CHARACTERIZATION OF GLOBALLY-UNIQUELY-SOLVABLE PROPERTY OF A CONE-PRESERVING Z-TRANSFORMATION ON EUCLIDEAN JORDAN ALGEBRAS

Abstract

Let V be a Euclidean Jordan algebra with a symmetric cone K. We show that for a Z-transformation L with the additional property L(K) ⊆ K (which we will call ’cone-preserving’), GUS ⇔ strictly copositive on K ⇔ monotone + P. Specializing the result to the Stein transformation S_A(X) := X - AXA^T on the space of real symmetric matrices with the property $S_A(S^n_+){\subseteq}S^n_+$, we deduce that S_A GUS ⇔ I ± A positive definite.

1. Introduction

Consider a real finite dimensional Hilbert space V and let K be a proper cone in V , i.e., K is a closed convex cone in V with K ∩ (−K) = {0} and K − K = V . Let the dual of K be defined by

Under this setting, the cone linear complementarity problem, which is denoted by LCP(L, K, q), is to find x ∈ V such that

There were many attempts in characterizing the transformation L so that the LCP(L, K, q) has a unique solution for any given q ∈ V, the so-called Globally-Uniquely-Solvable property (GUS-property, for short).

Restricting V to a Euclidean Jordan algebra and K to a symmetric cone in V, Gowda and Sznajder in 2007 [6] characterized GUS-property of an algebra automorphism L (an invertible mapping such that L(x ◦ y) = L(x) ◦ L(y) (p.469 of [6] where ’◦’ denotes the Jordan product). They showed that for such transformation, GUS = P (Theorem 5.1 of [6]).

In 2013, Yang and Yuan [14] characterized GUS-property when K is the second-order cone (also known as the Lorentz cone). They provided sufficient and necessary conditions on the matrix M so that M has the GUS-property (Theorem 2 of [14]).

In 2015, Balaji characterized the GUS-property of L when (V, ◦, ⟨·, ·⟩) is the Jordan spin algebra. He [1] showed that

GUS = P+ L positive semidefinite on the boundary of K.

In this paper, we focus our attention to a symmetric cone K in a Euclidean Jordan algebra V where L is a Z-transformation with resepect to K such that L(K) ⊆ K. The term, Z-transformation, is defined by Gowda and J. Tao [8] to designate a linear transformation L : V → V is such that

There are many examples of Z-transformations including

that ha applications in control theory [10]. For many other examples, we refer the readers to Section 3 of [8].

Gowda and Sznajder in 2007 [6] characterized GUS-property of L where both L and −L are Z-transformations (so-called Lyapunov-like transformation (Section 7 of [6])). They showed (Theorem 7.1 of [6]) that such L has the GUS-property if and only if L is positive stable (that is, every eigenvalue of L has positive real parts) and positive semidefinite. Recently, Kong, Tao, et al. characterized the GUS-property of a Z-transformation on a Lorentz cone (Theorem 3.2 of [9]).

We show in this paper that for a linear transformation L : V → V where V is a Euclidean Jordan algebra with a symmetric cone K such that L(K) ⊆ K:

The composition of the paper is as follows: In section 2, we provide notations, definitions and preliminaries. In section 3, the characterization of the GUS-property of a cone-preserving (i.e., L(K) ⊆ K) Z-transformation is established. In section 4, we specialize the result to the Stein transformation SA with and get some matrix theoretic result relating the GUS-property. A conclusion is given in section 5.

 

2. Definitions and Preliminaries

First, we describe all the notations which are used in this paper. The ’∗’ denotes the Hadamard product (entry-wise product) of matrices whereas ’◦’ denotes the Jordan product. For a vector d ∈ Rn, d ≥ (>)0 means every component of d is nonnegative (positive). The notation d ≤ 0 means −d ≥ 0. We write Diag(d) to mean a diagonal matrix whose diagonal is the vector d. The notation ║d║ denotes the Euclidean norm, that is,

For a diagonal matrix D, diag(D) means a vector whose entries are the diagonal of D. We write ρ(A) to denote the spectral radius of A, i.e., the maximum distance from the origin to an eigenvalue of A in the complex plane. Finally, tr(AB) means the sum of the diagonal elements of the matrix product AB.

We now list definitions and state preliminary results that go along with the corresponding definition.

