A characterization of nonnegativity relative to proper cones
Chandrashekaran Arumugasamy, Sachindranath Jayaraman, Vatsalkumar N. Mer
aa r X i v : . [ m a t h . F A ] M a y A CHARACTERIZATION OF NONNEGATIVITYRELATIVE TO PROPER CONES
Chandrashekaran Arumugasamy ∗ , Sachindranath Jayaraman ∗∗ , andVatsalkumar N. Mer ∗∗∗ Department of Mathematics, School of Mathematics and Computer Sciences,,Central University of Tamil Nadu, Neelakudi, Kangalancherry,, Thiruvarur -610 101, Tamilnadu, India ∗∗ School of Mathematics, Indian Institute of Science Education and ResearchThiruvananthapuram,, Maruthamala P.O., Vithura, Thiruvananthapuram – 695551, Kerala, IndiaEmail ids: ∗ [email protected], ∗∗ [email protected],[email protected] Abstract
Let A be an m × n matrix with real entries. Given two proper cones K and K in R n and R m , respectively, we say that A is nonnegative if A ( K ) ⊆ K . A issaid to be semipositive if there exists a x ∈ K ◦ such that Ax ∈ K ◦ . We prove that A is nonnegative if and only if A + B is semipositive for every semipositive matrix B . Applications of the above result are also brought out. Keywords : Semipositivity of matrices, nonnegative matrices, proper cones, semiposi-tive cone
MSC : We work with the field R of real numbers throughout. The following notations will beused. M m,n ( R ) denotes the vector space of m × n matrices over R . When m = n , thisspace will be denoted by M n ( R ). Let us recall that a subset K of R n is called a convexcone if K + K ⊆ K and αK ⊆ K for all α ≥
0. A convex cone K is said to be proper ifit is topologically closed, pointed ( K ∩ − K = { } ) and has nonempty interior (denotedby K ◦ ). The dual, K ∗ , is defined as K ∗ = { y ∈ R n : h y, x i ≥ ∀ x ∈ K } , where h ., . i denotes the usual Euclidean inner product on R n . Well known examples of proper cones1 emipositivity of matrices that occur frequently in the optimization literature are the nonnegative orthant R n + in R n , the Lorentz cone (also known as the ice-cream cone) L n + := { x = ( x , . . . , x n ) t ∈ R n : x n ≥ , x n − n − X i =1 x i ≥ } , the set S n + of all symmetric positive semidefinite matricesin S n , the copositive ( COP n ) and completely positive ( CP n ) cones in S n and finally,the cone of squares K in a finite dimensional Euclidean Jordan algebra (see [7] fordetails). Recall that M n ( R ) and S n have the trace inner product h X, Y i = trace ( Y t X )and h X, Y i = trace ( XY ), respectively. Assumptions:
All cones in this paper are proper cones in appropriate finite dimen-sional real Hilbert spaces.
Definition 1.1.
Let K and K be proper cones in R n . A ∈ M n ( R ) is1. nonnegative (positive) if A ( K ) ⊆ K ( A ( K \ { } ) ⊆ K ◦ ).2. semipositive if there exists a x ∈ K ◦ such that Ax ∈ K ◦ .3. eventually nonnegative (positive) if there exists a positive integer k such that A k is nonnegative (positive) for every k ≥ k .Let π ( K , K ) and S ( K , K ) denote, respectively, the set of all matrices that arenonnegative and semipositive relative to proper cones K and K . When K = K = K ,we use the notation π ( K ) and S ( K ) for these sets and elements of these sets are called K -nonnegative and K -semipositive matrices, respectively. We write x > x ∈ K ◦ , the interior of K .When K is a proper cone in R n , π ( K ) is a proper cone in M n ( R ). A proof of thisas well as an extensive study on the structure and properties of π ( K ) can be found in[12] and the references cited therein. Observe that in Definition 1.1, when K = R n + ,we retrieve back the notion of a nonnegative matrix. A good source of reference onvarious applications of nonnegative matrices is the book by Berman and Plemmons [3].Semipositivity occurs very naturally in dynamical systems, game theory and variousother optimization problems, most notably the linear complementarity problem overvarious proper cones. For instance, given A ∈ M n ( R ), asymptotic stability (that is,the trajectory of the system from any starting point in R n converges to the origin as t → ∞ ) of the continuous dynamical system ˙ x = Ax is equivalent to S n + -semipositivityof the Lyapunov map L A on S n induced by A , where L A ( X ) = AX + XA t , X ∈ S n .This is the famous Lyapunov theorem (for details, see the paper by Gowda and Tao[7]). We end this section by stating results that will be used later on. Theorem 1.2.
