positive semidefinite

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positive semidefinite

[′päz·əd·iv ¦sem·i′def·ə·nət]
(mathematics)
Also known as nonnegative semidefinite.
A square matrix A is positive semidefinite if for every choice of complex numbers x1, x2, …, xn , where x̄j is the complex conjugate of xj .
A linear operator T on an inner product space is positive semidefinite if 〈 Tu, u 〉 is equal to or greater than 0 for all vectors u in the space.
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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  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 :
In , 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).

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