Graphs and irreducible cone preserving maps

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Graphs and irreducible cone preserving maps a

Bit-Shun Tam & George Phillip Barker

b

a

Department of Mathematics, Tamkang University, Tamsui, Taiwan, 25137, Republic of China b

Department of Mathematics, University of Missouri-KC, Kansas City, MO, 64110-2499 Version of record first published: 01 Apr 2008.

To cite this article: Bit-Shun Tam & George Phillip Barker (1992): Graphs and irreducible cone preserving maps, Linear and Multilinear Algebra, 31:1-4, 19-25 To link to this article: http://dx.doi.org/10.1080/03081089208818118

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Linear and Mukilinear Algebra, 1992, Vol. 31, pp. 19-25 Reprints available directly from the publisher Photocopying permitted by license only @ 1992 Gordon and Breach Science Publishers S.A. Printed in the United States of America

Graphs and Irreducible Cone Preserving Maps BIT-SHUN TAM* Department of Mathematics, Tamkang University, Tamsui, Taiwan 25137, Republic of China

GEORGE PHILLIP BARKER

Downloaded by [Tamkang University] at 04:25 13 April 2013

Department of Mathematics, University of Missouri-KC, Kansas City, MO 647 10-2499

(Received October 25, 1990; in final form May 9, 1991)

Let K be a full, pointed closed cone in a fiiite dimensional real vector space. For any linear map A for which A K E K , denote by ( E , P ( A ) [ E , I ( A ) ) the ] directed graph whose vertex set consists of all the extreme rays of K such that there is an edge from F to G iff G aO[AF][GaO[(I + A)F]]. It is ] strongly proved that K is a simplicial cone if for any linear map A with A K C K , ( E , P ( A ) ) [ ( E , I ( A ) )is connected whenever A is irreducible [provided, in addition, that K is 2-neighborly].

INTRODUCTION

There is a well known method for associating a directed graph with an entrywise nonnegative matrix (cf. [3] and [5]). It turns out that this graph is strongly connected iff the matrix A is irreducible. In an earlier paper [2] the present authors defined four similar graphs for a cone preserving map. In contrast with the entrywise nonnegative case the graph which connects with irreducibility has as vertices the set of all faces of the cone. In the entrywise nonnegative case where the cone is the nonnegative orthant, a simplicia1 cone, the set of vertices is the set of extreme rays. The question addressed here is, loosely speaking, whether this phenomenon of irreducibility of operators being determined by the extreme rays alone is characteristic of simplicia1 cones. The first main theorem gives an affirmative answer to this question. 1.

DEFINITIONS AND NOTATION

The survey paper [I] is a general references for cones, faces, and cone preserving maps. Let V be a finite dimensional real vector space with the unique topology which makes V a topological vector space. (If V is identified with Rn, this is the usual topology.) -

--

*This author's research work was partially supported by the National Science Council of the Republic of

China.

19

20

B-S. TAM AND G. P. BARKER

DEHNITION 1 Let K be a subset of

V.

(a) K is a cone i f it is nonempty and satisfies: x , y c K , a , P > _ O (a,,BreaZ)

imply a x + p y ~ K .


(b) K is closed if it is closed in the topology of V. (c) K is full i f intK # 0 (or equivalently, span K = V). (d) K is pointed if K n (-K) = (0). (e) A subcone F of a closed pointed full cone K is a face of K i f x E K, y - x E K, and y E F imply x E F.

Notation. If K is a closed pointed full cone and if S 5 K, then the least face of K containing S is denoted by @ ( S ) . If S = {x}, we write @(x)for simplicity. If F is a face of K, then dimF is the dimension of the subspace spanned by F. If @(x)is a face of K and dim@(x)= 1, then @(x)is termed an extreme ray, and x an extremal, of K. The set F ( K ) of all faces is a lattice under the operations

where F, G E 3(K). In addition, the cone K is simplicia1 iff it has exactly n = dimV extreme rays. In this case K is the image of the nonnegative orthant under a nonsingular linear transformation. If there is no confusion about the cone K we shall write 3 for F(K). Note that the set F contains the trivial faces K and (0). We shall use F'to denote the set of all nontrivial faces. We let E denote the set of all extreme rays. Thus E E F'.If F E F we also write F a K . If F , G E 3 and F C G we write F a G a K . This notation is easily checked to be consistent, that is, F is a face of G where G is considered to be a cone in its span.

DEFINITION 2 Let K be a full pointed closed cone and let L(V) be the set of all linear transformations on V . K}. A linear map A E n ( K ) will be called nonnegative. (b) A E n ( K ) is irreducible fz A leaves no nontrivial face of K invariant. (c) A E JI(K) is primitive iff for all nonzero x E K there is a positive integer k such that A ~ EXintK. (a) Let II(K) = {A E L(V) : AK

Remark Since V is finite dimensional, we have, A is primitive iff there is an integer k > 0 such that for all nonzero x E K , A ~ EXintK. 2.