For proper cones K and K in R n and R m , respectively, and A ∈ M m,n ( R ) , A ∈ π ( K , K ) if and only if A t ∈ π ( K ∗ , K ∗ ) . handrashekaran, Sachindranath and Vatsalkumar Lemma 1.3.
Let K be a proper cone in R n and let x ∈ K . If h x, y i = 0 for some = y ∈ K ∗ , then x / ∈ K ◦ . Theorem 1.4. (Theorem 2.8, [4]) For proper cones K and K in R n and R m , re-spectively, and an m × n matrix A , one and only one of the following alternativesholds.1. There exists x ∈ K such that Ax ∈ K ◦ .2. There exists = y ∈ K ∗ such that − A t y ∈ K ∗ . We present the main results in this section. We divide this section into three parts: (1)A characterization of nonnegativity, (2) Applications to linear preserver problems and(3) Invariance of the semipositive cone of a matrix.
The following was proved recently by Dorsey et al in connection with strong linearpreservers of semipositive matrices.
Theorem 2.1. (Lemma 3.3, [6]) A square matrix A is R n + -nonnegative if and only iffor every R n + -semipositive matrix B , the matrix A + B is R n + -semipositive. Our main result of this note generalizes the above theorem to proper cones in R n .We prove the result below. The first result is obvious and we skip the proof. Theorem 2.2.
Let A ∈ M m,n ( R ) and let K , K be proper cones in R n and R m ,respectively. If A ∈ π ( K , K ) , then for any B ∈ S ( K , K ) , A + B ∈ S ( K , K ) . The following theorem gives the converse of Theorem 2.2.
Theorem 2.3.
Let A ∈ M m,n ( R ) and let K , K be proper cones in R n and R m ,respectively. If A + B ∈ S ( K , K ) for every B ∈ S ( K , K ) then A ∈ π ( K , K ) .Proof. Suppose
A / ∈ π ( K , K ). There exists 0 = x ∈ K ◦ such that Ax / ∈ K . By thedefinition of K ∗ and continuity of the inner product, there exists 0 = y ∈ ( K ∗ ) ◦ suchthat h Ax, y i = h x, A t y i < h x, − A t y i >
0. Let v ∈ K ◦ and define B = vz t y t v where z = − A t y . Note that y t v >
0. Then Bx = ( z t x ) vy t v ∈ K ◦ . We have thus verified that emipositivity of matrices B ∈ S ( K , K ). For u ∈ K consider ( A + B ) u and note that y T ( A + B ) u = 0. Thatis h y, ( A + B ) u i = 0, where y ∈ ( K ∗ ) ◦ . Thus ( A + B ) u / ∈ K ◦ (see Lemma 1.3 above).Hence A + B / ∈ S ( K , K ).The following corollary is immediate. Corollary 2.4.
Let K be a proper self-dual cone in R n . If for every K -semipositivematrix B , the matrix A + B is K -semipositive, then A is K -nonnegative. By taking K = R n + and K = R m + , we retrieve back the result of Dorsey et al as inTheorem 2.1.The following is an isomorphic version of Theorem 2.3. Theorem 2.5.
Let K be a proper cone in R n for which Theorem 2.3 holds and let K ⊂ R n be isomorphic to K through an isomorphism T with T ( K ) = K . Then T AT − ∈ M n ( R ) is K -nonnegative if and only if for every K -semipositive matrix B ∈ M n ( R ) , A + B is K -semipositive.Proof. The proof follows as π ( K ) = T − π ( K ) T , where T is an isomorphism between K and K .A few remarks are in order. Remark .
