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Article Excerpt
Generalizing the P-property of a matrix, Gowda et al. [Gowda, M. S., R. Sznajder, J. Tao. 2004. Some P-properties for linear transformations on Euclidean Jordan algebras. Linear Algebra Appl. 393 203-232] recently introduced and studied P- and globally uniquely solvable (GUS)-properties for linear transformations defined on Euclidean Jordan algebras. In this paper, we study the invariance of these properties under automorphisms of the algebra and of the symmetric cone. By means of these automorphisms and the concept of a principal subtransformation, we introduce and study ultra and super P-(GUS)-properties for a linear transformation on a Euclidean Jordan algebra.
Key words: Euclidean Jordan algebra; automorphism; P-property; globally uniquely solvable property; complementarity problem; super and ultra P-properties
MSC2000 subject classification: Primary: 90C33, 17C55; secondary: 17C30
OR/MS subject classification: Primary: programming; secondary: mathematics
History: Received August 25, 2004; revised May 18, 2005.
1. Introduction. A real n x n matrix M is said to be a P-matrix if every principal minor of M is positive. P-matrices have found numerous applications in various fields; see, e.g., Berman and Plemmons [3], Cottle et al. [4], and Facchinei and Pang [5]. It is well known that this property can be described in any one of the following ways:
(1) The implication x [member of] [R.sup.n], x * Mx [less than or equal to] [??] x = holds, where * denotes the componentwise product.
(2) For all q [member of] [R.sup.n], the linear complementarity problem LCP(M, q) has a unique solution, that is, there exists a unique x [member of] [R.sup.n] such that x [greater than or equal to] 0, Mx + q [greater than or equal to] 0, and = 0.
(3) The map q [??] SOL(M, q) is single valued and Lipschitzian on [R.sup.n], where SOL(M, q) denotes the solution set of LCP(M, q).
While there are numerous other ways of describing the P-property of a matrix, we will consider here two automorphism invariance properties that are relevant to our discussion. Consider [R.sup.n] with the usual inner product and the componentwise product defined by [(x * y).sub.i] = [x.sub.i][y.sub.i] for i = 1,2,..., n, where [x.sub.i] is the ith component of (column) vector x [member of] [R.sup.n]. Let Aut([R.sup.n]) denote the set of all invertible matrices A on [R.sup.n] satisfying the condition A(x * y) = Ax * Ay for all x, y [member of] [R.sup.n], and let Aut([R.sup.n.sub.+]) denote the set of all invertible matrices C on [R.sup.n] satisfying the condition C([R.sup.n.sub.+]) = [R.sup.n.sub.+], where [R.sup.n+.sub.] denotes the nonnegative orthant in [R.sup.n]. We say that elements of Aut([R.sup.n]) are automorphisms of the algebra [R.sup.n] and those of Aut([R.sup.n.sub.+]) are automorphisms of the cone [R.sup.n.sub.+]. It is easily seen that Aut([R.sup.n]) consists of permutation matrices, and any element in Aut([R.sup.n.sub.+]) is a product of a permutation matrix and a diagonal matrix with positive diagonal entries. Now we observe that if M is a P-matrix, then so are [A.sup.T] MA and [C.sup.T] MC for any A [member of] Aut([R.sup.n]) and C [member of] Aut([R.sup.n.sub.+]). In other words, the P-matrix property is invariant under automorphisms of the algebra and the (nonnegative) cone.
The space [R.sup.n] together with the usual inner product and the componentwise product is an example of a Euclidean Jordan algebra. A (general) Euclidean Jordan algebra is a finite-dimensional real inner product space V with a bilinear mapping (x, y) [right arrow] x [omicron] y satisfying certain properties (see [section]2.1). In recent times, Euclidean Jordan algebras have become important in the study of conic optimization; see, e.g., Schmieta and Alizadeh [17]. Two examples of Euclidean Jordan algebras that are heavily studied in the current literature are: (1) the space [L.sup.n] of all real symmetric n x n matrices with inner product := trace(XY) = [summation.sup.n.sub.i,j=1] [X.sub.ij][Y.sub.ij] and Jordan product defined by
X [omicron] Y := 1/2(XY + YX)
for X, Y [member of] [L.sup.n], and (2) the space [R.sup.n] (n > 1) with the usual inner product and the Jordan product defined by
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.],
where [x.sub.0] [member of] R and [bar.x] [member of] [R.sup.n-1].
In any Euclidean Jordan algebra V, there is the cone of squares K := {x [omicron] x: x [member of] V}, which is a self-dual closed convex cone. In such an algebra, one can define the automorphism groups Aut(V) and Aut(K) in the following way (Faraut and Koranyi [6]): [LAMBDA] [member of] Aut(V) if [LAMBDA] is an algebra automorphism, that is, [LAMBDA]: V [right arrow] V is an invertible linear transformation satisfying the condition [LAMBDA](x [omicron] y) = [LAMBDA](x) [omicron] [LAMBDA](y) for all x, y [member of] V, and [GAMMA] [member of] Aut(K) if [GAMMA] is a cone automorphism, that is, [GAMMA]: V [right arrow] V is an invertible linear transformation satisfying the condition [GAMMA](K) = K. In a Euclidean Jordan algebra V, K is a symmetric cone, which means that for any two objects x, y [member of] int(K), there is a [GAMMA] [member of] Aut(K) such that [GAMMA](x) = y.
