A graph-theoretic condition for irreducibility of a set of cone preserving matrices

0.16 MB·English

by  Murad Banaji

A graph-theoretic condition for irreducibility of a set of
cone preserving matrices
Murad Banajia,∗, Andrew Burbanksa
1
1 aDepartment of Mathematics, University of Portsmouth, Lion Gate Building, LionTerrace,
0 Portsmouth, Hampshire PO1 3HF, UK.
2

c
e
D
Abstract
7
Given a closed, convex and pointed cone K in Rn, we present a result which

] infers K-irreducibility of sets of K-quasipositive matrices from strong connect-
O
edness of certain bipartite digraphs. The matrix-sets are defined via products,
C
andthemainresultisrelevanttoapplicationsinbiologyandchemistry. Several
.
h examples are presented.
t
a Keywords: partial order, convex cone, irreducibility, strong monotonicity
m
2000 MSC: 15B48,15B35
[

1
v 1. Introduction
3
5 A digraph G is strongly connected, or irreducible, if given any vertices u
6
and v, there exists a (directed) path from u to v in G. It is well known that a
1
digraphisstronglyconnectedifandonlyifitsadjacencymatrixisirreducible[1].
.
2 Here, given a cone K, we present a result which infers K-irreducibility of sets
1
of K-quasipositive matrices from strong connectedness of associated bipartite
1
digraphs. Graph-theoretic approaches to K-irreducibility of sets of K-positive
1
: matrices have been described in earlier work [2, 3, 4]. These approaches are
v
somewhatdifferent in structure and philosophy to that described here. We will
i
X comment further on this in the concluding section.
r We will be interested in closed, convex cones in Rn which are additionally
a
pointed (i.e., if y ∈ K and y 6= 0, then −y 6∈ K). Closed, convex and pointed
coneswill be abbreviatedas CCP cones. We do not assumethe conesaresolid
(i.e., have nonempty interior in Rn) – however if a CCP cone is, additionally,
solid, then it will be termed a proper cone. For basic definitions and results
on cones in Rn the reader is referred to [1, 5].
Let K ⊂ Rn be a CCP cone. Consider an n×n matrix M. Recall that
M is K-positive if MK ⊆ K. Defining Rn to be the (closed) nonnegative
≥0
orthant in Rn, a nonnegative matrix is then Rn -positive. We will refer to M
≥0
as K-quasipositive if there exists an α ∈ R such that M +αI is K-positive.
∗Correspondingauthor: [e-mail].
Preprint submitted toElsevier December8, 2011
Rn -quasipositive matrices – generally referred to simply as quasipositive, or
≥0
Metzler – are just those with nonnegative off-diagonal entries. We define M
to be K-reducible if there exists a nontrivial face F of K such that M leaves
spanF invariant. This is a slight generalisation of the original definition of K-
reducibility for K-positive matrices [6] in order to allow us to apply the term
to matrices which are not necessarily K-positive. A matrix which is not K-
reducible is K-irreducible. Note that an irreducible matrix could be termed
Rn -irreducibleinthisterminology. AlternativelyanyotherorthantinRncould
≥0
be chosen as K.
Remark 1. Given a CCP cone K ⊂Rn, an n×n matrix M is K-irreducible if
and only if M+αI is K-irreducible for each α∈R. In one direction we choose
α =0. The other direction follows because given any face F of K, spanF is a
vector subspace of Rn.
Motivation from dynamical systems. Motivation for examining K-
irreducibility of a set of K-quasipositive matrices comes from the theory of
monotonedynamicalsystems[7,8]. Convex,pointedconesdefinepartialorders
in a natural way. Given a proper cone K and a C1 vector field f :Rn →Rn, if
theJacobianmatrixDf(x)isK-quasipositiveandK-irreducibleateachx∈Rn,
then the associated local flow is strongly monotone with respect to the partial
orderdefinedbyK. Thisresultandavarietyoftechnicalmodificationsprovide
useful conditions which can be used to deduce the asymptotic behaviour of
dynamical systems.
Remark 2. Although the main results on monotone dynamical systems require
the order cone to be solid, a CCP cone K which fails to be solid is as useful as
a proper one when the local flow or semiflow leaves cosets of spanK invariant.
