The eigenvalue map and spectral sets

Transcription

1 The eigenvalue map and spectral sets p. 1/30 The eigenvalue map and spectral sets M. Seetharama Gowda Department of Mathematics and Statistics University of Maryland, Baltimore County Baltimore, Maryland *************** International Conference on complementarity problems and related topics Beijing Jiaotong University May 18-19, 2016 (Joint work with Juyoung Jeong)

2 The eigenvalue map and spectral sets p. 2/30 Motivation Let S n = Space of all n n real symmetric matrices, S n + = the semidefinite cone. For X S n, let λ (X) and λ (X) denote respectively, the vectors of eigenvalues of X written in the decreasing and increasing order. Suppose (X,Y) is a complementary pair with respect to S+: n X S+, n Y S+, n and X,Y = 0. Then X and Y commute: X = U diag(x 1,x 2,...,x n )U T and Y = U diag(y 1,y 2,...,y n )U T.

3 The eigenvalue map and spectral sets p. 3/30 Here, U is orthogonal and x 1,x 2,...,x n are eigenvalues of x, etc. Then: x i 0, y i 0, and x i y i = 0 for all i. Thus, λ (X) 0, λ (Y) 0, and λ (X),λ (Y) = 0. So, if (X,Y) is a complementary pair in S+, n then (λ (X),λ (X)) is a complementary pair in R+. n Do we have results of this type in Euclidean Jordan algebras?

4 The eigenvalue map and spectral sets p. 4/30 To simplify notation, let λ(x) := λ (X). Note S+ n = λ 1 (R+) n and R+ n is a permutation invariant closed convex cone. The map λ is nonlinear, yet the inverse image of R+ n under λ is a closed convex cone. This motivates us to explore properties of sets E := λ 1 (Q), where Q is permutation invariant in R n.

5 The eigenvalue map and spectral sets p. 5/30 Outline Permutation invariant sets and cones in R n Euclidean Jordan algebras Spectral sets and cones The transfer principle A dimensionality result Commutation principles Cross complementarity concepts

6 The eigenvalue map and spectral sets p. 6/30 Permutation invariant set in R n A permutation matrix is a 0 1 matrix with exactly one 1 in each row and column. Σ n is the set of all n n permutation matrices. Σ n is a group. A set Q in R n is permutation invariant if σ(q) = Q σ Σ n.

7 The eigenvalue map and spectral sets p. 7/30 Majorization An n n real matrix A is doubly stochastic if is nonnegative and each row/column sum is one. Birkhoff s Theorem: A is doubly stochastic iff it is a convex combination of permutation matrices. For vectors x and y in R n, we say x is majorized by y and write x y if x = Ay for some doubly stochastic matrix A.

8 The eigenvalue map and spectral sets p. 8/30 Examples of permutation invariant sets R n and R+. n {x R n : x 1 +x 2 + +x n = 0}. For p = 1,2,...,n, let s p (x) = sum of smallest p entries of x. Q p := {x R n : s p (x) 0}. Note: R+ n = Q 1 Q 2 Q n. For any nonempty set S in R n, is permutation invariant. Q := Σ n (S)

9 The eigenvalue map and spectral sets p. 9/30 Euclidean Jordan algebra (V,,, ) is a Euclidean Jordan algebra if V is a finite dimensional real inner product space and the bilinear Jordan product x y satisfies: x y = y x x (x 2 y) = x 2 (x y) x y,z = x,y z K = {x 2 : x V} is the symmetric cone in V.

10 The eigenvalue map and spectral sets p. 10/30 Examples of Euclidean Jordan algebras Any EJA is a product of the following: S n = Herm(R n n ) - n n real symmetric matrices. Herm(C n n ) - n n complex Hermitian matrices. Herm(Q n n ) - n n quaternion Hermitian matrices. Herm(O 3 3 ) octonion Hermitian matrices. L n (n 3)- Jordan spin algebra. The above algebras are (the only) simple algebras.

11 The eigenvalue map and spectral sets p. 11/30 The eigenvalue map Spectral representation in an EJA of rank n: x = λ 1 e 1 +λ 2 e λ n e n, where {e 1,e 2,...,e n } is a Jordan frame, λ 1,λ 2,...,λ n are eigenvalues of x. The eigenvalue map λ : V R n is defined by λ(x) := (λ 1,λ 2,...,λ n ), where λ 1 λ 2... λ n. Note: The eigenvalue map is continuous, positive homogeneous, and nonlinear.