GRAPHS ASSOCIATED WITH A NONNEGATIVE OPERATOR

If A E II(K) we may associate with A four directed graphs. If F, G E F', we say there is an 2-edge from F to G, Z(F, G), iff G a @[(I+ A)F]. We say there is a 7'edge from F to G,P(F, G), iff G a@[AF].Note that every P-edge corresponds to an 2-edge since @[AF]a@[(I+ A)F]. Let Z and P denote the set of 2-edges and Pedges respectively. If necessary we write Z(A) or P(A) to indicate the dependence


CONE PRESERVING MAPS

21

upon A. Then (F1,Z) denotes the directed graph with vertex set 3' and edge set 2. The other pairings are defined analogously. These definitions differ from those in [2] by the use of 3' instead of 3. The strong connectedness of the directed graphs (F1,2) and (F, P ) are defined in the standard way. In other words, we say that (3',2) [respectively (F1,P)] is strong& connected iff for any F, G E 3' there is a path of 1-edges [respectively Pedges] from F to G . It is readily checked that the strong connectedness of (F1,Z) [respectively (3', P)] is equivalent to that of (3,Z) [respectively ( 3 , P ) l as defined in [2]. So the results in [2] remain valid using 3' in place of 3. In [2] we proved that A E II(K) is irreducible iff (F1,Z) is strongly connected and that A is primitive iff (F1,P) is strongly connected. Note that the statement of Lemma 2 there needs to be modified by adding the condition N(A) n K = (0). However, the result still holds. If K is a simplicial cone, in particular if K is the nonnegative orthant, then A E II(K) is irreducible iff (E, P) is strongly connected. For general cones K if (E, P ) is strongly connected, then A is irreducible. The converse is false as we will show shortly. The question addressed by the main results is whether the truth of the converse for all A E II(K) entails that K is simplicial. We give an affirmative answer.

EXAMPLELet K G R~ be defined by K = {x : (x;

+

< x3),

and let A be a rotation about the x3-axis through an angle which is not a multiple of 27r. Then A is irreducible, (£,I) is strongly connected, but (E, P ) is not strongly connected.

3. MAIN RESULTS

THEOREM1 Let K be a closed,pointed full cone in a finite dimensional real vector space V . Then K is simplicial if for any A E lI(K), (E,P(A)) is strongly connected whenever A is irreducible. Notation. If XI,...,x, are vectors we denote by pos{xl, ...,x,) the cone generated by X I ,...,x,, that is, pos{xl ,...,x,} = {Cajxj : aj 2 0, j = 1,. . . , r ) . Before giving the proof we make the following Observation. If C is any n-dimensional closed, pointed full non-simplicia] cone, then there exist linearly independent extreme vectors XI,...,x, with r < n such that xl+...+x, ~ i n t K . Indeed, first choose any n linearly independent extreme vectors XI,...,xn of C. Since C is not simplicial, there exists some vector y E C\pos{xl,. ..,x,}. As


there exists a > 0 such that

Call this vector w . Then clearly w can be expressed as the positive linear combination of fewer than n vectors from {xl, ...,xn). But w E int C as y E C and n

xi E int pos{xl,. ..,xn) E int C. i=l

Proof Suppose that K is not simplicial. Among all finite sets {el,. ..,ep) of extreme vectors of K with el ... e, E intK, choose one with minimal cardinality and call it {xl, ...,x,). (From the above observation, we have r < n, where n = dirnK.) Since r is minimal, it is clear that, for any j, 15 j 5 r, xl + + Zj + + Xr E bdyK, where Zj indicates that the term xj is to be omitted. In fact, pos{xi, ...,x,) is a simplicia1 cone, and for each j, we have @(xl+ . + Zj + .-. + x,) = pos{xl, ...,ij, ...,xr), where the left side denotes a face of K . This follows from the beginning observation. For instance, if @(x2+ + x,) is not equal to pos{xz,. ..,x, ), then @(x2+ . . . + x,) is not simplicia1 (since x2, ...,x, are extreme vectors forming a basis of span@(xl+ . . + x,)), and so from the beginning observation (and its proof), there exist 2 1 il < i2 < ... < it 5 r with t < r - 1 such that xik E relint @(x2+ .. . + x,). But then since


+ +

we have xl + xik E intK, contradicting the minimality of r. Let us recall the definitions of dual cone and dual face. The dual cone K * of K is the cone in the space V* of linear functionals on V given by K * = {f E V * : f(x)>O, all X E K). If F is a face of K, its dual face F~ is the face of K* given by FD= {f E K* : f(x) = 0, all x E F). Since we identify V** with V, we may talk about the dual face of a face of K*. In particular, we have FDD3 F. We say that F is an exposed face if = F (cf. [I] for a discussion). Proceeding with the proof we choose for each j, 15j 5 r, an fi E K * such that @(fi) = @(xl+ ..-+ Zj + - - .+ x , ) ~ .Then fj(xk) = 0 if j f k. But since XI + + x, E intK, it is clear that fj(xj) f 0. Normalizing fi, if necessary, we may assume that fj(xj) = 1.