Trivially, K has nonempty relative interior in spanK, and attention can be
restricted to cosets of spanK. This situation arises frequently in applications to
biology and chemistry.
2. Some background material
Sets of matrices. It is convenient to use some notions from qualitative
matrix theory. Let M be a real matrix.
1. Q(M), the qualitative class of M [9] is the set of all matrices with the
same dimensions and sign pattern as M, namely if N ∈ Q(M), then
M >0⇒N >0, M <0⇒N <0 and M =0⇒N =0.
ij ij ij ij ij ij
2. Q (M) will stand for the closure of Q(M), namely if N ∈ Q (M), then
0 0
M >0⇒N ≥0, M <0⇒N ≤0 and M =0⇒N =0.
ij ij ij ij ij ij
3. Q (M) will be the set of all matrices N with the same dimensions as M
1
and satisfying M N ≥0.
ij ij
Clearly, Q(M)⊆Q (M)⊆Q (M).
0 1
2
Remark 3. Suppose K is an orthant in Rn and M is an n×n K-quasipositive
matrix. It can easily be shown that each matrix in Q (M) is K-quasipositive.
0
Since in this case K-irreducibility is simply irreducibility, if M is K-irreducible
then each matrix in Q(M) is K-irreducible. However, the same is not true for
Q (M) which, after all, contains the zero matrix.
0
Notation for matrices. Given any matrix M, we refer to the kth column
of M as M and the kth row of M as Mk. We define the new matrix M(k) by
k
M(k) = M if i = k and M(k) = 0 otherwise: i.e., M(k) is derived from M by
ij ij ij
replacing all entries not in the kth row with zeros. We define a set of matrices
M to be row-complete if M ∈M⇒M(k) ∈M for each k. Clearly, givenany
matrixM,Q (M)isrow-complete;butmuchsmallersetscanberow-complete.
0
For example, given some fixed m×n matrix N,
M={DN : D is a nonnegative m×m diagonal matrix}
is row-complete.
Digraphs associated with square matrices. Givenann×n matrix M,
let G be the associated digraph on n vertices u ,...,u defined in the usual
M 1 n
way: the arc u u exists in G iff M 6=0.
i j M ij
Remark 4. Following on from Remark 3, if K is an orthant in Rn, then K-
irreducibility of an n × n matrix M is equivalent to strong connectedness of
G .
M
Bipartite digraphs associated with matrix-pairs. Given an n × m
matrix A and an m×n matrix B, define a bipartite digraph G on n+m
A,B
verticesasfollows: associateasetofnverticesu ,...,u withtherowsofA(we
1 n
will refer to these as the “row vertices” of G ); associate another m vertices
A,B
v ,...,v withthecolumnsofA(wewillrefertotheseasthe“columnvertices”
1 m
of G ); add the arc u v iff A 6=0; add the arc v u iff B 6=0.
A,B i j ij j i ji
Remark 5. Thisisaspecialisationofthegeneralconstructionofblock-circulant
digraphs from sets of appropriately dimensioned matrices in [10]. If A and B
are binary matrices (with AB a square matrix), then the adjacency matrix of
G is simply
A,B
0 A
.
(cid:18) B 0 (cid:19)
The context of the main result. Fundamental early results on con-
vergence in monotone dynamical systems [11] apply to systems with Jaco-
bian matrices which are quasipositive and irreducible (in our terminology Rn -
≥0
quasipositive and Rn -irreducible). Generalising from Rn to all orthants is
≥0 ≥0
straightforward: where K is an orthant, there is a simple graph-theoretic test
[7]todecideK-quasipositivityofagivenn×nmatrixM. ByRemarks3and4,
K-quasipositivity extends to all of Q (M), and K-irreducibility of some M′ ∈
0
Q0(M) is equivalent to strong connectedness of GM′.
3
WeareinterestedinhowthissituationgeneraliseswhenK isnotnecessarily
anorthant. Ingeneral,givenaK-quasipositivematrixM,wecanrarelyexpect
all matrices in Q (M) to be K-quasipositive. However the following situation
0
is notuncommon: there arematricesA andB˜ suchthat AB is K-quasipositive
for each B ∈ Q (B˜). The practical relevance is to applications in biology and
0
chemistry where Jacobian matrices often have a constant initial factor, but a
secondfactorwith variableentries whose signs are,however,known. A number
ofparticularexampleswerepresentedin[12]. Moregenerallyitmayalsohappen
thatAB isK-quasipositiveforeachB insomesetB,whereBisapropersubset
of the closure of some qualitative class.
