where matrix X [greater than or equal to] 0 represents X is symmetric

positive semidefinite.

Consider our problem in the space H = [R.sup.m] for the equilibrium bifunction f (x, y) = <Px + Qy + q, y - x> and the mapping Ux is defined by the form (2.3) in Section 2, where g : [R.sup.m] - R is a convex function such that [mathematical expression not reproducible] and P, [OMEGA] are two matrices of order m such that Q is symmetric

positive semidefinite and Q - P is symmetric negative semidefinite.

The conclusion is that -A is

positive semidefinite or A is negative semidefinite.

Let {[x.sub.1],[x.sub.2],...,[x.sub.l]) be a set of l points in any given [R.sup.N]; we just need to prove that the Gram matrix in (9) is

positive semidefinite matrix.

Ran and Reurings in [2] studied the solutions and perturbation theory for a general matrix equation X + [A.sup.*] F(X)A = Q, where F represent a map from the set of all

positive semidefinite matrices into a space of complex matrices and satisfy some monotonicity properties.

The matrix [mathematical expression not reproducible] is nonsmgular since all the three diagonal blocks of are

positive semidefinite. The Hessian matrix H can be rewritten into D--L--U, where D is a diagonal block matrix,--L is a strictly lower block matrix and--U is a strictly upper block matrix of H.

Given a set of person image pairs, metric learning based methods are to learn an optimal

positive semidefinite matrix for the validity of metric that maximizes the probability of true matches pair having smaller distance than wrong match pairs.

If A d is positive definite, E is

positive semidefinite ([A.sub.d] > 0, E [??] : 0), and none of the element of [[bar.X].sub.d] is zero ([[bar.x].sub.i] [not equal to] 0, i = l, ..., n), then the system matrix [A.sub.d] [??] E is positive definite.

Therefore the total energy is E [greater than or equal to] [+ or -]([Q.sub.E] sin [alpha] + [Q.sub.B] cos [alpha]) since the other terms are

positive semidefinite. The total energy is saturated if the BPS equations are satisfied as follows [24]:

In [5], Aubrun studied bipartite random quantum states from the induced ensemble, and determined which values of the ratio environment size/system size the random states are, with high probability, PPT (i.e., they have a

positive semidefinite partial transpose).

where [M.sub.xx] and [M.sub.yy] are

positive semidefinite matrices and [[zeta].sub.1] and [[zeta].sub.2] are positive numbers.

P >0 (P [greater than or equal to] 0, P < 0, and P [less than or equal to] 0) means that the matrix P is positive definite (

positive semidefinite, negative definite, and negative semidefinite).