A Euclidean Jordan algebra is a triple (V, ◦, ⟨·, ·⟩) where (V, ⟨·, ·⟩) is a finite dimensional inner product space over R and (x, y) ↦ x ◦ y : V × V → V is a bilinear mapping satisfying the following conditions:

In a Euclidean Jordan algebra V , the set of squares

is a symmetric cone (Faraut and Korányi [3], p.46). This K is a self-dual (i.e., K = K∗) closed convex cone, and proper. For standard examples of Euclidean Jordan algebras, we refer the readers to p. 465 of [6].

This symmetric cone K induces a (partial) order on V (section 2.1 of [7]):

Since K is self-dual, closed and convex cone, let ΠK denote the metric projection onto K, that is, for an x ∈ V, x+ := ΠK(x) if and only if x+ ∈ K and ║x − x+ ║ ≤ ║x − y║ for all y ∈ K. It is well known that x+ is unique (by Moreau decomposition [11]) and any x ∈ V can be written as

with x+, x− ≥ 0 and ⟨x+, x− ⟩ = 0.

In a Euclidean Jordan algebra V, for a given x ∈ V, Gowda and Sznajder (p.464 of [6]) defined the Lyapunov transformation Lx : V → V as

and called the elements x and y operator commute if Lx Ly = Ly Lx. Moreover, for x, y ∈ K, if ⟨x, y⟩ = 0, then x and y operator commute and x ◦ y = 0 (Proposition 2.2 of [6]).

A linear transformation L has the

A linear transformation L : V → V is called

Specializing Corollary 6 of [4] to our case, we get that if L ∈ Z(K) is strictly copositive on K (where K is a symmetric cone in a Euclidean Jordan algebra), then L is monotone.

In addition, Theorem 22 [7] states that for a monotone transformation,

 

3. GUS-property of a cone-preserving Z-transformation on a Euclidean Jordan Algebra

We will call a Z-transformation with respect to K as simply Z-transformation if not otherwise specified. We first show that for a Z-transformation L : V → V with L(K) ⊆ K, GUS-property is equivalent to L being strictly copositive on the symmetric cone K.

Theorem 3.1. Consider a Euclidean Jordan algebra (V, ◦, ⟨·, ·⟩) and a symmetric cone K in V. For a Z-transformation L : V → V with L(K) ⊆ K, the following are equivalent:

Proof. For (a) ⇒ (b): Note that our assumption leads to L being monotone. Suppose there exists q ∈ V with two distinct solutions x1 and x2 in K. Let yi = L(xi) + q ∈ K for i = 1, 2, and let z = x1 − x2. Linearity and monotonicity of L together imply

whence ⟨z, L(z)⟩ = 0. Then, by writing z = z+ − z−, we get

Since both z+ and z− are in K with at least one of them nonzero, the sum of the first two terms is positive by strict copositivity of L on K. Moreover, since the symmetric cone K is self-dual and L ∈ Z(K), LT ∈ Z(K). Then ⟨z+, L(z−)⟩ = ⟨LT (z+), z−⟩ with z+ and z− orthogonal, therefore, ⟨z+, L(z−)⟩ and ⟨z−, L(z+)⟩ are both nonpositive by Z-property of LT. However, these observations lead to an absurd conclusion, namely,

Hence L has the GUS-property.

For (b) ⇒ (a): Suppose there exists 0 ≠ x ∈ K such that

Since L has the GUS-property, it has the Q-property. So L−1 exists and L−1 (K) ⊆ K. Since L(K) ⊆ K and 0 ≠ x ∈ K, one can find 0 ≠ y ∈ K such that x = L−1 (y). Then (2) becomes ⟨L−1 (y), y⟩ = ⟨x, y⟩ ≤ 0. Since both x, y are elements of K, this means x ◦ y = 0 = x ◦ L(x) for some nonzero x, and x and L(x) operator commute. But note that since GUS ⇒ P for L, this is a contradiction. This completes the proof. □

Now we establish the main result of the paper.

Corollary 3.2. For a Z-transformation L on a Euclidean Jordan algebra with a symmetric cone K with L(K) ⊆ K, the following are equivalent:

Proof. The equivalence of (a) and (b) follows from Theorem 3.1. For (b) ⇒ (c): if L has the GUS-property, then by Theorem 3.1, L is strictly copositive on K, and hence L is monotone, and GUS ⇒ P. Hence (c) holds. For the converse, P = GUS for a monotone transformation. This completes the proof. □

 

4. GUS-property of the Stein Transformation SA : Sn → Sn when

In this section, we restrict our attention to the space of real symmetric matrices Sn and the Stein transformation L = SA : Sn → Sn, defined by

We recall below some known results for SA:

Using Corollary 3.2, we try to characterize the GUS-property of SA in terms of the matrix A when .