In Gowda and Song [7], the properties (1)-(3) of a P-matrix were extended to a linear transformation defined on [L.sup.n]; they were further extended to Euclidean Jordan algebras in Gowda et al. [10]. It was shown in Gowda et al. that the generalizations of properties (1)-(3), respectively called the P-property, the globally uniquely solvable (GUS)-property, and the Lipschitzian GUS-property, are all different. A generalization of the positive principal minor property was also introduced in that reference.
One objective of this paper is to study the invariance properties of the above P- and GUS-properties under the algebra and cone automorphisms. We will show that the properties that are based on the inner product (such as the GUS- and Lipschitzian GUS-properties) remain invariant under cone automorphisms, but that the P-property (which is based on the Jordan product) fails to have this invariance property. However, all the properties that we study remain invariant under algebra automorphisms.
There is another motivation for the present work. It has been shown (see Gowda and Song [7]) that the GUS-property implies the P-property but that the converse does not hold. As a sufficient condition for the GUS-property, Gowda and Song introduced the concept of the [P.sub.2]-property of a linear transformation on [L.sup.n] by means of the following condition:
X [greater than or equal to] 0, Y [greater than or equal to] 0, (X-Y)L(X-Y)(X+Y) [less than or equal to] [??] X = Y.
Parthasarathy et al. [16] show that a strongly monotone transformation satisfies this property. Gowda et al. [9] show that L has this property if and only if for every invertible Q [member of] [R.sup.nxn], every principal subtransformation of L, defined by L(X) := [Q.sup.T] L(QX[Q.sup.T])Q, has the P-(also the GUS) property. Because the above [P.sub.2]-condition is based on the associative property of the ordinary matrix product in [L.sup.n], this property cannot be extended to the context of a (general) Euclidean Jordan algebra. However, noting that L = [[GAMMA].sup.T] L[GAMMA], where [GAMMA] (defined by [GAMMA](X) := QX[Q.sup.T]) belongs to Aut([L.sup.n.sub.+]), we extend this property to general Euclidean Jordan algebras via automorphisms of the symmetric cone K. By calling this property the "ultra P-property," we show that the ultra P-property implies the GUS-property. In addition, we show that the Lipschitzian GUS-property implies the ultra P-property under certain conditions. We also define a related "super P-property" by using algebra automorphisms and study some of its properties.
2. Preliminaries.
2.1. Euclidean Jordan algebras. In this subsection, we briefly recall some concepts, properties, and results from Euclidean Jordan algebras. Most of these can be found in Faraut and Koranyi [6], Schmieta and Alizadeh [17], and Gowda et al. [10].
A Euclidean Jordan algebra is a triple (V, [omicron], ), where (V, ) is a finite-dimensional inner product space over R and (x, y) [??] x [omicron] y: V x V [right arrow] V is a bilinear mapping satisfying the following conditions:
(i) x [omicron] y = y [omicron] x for all x, y [member of] V,
(ii) x [omicron] ([x.sup.2] [micron] y) = [x.sup.2] [micron] (x [micron] y) for all x, y [member of] V, where [x.sup.2] := x [micron] x, and
(iii) (x [omicron] y, z) = (y, x [omicron] z) for all x, y, z [member of] V.
In addition, we assume that there is an element e [member of] V (called the unit element) such that x [omicron] e = x for all x [member of] V.
Henceforth, we assume that V is a Euclidean Jordan algebra and call x [omicron] y the Jordan product of x and y. In V, the set of squares
K := {x [omicron] x: x [member of] V}
is a symmetric cone (see, Faraut and Koranyi [6, p. 46]). This means that K is a self-dual closed convex cone and for any two elements x, y [member of] int(K), there exists an invertible linear transformation [GAMMA]: V [right arrow] V such that [GAMMA](K) = K and [GAMMA](x) = y.
For an element z [member of] V, we write
z [greater than or equal to] if and only if z [member of] K,
and z [less than or equal to] when -z [greater than or equal to] 0. We write z > if z [member of] int(K).
For x [member of] V, we define m(x) := min{k > 0: {e, x, ..., [x.sup.k]} is linearly dependent} and rank of V by r = max{m(x): x [member of] V}. An element c [member of] V is an idempotent if [c.sup.2] = c; it is a primitive idempotent if it is nonzero and cannot be written as a sum of two nonzero idempotents. We say that a finite set {[e.sub.1], [e.sub.2],.... [e.sub.m]} of primitive idempotents...
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