12 The eigenvalue map and spectral sets p. 12/30 Spectral sets Let V be a Euclidean Jordan algebra of rank n. A set E in V is a spectral set if for some permutation invariant set Q in R n, E = λ 1 (Q). We will see that λ 1 behaves more like a linear transformation. Examples: A symmetric cone in a EJA, its interior, its boundary are spectral sets.

13 The eigenvalue map and spectral sets p. 13/30 Characterization of spectral sets In V, let x y if λ(x) = λ(y). For a set Q in R n, Q := λ 1 (Q). For a set E in V, let E := Σ n (λ(e)). Theorem: For a set E in V, the following are equivalent: 1. E is a spectral set. 2. x y, y E x E. 3. E = E. Note: Product of spectral sets need not be spectral. Example: R S+ 2 in R S 2.

14 The eigenvalue map and spectral sets p. 14/30 Automorphism invariance A : V V is an automorphism if A is linear, invertible, and A(x y) = Ax Ay. In V = S n, automorphisms are of the form for some orthogonal U. X UXU T Theorem: Every spectral set is automorphism invariant. Converse holds ifv is simple. Note: Converse may not hold in general V.

15 The eigenvalue map and spectral sets p. 15/30 The transfer principle Let Q be permutation invariant set in R n and The transfer principle: E = λ 1 (Q). Describes properties of Q that get transferred to E. We will look at topological, convexity, and linear properties.

16 The eigenvalue map and spectral sets p. 16/30 Topological properties Q is permutation invariant in R n. Then, 1. λ 1 (Q) = λ 1 (Q). (closure) 2. [λ 1 (Q)] = λ 1 (Q ). (interior) 3. (λ 1 (Q)) = λ 1 ( (Q)). (boundary) Note: f 1 (A) f 1 (A) when f and A are arbitrary. Cor: If Q is closed/open/compact, then so is E. Note: If E is is closed/open/compact, then so is Q.

17 The eigenvalue map and spectral sets p. 17/30 Convexity/conic properties Let Q be permutation invariant in R n. 1. If Q is convex, then so is λ 1 (Q). 2. conv(λ 1 (Q)) = λ 1 (conv(q)). (convex hull) 3. If Q is a cone, then so is λ 1 (Q). We will prove the first item. The proof relies on majorization: In any simple algebra, λ(x+y) λ(x)+λ(y). Assume V is simple and let Q be convex.

18 The eigenvalue map and spectral sets p. 18/30 Let x,y λ 1 (Q). For 0 t 1, λ(tx+(1 t)y) tλ(x)+(1 t)λ(y), so λ(tx+(1 t)y) = A[tλ(x)+(1 t)λ(y)] for some doubly stochastic matrix A. By Birkhoff s theorem, this A is a convex combination of permutation matrices. Since P[tλ(x)+(1 t)λ(y)] Q for any permutation matrix P, we see that A[tλ(x)+(1 t)λ(y)] Q. Hence, λ(tx+(1 t)y) Q, so tx+(1 t)y λ 1 (Q).

19 The eigenvalue map and spectral sets p. 19/30 Linear properties For Q permutation invariant in R n, 1. αλ 1 (Q) = λ 1 (αq) (α R). 2. λ 1 (Q 1 +Q 2 ) = λ 1 (Q 1 )+λ 1 (Q 2 ). 3. [λ 1 (Q)] = λ 1 (Q ) ( -dual operation). Proof of second item relies on: λ(x+y) λ(x)+λ(y). Proof of third item relies on Schur type majorization: diag(x) x. Cor: If Q is a (closed) convex cone in R n, then λ 1 (Q) is a (closed) convex cone in V.

20 The eigenvalue map and spectral sets p. 20/30 Spectral cones Let Q be a permutation invariant convex cone in R n. Then K := λ 1 (Q) is called a spectral cone. Symmetric cones in EJAs are spectral cones. What is the dimension of a spectral cone? Recall: If K is a convex cone, dim(k) := dim(k K). Theorem: dim(k) {0,1,m 1,m}, where m = dim(v). What are spectral cones with these dimensions?

