Spectral classification of convex cones
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Positivity
Spectral classification of convex cones Alberto Seeger1 Received: 11 June 2019 / Accepted: 12 December 2019 © Springer Nature Switzerland AG 2020
Abstract Closed convex cones can be classified according to their capacity to produce complementarity eigenvalues. Let K be a closed convex cone in a Euclidean space E. The K - spectrum of a linear map A : E → E is the set σ K (A) of all λ ∈ R for which the complementarity problem K x ⊥ (Ax − λx) ∈ K ∗ has a nonzero solution x ∈ E. Here, K ∗ is the dual cone of K and ⊥ stands for orthogonality. It is known that if K is a polyhedral cone, then σ K (A) has finite cardinality for all A. This work identifies a special class of non-polyhedral cones with the same property. It also identifies some classes of cones for which σ K (A) contains an interval for a particular A. The cardinality analysis of the spectral map σ K induced by K provides valuable information on the structure of the cone itself. Keywords Complementarity eigenproblem · Cone spectrum · Ellipsoidal cone · Cone of matrices Mathematics Subject Classification 15A18 · 15B48 · 47A75 · 90C33
1 Introduction Let E be a real Euclidean space of dimension at least three and L(E) be the vector space of linear endomorphisms on E. Eigenvalue problems involving nonnegativity constraints on eigenvectors are usually formulated in terms of complementarity conditions relative to a closed convex cone. Let C(E) be the set of nontrivial closed convex cones in E. That a closed convex cone is nontrivial means that it is different from the singleton {0 E } and different from the underlying space E. The dual cone of a closed
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Alberto Seeger [email protected] Department of Mathematics, University of Avignon, 33 rue Louis Pasteur, 84000 Avignon, France
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A. Seeger
convex cone K is defined by K ∗ := {y ∈ E : y, x ≥ 0
for all x ∈ K },
where ·, · is the inner product of E. The formulation of a complementarity eigenproblem in E reads as follows: given K ∈ C(E) and A ∈ L(E), we search for a scalar λ ∈ R and a nonzero vector x ∈ K such that the vector Ax − λx lies in the dual cone K ∗ and is orthogonal to x, i.e., K x ⊥ (Ax − λx) ∈ K ∗ .
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Such a nonzero vector x is called a K -eigenvector of A and the associated λ is called a K -eigenvalue of A. The set of K -eigenvalues of A is denoted by σ K (A) and it is called the K -spectrum of A. The spectral map σ K induced by K provides valuable information on the structure of the cone itself. In this work we explore the cone spectral map σ K from a cardinality point of view: we wish to know whether σ K (A) is a set of finite cardinality for all A, or whether there exists a particular A such that σ K (A) contains an interval. Of course, these two cases do not cover all the possibilities because a K -spectrum could be infinite and have empty interior at the same time. The later situation occurs however very rarely in practice. To proceed with the exposition at a more rigorous level, we introduce the following definition. Recall that a tensor product on
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