(1)2 Preliminary For any x = (x1, x2

2 Preliminary

For any x = (x₁, x₂) ∈ IR × IRⁿ⁻¹ and y = (y₁, y₂) ∈ IR × IRⁿ⁻¹, we define their Jordan product as

x ◦ y = (x^Ty , y₁x₂+ x₁y₂). (7) We write x² to mean x ◦ x and write x + y to mean the usual componentwise addition of vectors. Then ◦, +, together with e = (1, 0, . . . , 0)^T ∈ IRⁿ have the following basic properties (see [6, 7]):

(1) e ◦ x = x , ∀ x ∈ IRⁿ. (2) x ◦ y = y ◦ x , ∀ x, y ∈ IRⁿ.

(3) x ◦ (x²◦ y) = x²◦ (x ◦ y) , ∀ x, y ∈ IRⁿ. (4) (x + y) ◦ z = x ◦ z + y ◦ z , ∀ x, y, z ∈ IRⁿ.

Notice that the Jordan product is not associative in general. Besides , Kⁿ is not closed under Jordan product.

The Jordan product (7) is associated with SOC Kⁿ via the following useful facts.

1. For each x = (x1, x2) ∈ IR × IRⁿ⁻¹, the determinant and the trace of x are defined by det(x) = x²₁− kx₂k² , tr(x) = 2x₁. (8)

In general , det(x ◦ y) 6= det(x)det(y) unless x2 = y2.

2. A vector x = (x₁, x₂) ∈ IR × IRⁿ⁻¹ is said to be invertible if det(x) 6= 0. If x is invertible , then there exists a unique y = (y1, y2) ∈ IR × IRⁿ⁻¹ satisfying x ◦ y = y ◦ x = e. We call this y the inverse of x and denote it by x⁻¹.In fact, we have

x⁻¹ = 1

x²₁− kx₂k²(x1 , −x2) = 1 det(x)

µ

tr(x)e − x

¶

. (9)

Therefore, x ∈ int(Kⁿ). Moreover, if x ∈ int(Kⁿ), then x^−k = (x^k)⁻¹ is also well- defined.

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3. For any x ∈ Kⁿ, it is know that there exists a unique vector in Kⁿ denoted by x^1/2 such that (x^1/2)² = x^1/2◦ x^1/2 = x. Indeed,

x^1/2 =

µ

s, x2

2s

¶

, where s =

s1 2

µ

x1+

q

x²₁− kx2k²

¶

. (10)

In the above formula, the term x₂/s is defined to be the zero vector if x₂ = 0 and s = 0, i.e., x = 0.

4. For any x ∈ IRⁿ, we always have x² ∈ Kⁿ (i.e.,x² º_Kn 0). Hence, there exists a unique vector (x²)^1/2 ∈ Kⁿ denoted by |x|. It is easy to verify that |x| º_Kn 0 and x² = |x|² for any x ∈ IRⁿ. It is also known that |x| º_Kn x.

The SOC-functions are defined via the above spectral decomposition. More specifically, for any function f : IR → IR, the following vector-valued function associated with Kⁿ (n ≥ 1) was considered [5, 7]:

f^soc(x) = f (λ₁(x))u⁽¹⁾_x + f (λ₂(x))u⁽²⁾_x , ∀x = (x₁, x₂) ∈ IR × IRⁿ⁻¹. (11) If f is defined only on a subset of IR, then f^soc is defined on the corresponding subset of IRⁿ. The definition (11) is unambiguous whether x₂ 6= 0 or x₂ = 0. The cases of f^soc(x) = x^1/2, x², exp(x) are discussed in the book of [6]. In fact, the above definition (11) is analogous to one associated with the semidefinite cone S₊ⁿ; see [12, 13].

Next, the concepts of SOC-monotone and SOC-convex functions are introduced in [4].

Let f : IR → IR ,

(a) f is said to be SOC-monotone of order n if its corresponding vector-valued function f^soc satisfies

x º_Kn y =⇒ f^soc(x) º_Kn f^soc(y), (12) f is said to be SOC-convex of order n if its corresponding vector-valued function

(b) f^soc satisfies

f^soc

µ

(1 − λ)x + λy

¶

¹_Kn (1 − λ)f^soc(x) + λf^soc(y) , (13) 5

for all x, y ∈ IRⁿ and 0 ≤ λ ≤ 1.