Given a set of K-quasipositive matrices of the form {AB : B ∈ B}, we
would hope that there is a natural graph-theoretic test to decide which mem-
bers of this set are also K-irreducible. Theorem 1 provides precisely such a
condition: provided B is row-complete and the initial factor A satisfies a mild
genericity condition, K-irreducibility of AB follows from strong connectedness
of the bipartite digraph G . In the special case where K is the nonnegative
A,B
orthant, A is the identity matrix, and B is the set of nonnegative matrices, the
results reduce to well known ones.
Remark 6. Our motivation for considering row-complete sets of matrices is as
follows: consider a set of K-quasipositive matrices of the form {AB : B ∈B},
where B is row-complete. Clearly
AB =A B(k) = AB(k) = A Bk.
k
Xk Xk Xk
So any matrix in the set can be written as a sum of rank 1 K-quasipositive
matrices.
3. The main result
From now onK ⊂Rn will be a CCP cone in Rn, A an n×m matrix and B
a row-complete set of m×n matrices. For any B ∈B, AB is an n×n matrix.
The main result of this paper is:
Theorem 1. Assumethat ImA6⊆spanF for any nontrivial face F of K. Sup-
pose that for each B ∈ B, AB is K-quasipositive. Then whenever G is
A,B
strongly connected, AB is also K-irreducible.
An immediate corollary of Theorem 1 is:
Corollary 2. Assume that ImA 6⊆ spanF for any nontrivial face F of K.
SupposethatforeachB ∈B,AB isK-positive. ThenwheneverG isstrongly
A,B
connected, AB is also K-irreducible.
Proof. K-positivity of AB implies K-quasipositivity of AB. The result now
follows from Theorem 1. (cid:3)
4
Remark 7. Note that if ImA⊆spanF for some nontrivial face F of K, then
trivially AB is K-reducible. To see that the assumption that ImA 6⊆ spanF is
in general necessary in Theorem 1, consider the matrices
1 2 1 1 a b
Λ= , A= , B =
(cid:18) 2 1 (cid:19) (cid:18) 2 2 (cid:19) (cid:18) c d (cid:19)
where a,b,c,d≥0. Let K ={Λz : z ∈R2 }. Then
≥0
ABΛ =(a+c+2(b+d))Λ , ABΛ =(2(a+c)+b+d)Λ
1 1 2 1
which are both clearly in K for any a,b,c,d≥0. So AB is K-positive. On the
other hand F = {rΛ : r ≥ 0} satisfies (AB)F ⊆ F for all B, so AB is K-
1
reducible for all nonnegative values of a,b,c,d. However, for a,b,c,d>0, G
A,B
is the complete bipartite digraph K , which is obviously strongly connected.
2,2
4. Proofs
We need some preliminary lemmas in order to prove Theorem 1. The fol-
lowing is proved as Lemma 4.4 in [12]:
Lemma 3. Let F be a face of K, v ,v ∈ F, and w ∈ Rn. If there exist
1 2
α,β >0 such that v +αw ∈K and v −βw ∈K, then w ∈span(F).
1 2
Proof. Define y ≡ v +αw and y ≡ v −βw. Then y ≡ y +(α/β)y =
1 1 2 2 3 1 2
v +(α/β)v ∈ F. Since y ∈ F, y −y = (α/β)y ∈ K, and y ∈ K, by the
1 2 3 3 1 2 1
definition of a face, y ∈F. So w =(y −v )/α∈span(F). (cid:3)
1 1 1
Extremals. A one dimensional face of K will be termed an extremal ray or
an extremal for short, and any nonzero vector in an extremal ray will be an
extremal vector of K.
Lemma 4. Suppose that for each B ∈ B, AB is K-quasipositive. Let E be an
extremal vector of K. Then for each (fixed) j either A = rE for some real
j
number r or BjE ≥0 for all B ∈B or BjE ≤0 for all B ∈B.
Proof. Suppose there exist j and B,B ∈ B such that p ≡ BjE > 0 and
1
p ≡ −BjE > 0. Note that AB(j) = A Bj and AB(j) = A Bj. Since B is
2 j j
row-complete, B(j),B(j) ∈B and so AB(j), AB(j) are K-quasipositive. Let α
1
and α be such that AB(j)E +α E ∈ K and AB(j)E +α E ∈ K respectively.
2 1 2
We can assume (w.l.o.g.) that α ,α >0. Then:
1 2
(j) j
AB E +α E =A B E +α E =α E +p A ∈K
1 j 1 1 1 j
and
AB(j)E +α E =A BjE +α E =α E −p A ∈K.
2 j 2 2 2 j
Now, by Lemma 3, these two equations imply that A =rE for some r. (cid:3)
j
5
Remark 8. An alternative phrasing of Lemma 4 is that if A is not collinear
j
with E then either Bj ∈Q (ET) for all B ∈B or Bj ∈Q (−ET) for all B ∈B.
1 1
Lemma 5. Let F be a nontrivial face of K, {J(k)} be a finite set of n×n
K-quasipositive matrices and J = J(k). If there exists x ∈ F such that
Jx∈spanF, then J(k)x∈spanF foPr each k.
Proof. By K-quasipositivity of each J(k) we can write
J(k)x=p +q
k k