First we recall useful results for nonnegative matrices:

Lemma 4.1 (Part of Theorem (1.1) of [2]). If A is a nonnegative square matrix, then

We characterize the strict copositivity of SA in terms of the matrix A, and relate it to the GUS-property.

Theorem 4.2. For A ∈ Rn×n, consider the Stein Transformation SA(X) = X − AXAT with . Then the following are equivalent.

Proof. For (a) ⇒ (b): suppose SA is strictly copositive on , i.e., ⟨X, SA(X)⟩ > 0 for all 0 ≠ X ∈ . Fix an arbitrary orthogonal matrix U and an arbitrary vector d ≥ 0. Let B = UT AU, D = Diag(d), and X = U DUT. Then,

So,

From Lemma 4.1, the matrix B ∗ B has a nonnegative (hence real) eigenvector corresponding to the real eigenvalue ρ(B ∗ B). If x is such an eigenvector, then ⟨x, (B ∗ B)x⟩ = ρ(B ∗ B)⟨x, x⟩ < ⟨x, x⟩ by (3). Hence ρ(B ∗ B) < 1. Since U is arbitrary, we get the desired result.

For (b) ⇒ (c): suppose SA is not GUS. Then there exists Q ∈ Sn such that two distinct symmetric positive semidefinite matrices X1 and X2 solve SDLCP(SA, Q). Let Yi = SA(Xi) + Q for i = 1, 2, and let Z = X1 − X2. Then by linearity of SA,

Let us write Z = U DUT in a way that diagonal of D is the eigenvalues of Z in decreasing order. That is, D = D+ − D− where D+ has only positive eigenvalues of Z (if any) or zero on its diagonal and D− has absolute values of negative eigenvalues of Z (if any) or zero on its diagonal. Note that both D+, D− ≥ 0, D+ D− = 0, and at least one of D+ or D− is nonzero since Z ≠ 0. Now let B = UT AU, and d = diag(D), to get

Note that

⟨d+, (I − B∗B)d+⟩ = ⟨U D+UT, U D+UT − AU D+UT AT⟩, and a similar equation holds for ⟨d−, (I − B ∗ B)d−⟩ as well. Since both d+ and d− are nonnegative with at least one of d+ or d− is nonzero, the sum of these two terms is positive by strictly copositivity of SA. Moreover,

⟨d+, (I − B ∗ B)d−⟩ = ⟨d+, d−⟩ − ⟨d+, (B ∗ B)d−⟩ ≤ 0 because d+ and d− are orthogonal and (B ∗ B) is a nonnegative matrix. Same conclusion holds for ⟨d−, (I − B ∗ B)d+⟩. However, these observations lead to an absurd conclusion, namely, 0 ≥ ⟨Z, SA (Z)⟩ > 0. Hence SA is GUS and (b) ⇒ (c) is established.

Finally, the implication (c) ⇒ (a) is obtained by applying the result of Theorem 3.1 for V = Sn, K = and L = SA. □

Now, We give a matrix-theoretic characterization of the GUS-property of the Stein Transformation SA(X) = X − AXAT with below and show that, interestingly, strict copositivity and strict monotonicity are equivalent in such case, i.e., if we let K = and V = Sn, then

That is, SA positive definite on the symmetric cone K implies SA positive definite for the whole space V .

Corollary 4.3. For A ∈ Rn×n, consider the Stein Transformation SA(X) = X − AXAT with . Then the following are equivalent.

Proof. The equivalence of the statements (a), (b), and (c) is from Theorem 2.5 (p233) of [13]. The equivalence of the statements (c) and (f) is from Theorem 2.4 (p232) of [13]. On the other hand, the equivalence of the statements (d) and (e) is from Theorem 4.2 of this paper. Moreover, (d) ⇒ (b) (GUS ⇒ P), and (f) ⇒ (d) (strict monotonicity ⇒ GUS). Hence, the statements (a) through (f) are all equivalent. Finally, the equivalence of (g) and (d) follows from Theorem 4.2 of this paper. □

 

5. Conclusion

In this paper, we showed that for a Z-transformation L with respect to a symmetric cone K in a Euclidean Jordan algebra such that L(K) ⊆ K, the following implications hold:

By specializing the result to the Stein transformation SA on the space of real symmetric matrices such that , we have