The function f is SOC-monotone (respectively, SOC-convex) if f is SOC-monotone of all order n (respectively, SOC-convex of all order n). In particular , we notice that if f is continuous , then the condition (13) can be replace by the more special condition :

f^soc

µx + y 2

¶

¹_Kn 1 2

µ

f^soc(x) + f^soc(y)

¶

. (14)

The concepts of SOC-monotone and SOC-convex functions are analogous to matrix mono- tone and matrix convex functions [2, 8, 9, 11], and are special cases of operator monotone and operator convex functions [1, 3, 10]. Examples of SOC-monotone and SOC-convex functions are given in [4]. It is clear that the set of SOC-monotone functions and the set of SOC-convex functions are both closed under linear combinations and under pointwise limits.

Some interesting properties of λ₁, λ₂and u⁽¹⁾, u⁽²⁾ with the Jordan product associated with SOC are summarized below . We only state the properties without proof because they can found in [4, 7].

Property 2.1 ([7, Property 2.2])For any x = (x₁, x₂) ∈ IR × IRⁿ⁻¹ with the spectral values λ1, λ2 and spectral vectors u⁽¹⁾, u⁽²⁾ given as in (5) and (6), we have

(a) u⁽¹⁾ and u⁽²⁾ are orthogonal under Jordan product and have length 1/√ 2, i.e., u⁽¹⁾◦ u⁽²⁾ = 0 , ku⁽¹⁾k = ku⁽²⁾k = 1

√2. (15)

(b) u⁽¹⁾ and u⁽²⁾ are idempotent under Jordan product,i.e.,

u⁽ⁱ⁾◦ u⁽ⁱ⁾ = u⁽ⁱ⁾ , i = 1, 2 . (16)

(c) λ₁, λ₂ are nonnegative (positive) if and only if x ∈ Kⁿ (x ∈ int(Kⁿ)), i.e.,

λ_i º_Kn 0 , ∀i = 1, 2 ⇐⇒ x º_Kn 0 , (17) λ_i Â_Kn 0 , ∀i = 1, 2 ⇐⇒ x Â_Kn 0. (18)

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(d) The determinant,the trace and the Euclidean norm of x can all be represented in terms of λ₁, λ₂ :

det(x) = λ1λ2 , tr(x) = λ1+ λ2 , kxk² = 1

2(λ²₁+ λ²₂). (19)

Property 2.2 ([4, Property 2.2])For any x = (x₁, x₂) ∈ IR × IRⁿ⁻¹ with the spectral values λ₁, λ₂ and spectral vectors u⁽¹⁾, u⁽²⁾ given as in (5) and (6), we have

(a) x² = λ²₁u⁽¹⁾+ λ²₂u⁽²⁾, (b) If x ∈ Kⁿ , then x^1/2 =√

λ₁u⁽¹⁾+√ λ₂u⁽²⁾,

(c) |x| = |λ₁|u⁽¹⁾+ |λ₂|u⁽²⁾.

Property 2.3 ([4, Property 2.3]) (a) Any x ∈ IRⁿ satisfies |x| º_Kn x,

(b) For any x, y º_Kn 0 , if x º_Kn y , then x^1/2º_Kn y^1/2, (c) For any x, y ∈ IRⁿ , if x² º_Kn y² , then |x| º_Kn |y|, (d) For any x ∈ IRⁿ, x º_Kn 0 ⇐⇒ hx, yi ≥ 0, ∀y º_Kn 0, (e) For any x º_Kn 0 and y ∈ IRⁿ, x² º_Kn y² ⇐⇒ x º_Kn y.

Property 2.4 ([4, Proposition 2.1])For any x Â_Kn 0 and y Â_Kn 0, the follow results hold.

(a) If x º_Kn y , then det(x) ≥ det(y), tr(x) ≥ tr(y).

(b) If x º_Kn y , then λ_i(x) ≥ λ_i(y), ∀i = 1, 2.

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