Claim. fl + ..-+ fr E intK*

CONE PRESERVING MAPS

First assume that every face of K is exposed. Then

r

=

/\ m(xl,. ..,i j , ...,xrlDD


j=1

+ +

where fi .. f, E int K*. For the general case we may need to modify the choice of the x . For the previous argument to work we need only require, say, that @(x2,...,x r F = @(x2,...,xr). . the only vector in b(x2,. ..,x,) which For then @(fl ... + fr)D E @(x2,...,~ r ) But is orthogonal to fl + + f, is the zero vector since fj(xj) = bij. So we have @(fl . . + f,)D = (0). In order to choose the xj so that @(x2,...,x,) is exposed we use the following lemma (Lemma 4.1 in [4]).

+

+

LEMMALet K be a closed pointed full cone, If y E K is such that dim@(y)= k , then there is a sequence of V ~ C ~ O{yj) ~ SjENin K converging to y such that each @(yj) is an exposed face of K of dimension at most k. If @(x2,...,xr) (which is of dimension r - 1) is not an exposed face of K , then according to the lemma we can choose some vector w E K sufficiently close to x2 ... + x, such that xl + w E intK and @(w) is an exposed face of K of dimension at most r - 1. By the minimality of r and the observation preceding the proof of Theorem 1, * ( w ) is a simplicia1 face of dimension r 1. Choose extreme vectors 12,. ..,zr of @(w) in place of x2,. ..,x, and choose the f j as before. So in all cases we can guarantee that fl + . - .+ fr E intK*. We next construct an irreducible A E II(K). Recall that for any z E V, g E V* the rank one operator z @ g is defined by ( z @g)(x)= (g(x))z for all x E V. Now put A = C;=lX j @ fj+l, where fr+l stands for tl. Clearly A E JI(K). Note that the index of A is one, since ImA = span(x1,. ..,xr) (r < n ) and A takes the space span(x1,. ..,x,) onto itself. Since xl + . . + x, E int K we have that Im A n K g bdyK. As fl + . + fr E intK* it is also clear that N(A) n K = (0). Also, I m A n K is the simplicial cone pos{xl,, ..,x,); otherwise the minimality of r will be contradicted. As A permutes the extreme vectors xl, ...,xr cyclically (in the backwards order), it is clear that A ~ I is~ irreducible A with respect to ImA n K. So by Lemma 2 of [2] (as corrected in the previous section), A is irreducible. If xr+l is an extreme vector of K distinct from XI,...,xr, it is clear that there is no path in (E,P(A)) from xl to xr+l. So the graph (E, P(A)) is not strongly connected.

+

-

24


The definition and general properties of k-neighborly polytopes are well known in the theory of convex sets. We can extend these ideas to (not necessarily polyhedral) cones. In particular we say that K is 2-neighborly iff for any two extreme rays @(xl) and @(x2)the face @(XI)V @(x2)is in the boundary of K .

THEOREM2 Let K be closed, pointed full cone in a finite dimensional real vector for any A E II(K), space. Suppose that K is 2-neighborly. Then K is simplicial if, (&?(A)) is strongZy connected whenever A is irreducible. Proof Suppose that K is not simplicial. We are going to construct some A E II(K) which is K-irreducible, but for which (E,Z(A)) is not strongly connected. The proof is almost exactly the same as that for the previous theorem. We work with the same minimal set of extreme vectors {xl,. ..,x,) with the property that xj E intK. Construct the same K-irreducible matrix A as before. It remains to show that the directed graph (E,Z(A)) is not strongly connected. Note that since K is Zneighborly, we have r 2 3. For any j, 15 j 5 r, (Z + A) xi = xi + xi-1, where xo stands for Xr and x,+l stands for xl. But @(xi+ E @(xl+ . . . + 2j+l + ... + x,), and as mentioned in the beginning part of the proof of . . + xr) is equal to the simplicial cone pos{xl, ..., Theorem 1 @(xl + ... + ..,xr). Hence, @(xi+ = pos{xi, xi-1). Thus in the directed graph (E,Z(A)), the only Z-edges with initial vertex @(xi) are the loop at @(xi) and the edge from @(xi) to @(xi-1). Take any extreme ray F of K, distinct from @(xi), 1_< i 5 r. It is now quite clear that in (E,Z(A)) there is no path from @(xi) to F. So (E,Z(A)) is not strongly connected.


xi,l

+

a

Remark A Zneighborly cone may not be polyhedral. For instance, consider the 4-dimensional proper cone which comes from the following 3-dimensional compact convex set C in the usual way: C = conv[{(O,O, 1)) U {(51,52,0) : 5: + 5; I].