where p ∈(K\F)∪{0} and q ∈spanF. Summing, we get
k k
Jx=p+q
where q = q ∈spanF and p= p ∈(K\F)∪{0}. Now if Jx∈spanF
k k k k
then p=0.PSince pk ∈K andK is pPointed, this implies that pk =0 for eachk.
So J(k)x=q ∈spanF for each k, proving the lemma. (cid:3)
k
Lemma 6. LetAB beK-quasipositive for eachB ∈BandletF beanontrivial
face of K spanned by (pairwise independent) extremal vectors {Λ } . Choose
i i∈I
and fix some B ∈ B. Then given any nonempty set R ⊆{1,...,m}, either (i)
A ∈ spanF for some k ∈ R or (ii) BkΛ = 0 for each k ∈ R,i ∈ I, or (iii)
k i
there exists Λ ∈F such that A BkΛ 6∈spanF.
i k∈R k i
P
Proof. Foreachk,recallthatB(k) ∈B,soAB(k) =A Bk isK-quasipositive.
k
Suppose (iii) does not hold, i.e., A BkΛ ∈ spanF for each Λ ∈ F.
k∈R k i i
Applying Lemma 5 with J(k)=AkPBk, we get for each k ∈R and each Λi ∈F
thatA BkΛ ∈spanF. Soforeachfixedk∈R,eitherA ∈spanF orBkΛ =0
k i k i
for all i∈I. (cid:3)
Proof of Theorem 1. We show that if AB is K-reducible for some B ∈
B, then G cannot be strongly connected. Let {Λ } be a set of pairwise
A,B i
independent extremal vectors of K. Let F be a nontrivial face of K such that
spanF is left invariant by AB, and let I be the indices of vectors Λ in F, i.e.,
i
i ∈ I ⇔ Λ ∈ F. Let R ⊂ {1,...,m} be defined by k ∈ R ⇔ A ∈ spanF.
i k
R may be empty, but by assumption cannot be all of {1,...,m} since ImA 6⊆
spanF. So Rc, the complement of R, is nonempty.
Choose any x∈F. We have
ABx= A Bkx= A Bkx+ A Bkx (1)
k k k
Xk kX∈R kX∈Rc
Clearly A Bkx ∈ spanF. By assumption, ABx ∈ spanF, and so
k∈R k
k∈RcAPkBkx ∈ spanF. Now since x ∈ F was arbitrary and Ak 6∈ spanF
Pfor any k ∈ Rc, by Lemma 6 we must have BkΛi = 0 for each k ∈ Rc,i ∈ I.
But from Lemma 4 we know that either (i) A = rΛ for some scalar r or (ii)
k i
Bk ∈Q (ΛT)orBk ∈Q (−ΛT). SinceA 6∈spanF thefirstpossibilityisruled
1 i 1 i k
out, and (ii) must hold. Consequently BkΛ =0 implies B Λ =0 for each l.
i kl li
The above is true for each i∈I, k ∈Rc. Now there are two possibilities:
6
1. Suppose that R is empty. Then, for each i ∈ I, and all k,l, B Λ = 0.
kl li
Fix some i ∈I and some l such that Λ 6= 0; then B = 0 for all k (the
li kl
lthcolumnofB iszero). By thedefinitionofG ,this meansthatthere
A,B
are no arcs incident into the row vertex u . Thus G is not strongly
l A,B
connected.
2. SupposethatRisnonempty. Givenk′ ∈RwecanwriteAk′ = i∈IqiΛi
for some constants qi, so for any k ∈Rc, P
BkAk′ = qiBkΛi =0.
Xi∈I
Moreover,since for eachi∈I andeachl, BklΛli =0,we getBklAlk′ =0.
IntermsofG ,thismeansthatthereisnodirectedpathoflength2from
A,B
any column vertex vk with k ∈ Rc to a column vertex vk′ with k′ ∈ R.
Thus there is no directed path (of any length) of the form vk···vk′ with
k ∈Rc, k′ ∈R, and G is not strongly connected.
A,B
(cid:3)
5. Examples
The examples below illustrate application of Theorem 1 and Corollary 2.
Example 1. Consider the special case where Corollary 2 is applied with
A = I, B the nonnegative matrices and K = Rn . Since ImI = Rn, clearly
≥0
ImI 6⊆ spanF for any nontrivial face F of Rn . It is also immediate that for
≥0
each B ∈ B, the n×n matrix IB is K-positive. Now we show that that G
I,B
is strongly connected if and only if G is strongly connected. (i) Suppose G
B B
is strongly connected. An arc from vertex i to vertex j in G implies that
B
B 6= 0. But B = I B , and since I = 1 this implies that there exists a
ij ij ii ij ii
path u v u in G . Thus a path from vertex i to vertex j in G implies a
i i j I,B B
path from vertex u to vertex u in G . Thus strong connectedness of G
i j I,B B
implies a path between any two row vertices u and u of G . On the other
i j I,B
hand since all arcs u v exist in G , the path u ···u implies the existence of
i i I,B i j
paths u ···v , v ···u and v ···v . (ii) Suppose G is strongly connected.
i j i j i j I,B
The path v ···v in G immediately implies a path from vertex i to vertex j
i j I,B
inG . ThuswerecoverfromCorollary2the factthatforanonnegativematrix
B
B, strong connectedness of G implies irreducibility of B.
B
Example 2. Let
1 0 −1 −1 0 1
A=Λ= 1 −2 0  and B˜ = 0 1 0 
−1 1 0 1 0 0
   