1. Theorem 2.3 can be suitably modified for linear maps between finite dimensionalreal Hilbert spaces V and W , equipped with proper cones K and K , respectively.2. Let Q be a nonsingular symmetric matrix with inertia ( n − , , λ n bethe single negative eigenvalue of Q with a normalized eigenvector u n . Define K := K ( Q, u n ) = { x ∈ R n : x t Qx ≤ , x t u n ≥ } . Let − K = K ( Q, − u n ). It canbe seen that K ( Q, ± u n ) is a proper cone, known as an ellipsoidal cone. It is noweasy to see that the Lorentz cone L n + is an example of an ellipsoidal cone. Thiscan be seen by taking Q = " I n − − and u n = e n , the n th unit vector in R n .The following result from [11] will be used below. Lemma . (Lemma 2.7, [11]) A cone K is ellipsoidal if and only if K = T ( L n + ) for some invertible matrix T .
3. It now follows from Theorem 2.5 and Lemma 2.7 that over any ellipsoidal cone K , the square matrix T AT − is K -nonnegative if and only if for every L n + -semipositive matrix B, A + B is L n + -semipositive. handrashekaran, Sachindranath and Vatsalkumar In what follows, we illustrate Theorem 2.3 for the Lorentz cone, as the computationsare nontrivial and interesting.
Proof.
Suppose A is not L n + -nonnegative, but for every L n + -semipositive matrix B , thematrix A + B is L n + -semipositive. Choose 0 = x ∈ L n + such that A t x / ∈ L n + . Let usdiscuss two cases.Case-1: Suppose y := − A t x ∈ L n + . Notice that y = 0. For, otherwise, A t x = 0 ∈ L n + .There exist α i ∈ R , ≤ i ≤ n − α n > α i x n = y i and α n x n
0. Now − ( B + A ) t x = − α n x n + y n ∈ L n + , since 0 < − α n x n + y n . It is now easy to see that A + B is not L n + -semipositive.Case-2: Suppose y := − A t x / ∈ L n + . Consider the matrix B = y t x n . Suppose for every x ∈ L n + , y t x ≤
0. Then, we must have − y = A t x ∈ L n + . Since A t x / ∈ L n + , this is acontradiction to our assumption. Therefore B is L n + -semipositive. Now − ( B + A ) t x = − B t x − A t x = − B t x + y . Note that B t = h . . . y/x n i . Therefore, − B t x + y = − y + y = 0 ∈ L n + . As before, it follows that A + B is not L n + -semipositive.Let us also mention that a characterization of nonnegativity over the Lorentz cone L n + was given earlier by Loewy and Schneider (Theorem 2.2, [9]). Besides independent interest in generalizing Lemma 3.3 of [5] over proper cones,one can use Theorem 2.3 to prove that a strong linear preserver of semipositivity withrespect to a proper cone K also preserves the set of nonnegative matrices over K . Letus mention a useful result before proceeding further. If S is a collection of matrices, alinear map L on M n ( R ) is called an into preserver of S if L ( S ) ⊂ S and a strong/ontolinear preserver if L ( S ) = S . If S ⊂ M n ( R ) contains a basis for M n ( R ), then a strong emipositivity of matrices linear preserver L of S is an into linear preserver that is invertible with L − being aninto linear preserver of S (see [5] for details). The following results will be used in thetheorem that follows. Lemma 2.8.
Given any proper cone K in R n , there is a basis for M n ( R ) from the(proper) cone π ( K ) . Lemma 2.9.
Let K be a proper cone in R n and let S , S ∈ π ( R n + , K ) be such that S (( R n + ) ◦ ) ⊆ K ◦ and S invertible. If B ∈ S ( R n + ) , then S BS − ∈ S ( K ) . Consequently, S ( K ) contains a basis for M n ( R ) .Proof. To prove the first statement, choose x ∈ ( R n + ) ◦ such that Bx ∈ ( R n + ) ◦ . Take y := S x ∈ K . We then have S BS − y ∈ K ◦ . This finishes the proof as K is a propercone. To prove the second statement, take a basis { B , . . . , B n } for M n ( R ) from S ( R n + )(for example, the set of matrices with 2 in one entry and remaining entries being 1).Consider the collection { A i : i = 1 , . . . n } , where A i = S B i S with S , S ∈ π ( R n + , K )are both invertible and S (( R n + ) ◦ ) ⊆ K ◦ . From the first statement of this result, weknow that each A i ∈ S ( K ). It is now obvious that the A i s form a basis for M n ( R ). Theorem 2.10.