21 The eigenvalue map and spectral sets p. 21/30 Dimensions of spectral cones 1. K = {0}: dimension zero. 2. K nonzero, K {te : t R}: dimension one. 3. K = {x V : trace(x) = 0}: dimension n K is a spectral cone with interior: dimension n.

22 The eigenvalue map and spectral sets p. 22/30 Commutation principle in EJA Suppose E is a spectral set in V and f : V V is Frechet differentiable. Theorem (Ramirez, Seeger, and Sossa, 2013) Ifais a local minimizer of min f(x), x E then,aoperator commutes withf (a). Cor: If K is a spectral cone in V and a K, b K, and a,b = 0, then, a and b operator commute.

23 The eigenvalue map and spectral sets p. 23/30 Recall that a and b operator commute in V means: a = a 1 e 1 +a 2 e a n e n and b = b 1 e 1 +b 2 e b n e n, where {e 1,e 2,...,e n } is a Jordan frame, and a 1,a 2,...,a n are eigenvalues of a, etc. When K = λ 1 (Q), ā := (a 1,a 2,...,a n ) Q, b := (b 1,b 2,...,b n ) Q and ā, b = 0.

24 The eigenvalue map and spectral sets p. 24/30 This means that if (a,b) is a complementary pair of a spectral cone K = λ 1 (Q) in V, then some eigenvalue vectors ā, b form a complementarity pair of the cone Q in R n. This result can be sharpened. Based on Normal decomposition systems introduced by A. Lewis in 1996, we state a specialized commutation principle.

25 The eigenvalue map and spectral sets p. 25/30 Theorem (Gowda-Jeong, 2016) Suppose V is R n or a simple EJA. Let K = λ 1 (Q) be a spectral cone in V with Q permutation invariant in R n. If a K, b K, and a,b = 0, then, there exists a Jordan frame {e 1,e 2,...,e n } such that a = λ 1 (a)e 1 +λ 2 (a)e λ n(a)e n and b = λ 1 (b)e 1 +λ 2 (b)e λ n(b)e n. Hence, λ (a) Q, λ (b) Q, λ (a),λ (b) = 0.

26 The eigenvalue map and spectral sets p. 26/30 Summarizing: If V is R n or a simple EJA, K = λ 1 (Q) is a spectral cone in V, and (a,b) is a complementary pair with respect to K, then, (λ (X),λ (X)) is a complementary pair with respect to Q. This may lead to a new type of problem: Cross complementarity problem: Given permutation invariant Q in R n, f : R n R n, Find x R n such that x Q, y = f(x) Q, x,y = 0.

27 The eigenvalue map and spectral sets p. 27/30 Spectral functions f : R n R is a permutation invariant function if f(σ(x)) = f(x) for all x R n and σ Σ n. F : V R is a spectral function if there exists f permutation invariant such that F = f λ. Example: A set E in V is spectral iff its characteristic function is a spectral function. Many properties of f get transferred to F.

28 The eigenvalue map and spectral sets p. 28/30 Transfer of convexity property Theorem: SupposeQis convex and permutation invariant inr n, f : Q R is convex and permutation invariant. Then,F := f λis convex. Moreover, x y inv impliesf(x) F(y). Illustration: For 1 p, x sp,p := λ(x) p is a norm on V.

29 The eigenvalue map and spectral sets p. 29/30 Concluding remarks In this talk, we introduced spectral sets and cones in Euclidean Jordan algebras. We showed that many properties of underlying permutation invariant sets transfer to spectral sets/cones. Because these are generalizations of symmetric cones, it may be useful and worthwhile to look at complementarity problems on spectral cones.

30 The eigenvalue map and spectral sets p. 30/30 References (1) M.S. Gowda, Positive maps and doubly stochastic transformations on EJAs, to appear in LAA. (2) M.S. Gowda and J. Jeong, Commutation principles in EJAs and normal decomposition systems, Research Report, UMBC, April (3) J. Jeong and M.S. Gowda, Spectral sets and functions in EJAs, Research Report, UMBC, February (4) A.S. Lewis, Group invariance and convex matrix analysis, SIMAX, 17 (1996), (5) H. Ramirez, A. Seeger, and D. Sossa, Commutation principle in EJAs, SIOPT, 2013.