Let B=Q (B˜) and define
0
K ={Λz : z ∈R3 }.
≥0
7
K is a CCP cone in R3. (It is easy to show that any nonsingular n×n matrix
defines apropersimplicialconeinRn inthis way.) Since A=Λ itis immediate
that ImA does not lie in the span of any nontrivial face of K. Consider any
B ∈B, i.e., any matrix of the form
−a 0 b
B = 0 c 0 
d 0 0
 
where a,b,c,d≥0. Since A=Λ,
ABΛ+(a+b+2c+d)Λ=Λ(BΛ+(a+b+2c+d)I)
and it can be checked that BΛ+(a+b+2c+d)I is nonnegative. Thus AB
is K-quasipositive for all B ∈ B. On the other hand G is illustrated in
A,B
Figure 1 for any B ∈relintB andcan be seento be stronglyconnected. So AB
is K-irreducible for any B ∈relintB.
u v u
1 1 2
v u v
3 3 2
Figure1: GA,B forthesysteminExample2withB∈relintB. Byinspectionthedigraphis
stronglyconnected.
Example 3. We consider cases where B is not the closure of a qualitative
class,butisasmallerset. SupposeΛisann×mmatrix,A=ΛandK ={Λz :
z ∈ Rm } is a CCP cone in Rn. As in the previous example, it is immediate
≥0
that ImA does not lie in the span of any nontrivial face of K. For some m×n
matrix B˜, define
B={DB˜ : D is a nonnegative diagonal matrix}.
Clearly B is row-complete. Suppose, further, that B˜Λ is nonnegative, and
consequentlyDB˜ΛisnonnegativeforanynonnegativediagonalmatrixD. Then
givenanyB =DB˜ ∈B,ABΛ=Λ(DB˜Λ),andsoABisK-positive. Corollary2
applies andcan be usedto deduce K-irreducibilityof AB. For example,choose
1 1 −1 3 0 −1
A=Λ= 0 2 4  and B˜ = 1 3 0 .
1 2 −3 −2 1 2
   
