ ⎥ ⎥ 0 0⎥ ⎥ ⎥ 0 0⎥ ⎥ ⎥ 1 ⎥ 0 6 ⎦ 0 0

= J (A).

Acknowledgment The authors thank the anonymous referee for his/her helpful suggestions which improved the presentation of this paper. References [1] D.A. Cardon, B. Tuckfield, The Jordan canonical form for a class of zero-one matrices, Linear Algebra and its Applications 435 (2011) 2942–2954. [2] R.A. Horn, C.R. Johnson, Matrix Analysis, Cambridge University Press, Cambridge, 1990. [3] P. Lancaster, M. Tismenetsky, The Theory of Matrices. Computer Science and Applied Mathematics, Academic Press, FL, 1985.