Let K be a proper cone in R n . Suppose L is a strong linear preserverof S ( K ) . Then L is a linear automorphism of π ( K ) .Proof. Let A ∈ π ( K ). To begin with, let us observe that L is an invertible mapas M n ( R ) contains a basis from S ( K ) (refer Lemma 2.9 above). Then for any B ∈ S ( K ) , A + B ∈ S ( K ). Now L ( A + B ) = L ( A )+ L ( B ) ∈ S ( K ). Note that L ( B ) ∈ S ( K ).Further, for any C ∈ S ( K ) there exists a B ∈ S ( K ) such that L ( B ) = C . Thus, forany C ∈ S ( K ) , L ( A ) + C ∈ S ( K ). It follows from Theorem 2.3 that L ( A ) ∈ π ( K ).A similar argument works for L − as well. Since π ( K ) contains a basis for M n ( R ),the desired result follows. A and its invariance We discuss in this section, invariance of the semipositive cone of A . Whereverpossible, examples are provided to substantiate our results. Given a proper cone K in R n and a square matrix A , one can consider the set K A = { x ∈ K : Ax ∈ K } . This set is a closed convex cone called the semipositive cone of A . Note that K A = K ∩ A − ( K ), where A − ( K ) = { x : Ax ∈ K } . It can be proved that K A is a proper handrashekaran, Sachindranath and Vatsalkumar cone if A is K -semipositive. Recently, in [10], Sivakumar and Tsatsomeros posed thequestion of when A will leave the cone K A invariant. In [1], the authors answered thisquestion affirmatively when K = R n + and A is semipositive. A complete answer in therank one operator case and a partial answer in the case of an invertible matrix werealso given. The proofs of these statements can be found in Theorems 2.30 and 2.31 of[1]. We discuss yet another case below.We now have the following theorem. Theorem 2.11.
Let A ∈ M n ( R ) be K -semipositive. Assume that for some j ≥ both A j and A j +1 map K ◦ A onto itself. Then, A leaves K A invariant.Proof. Suppose A ( K A ) * L A . Then, by Theorem 2.3, there exists a K A -semipositivematrix B such that A + B is not K A -semipositive. Therefore, by Theorem 1.4, thesystem ( A + B ) x ∈ K ◦ A , x ∈ K A has no solution. This means that the system − ( A + B ) t y ∈ K ∗ A , = y ∈ K ∗ A has a solution. Therefore, for every z ∈ K ∗∗ A = K A , h− ( A + B ) t y, z i ≥ h ( A + B ) t y, z i ≤ z ∈ K A . Let j be a natural number such that both A j and A j +1 map K ◦ A onto itself. Let z = A j x, x ∈ K ◦ A . Then, h y, ( A + B ) A j x i = h y, A j +1 x i + h y, BA j x i ≤
0. Notice that BA j x ∈ K ◦ A . Then, the second inner product is positive, as y ∈ K ∗ A and BA j x ∈ K ◦ A . The first inner product is certainly positive as A j +1 x ∈ K ◦ A and 0 = y ∈ K ∗ A . This contradiction proves the result. It is worth pointing out that when K is a proper self-dual cone in R n and if A isan invertible matrix such that A ( K ) ⊆ K , then A ( K A ) ⊆ K A (see Theorem 2.37, [1]),although the converse is not true. In Examples 2.12 and 2.14 below, we work with theLorentz cone L n + and we shall denote in this case the semipositive cone by L A . Example2.12 illustrates that there are matrices that are K -semipositive, but some power of A is not K -semipositive and A does not leave the semipositive cone invariant. Example 2.12.
Let A = " −
11 1 and K = L . Then, A is L -semipositive, whereas A is not. The cone L A = { x = ( x , x ) t ∈ R : x ≥ , x ≥ , x ≥ x } . Taking u = (0 , t ∈ L A , we see that Au = ( − , t / ∈ L A . emipositivity of matrices The following is an example of a matrix A for which A j is L A -semipositive for some j ≥ A j is L -semipositive for all such j ≥ A does not leave thesemipositive cone L A invariant. Let us recall a definition before proceeding with theexample. Definition 2.13.