AsinExample2,SinceΛisnonsingular,the coneK generatedbyΛisaproper
cone in R3. It can be checked that B˜Λ is a nonnegative matrix, so AB is
K-positive for each B ∈ B (this is not the case for every B ∈ Q (B˜)). So
0
Corollary 2 applies. G is illustrated in Figure 2 for any B ∈ relintB (i.e.,
A,B
8
u v u
1 2 3
v u v
1 2 3
Figure 2: GA,B for the system in Example 3 with B ∈ relintB. The digraph contains the
Hamiltoniancycle(u1v3u2v2u3v1)andsoisstronglyconnected.
any B = DB˜ where D has positive diagonal entries). The digraph is strongly
connected and so, by Corollary 2, AB is K-irreducible.
Example 4. The following is an example with a cone which is not solid. Let
−1 −1 0 1 0 1 1 −1
A= −1 0 −1 , Λ= 0 −1  and B˜ = 1 0 −1 
2 1 1 −1 1 0 1 −1
     
Let B=Q (B˜) and define
0
K ={Λz : z ∈R2 }.
≥0
K is a CCP cone of dimension2 in R3. ClearlyImA does notlie in the spanof
any nontrivial face (i.e., any extremal) of K. Defining
−1 −1 0
T = ,
(cid:18) 1 0 1 (cid:19)
note that A=ΛT. Consider any B ∈B, i.e., any matrix of the form
a b −c
B = d 0 −e 
0 f −g
 
where a,b,c,d,e,f,g≥0. Then
ABΛ+(a+b+c+d+e+f +g)Λ=Λ(TBΛ+(a+b+c+d+e+f +g)I)
where I is the 2×2 identity matrix. It can be checkedthat TBΛ+(a+b+c+
d+e+f+g)I isnonnegative. ThusAB isK-quasipositiveforallB ∈B. G
A,B
is illustrated in Figure 3 for any B ∈ relintB and can be seen to be strongly
connected. So AB is K-irreducible for any B ∈relintB.
Example 5. As a final, nontrivial, example, let
−1 0 0 0 0 0 0 1 1 1 1
 1 −1 0   0 0 1 1 −1 −1 0 0 
A= , Λ= ,
0 1 −1 1 0 0 −1 1 0 0 −1
   
 1 0 1   0 1 0 1 −1 0 −1 0 
   
(2)
9
u v u
1 1 2
v u v
2 3 3
Figure3: GA,B forthesysteminExample4withB∈relintB. Byinspectionthedigraphis
stronglyconnected.
and B=Q (−AT). Define
0
K ={Λz : z ∈R8 }.
≥0
Various facts can be confirmed either theoretically or via computation:
1. K is a proper cone in R4.
2. ImA6⊆spanF for any nontrivial face F of K.
3. For each B ∈B, AB is K-quasipositive.
Some insight into the structure of K and the proof of these facts is provided in
the Appendix. It now follows from Theorem 1 that whenever G is strongly
A,B
connected, AB (and hence AB+αI for each α∈R) is also K-irreducible. For
example, it can easily be checked that for B ∈relintB (namely B ∈Q(−AT)),
G is strongly connected and so AB is K-irreducible. The condition that
A,B
B ∈ relintB can be relaxed while maintaining K-irreducibility. The digraphs
G for two choices of B ∈B are illustrated in Figure 4.
A,B
u u
2 2
u u
1 1
v v v v
3 2 3 2
u u u u
4 3 4 3
v v
1 1
Figure4: Left. ThedigraphGA,BwhereAisasshownin(2)andBisanymatrixinQ(−AT).
ui is the row vertex corresponding to row i in A, while vi corresponds to column i. Right.
SettingB22=B13=B34=0removesthearcsv2u2,v1u3,v3u4 fromGA,B,butstillgivesa
stronglyconnected digraph.
Confirming K-irreducibility for choices of B without the aid of Theorem 1
is possible but tedious, requiring computation of the action of AB on each of
the 26 nontrivial faces of K.
6. Concluding remarks
Rather different graph-theoretic approaches to questions of irreducibility of
matrices are taken in [2, 3, 4]. In [4], for example, polyhedral cones with n
K
extremals were considered, and digraphs on n vertices constructed. Results
K
10