Given a proper cone K in R n , we say that A ∈ M n ( R ) has • the K -Perron-Frobenius property if ρ ( A ) > , ρ ( A ) ∈ σ ( A ) and there is aneigenvector x ∈ K corresponding to ρ ( A ). • the strong K -Perron-Frobenius property if, in addition to having the K -Perron-Frobenius property, ρ ( A ) is a simple eigenvalue such that ρ ( A ) > | λ | for all | λ | ∈ σ ( A ) , | λ | 6 = ρ ( A ), as well as there is an eigenvector x ∈ K ◦ correspondingto ρ ( A ). Example 2.14.
Let A = − − . It can be easily seen that this matrix is L -semipositive and its associated semipositive cone L A = { x = ( x , x , x ) t ∈ L :2( x + x − x ) + 8 x x + 4 x x ≥ } is a proper cone in R . The matrix A has thefollowing properties. • It is L A -semipositive, as x ◦ = (0 , , t ∈ L ◦ A and Ax ◦ = ( − , , t ∈ L ◦ A . • The spectral radius of
A, ρ ( A ) = 3 . A with acorresponding eigenvector x = ( − . , . , t in the interior of L A and A t hasas eigenvector y = ( − . , . , . t in the interior of L ∗ A , corresponding tothe dominant eigenvalue ρ ( A ) = 3 . A has the strong Perron-Frobenius property relative to L A and so is eventually L A -positive (see Theorem7, [8]; also see the references cited therein for details about the Perron-Frobeniusproperty). Therefore, there exists a natural number j such that for every j ≥ j , A j is L A -positive. • It is also eventually positive over L .However, for x ◦ = (1 , − , t ∈ L A , Ax ◦ = ( − , , t , which is not an elementof L A . Thus, A does not leave the cone L A invariant. Note that A cannot be L -nonnegative. In fact, taking the above vector x ◦ ∈ L , we see that A ( x ◦ ) / ∈ L .The above example illustrates that semipositivity of a matrix with respect to aproper cone K together with strong Perron-Frobenius property of A with respect tothe semipositive cone K A does not imply that A ( K A ) ⊆ K A . Other cone invariance handrashekaran, Sachindranath and Vatsalkumar properties of the matrix A in the above example can be found in Examples 5 and 10of [8]. It now follows from Theorem 2.11 that there is no j for which both A j and A j +1 map L ◦ A onto itself. A direct proof of this is not obvious. We end with a fewremarks. Stern and Tsatsomeros proved that if a square matrix A is semipositive aswell as has the Z -property with respect to a proper cone K , then ( A + ǫI ) − ∈ π ( K )for all ǫ ≥
0. Moreover, the determinant of A is positive and if A is also symmetric,then A is positive definite (Corollary 1, [7]). The following examples illustrate that inTheorem 2.3, one cannot restrict the matrix B to positive diagonal matrices or evensemipositive M -matrices. Example 2.15.
Consider A = " − , which is not nonnegative. However, for any α >
0, the matrix A + αI is semipositive, as for u = ( x , t ∈ R with x > A + αI ) u = ((1 + α ) x , x ) t ∈ ( R ) ◦ . The same calculation proves that A + B issemipositive for any positive diagonal matrix B , although A is not nonnegative. Now let B be a semipositive M -matrix of the form " α p p β , with α > , β > , p , p ≤
0. Let x = ( x , x ) t ∈ ( R ) ◦ be such that Bx ∈ ( R ) ◦ . Then, αx + p x > , p x + βx > x ≤ x , then take A = " − and if x > x , then take A = " − . In boththese cases, A is not nonnegative, but A + B is semipositive.As a final remark, it is worth pointing out that several interesting results on eventualcone invariance have been obtained recently by Kasigwa and Tsatsomeros [8]. Acknowledgements:
The first author acknowledges the SERB, Government of In-dia, for partial support in the form of a grant (Grant No. ECR/2017/000078). Thethird author acknowledges the Council of Scientific and Industrial Research (CSIR),India, for support in the form of Junior and Senior Research Fellowships (Award No.09/997(0033)/2015-EMR-I).
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