Positivity conditions matrices

Probability that the analyte A is present in the test sample Conditional probability probability of an event B on the condition that another event A occurs Probability that the analyte A is present in the test sample if a test result T is positive Score matrix (of principal component analysis)... 

While all three matrices are interconvertible, the nonnegativity of the eigenvalues of one matrix does not imply the nonnegativity of the eigenvalues of the other matrices, and hence the restrictions Q>0 and > 0 provide two important 7/-representability conditions in addition to > 0. These conditions physically restrict the probability distributions for two particles, two holes, and one particle and one hole to be nonnegative with respect to all unitary transformations of the two-particle basis set. Collectively, the three restrictions are known as the 2-positivity conditions [17]. 

Because the addition of any two positive semidefinite matrices produces a positive semidefinite matrix, the four 3-positivity conditions [17] imply the following two less stringent constraints ... 

The three complementary representations of the reduced Hamiltonian offer a framework for understanding the D-, the Q-, and the G-positivity conditions for the 2-RDM. Each positivity condition, like the conditions in the one-particle case, correspond to including a different class of two-particle reduced Hamiltonians in the A-representability constraints of Eq. (50). The positivity of arises from employing all positive semidefinite in Eq. (50) while the Q- and the G-conditions arise from positive semidefinite and B, respectively. To understand these positivity conditions in the particle (or D-matrix) representation, we define the D-form of the reduced Hamiltonian in terms of the Q- and the G-representations ... 

If the G-matrix is positive semidefinite, then the above expectation value of the G-matrix with respect to the vector of expansion coefficients must be nonnegative. Similar analysis applies to G, operators expressible with the D- or Q-matrix or any combination of D, Q, and G. Therefore variationally minimizing the ground-state energy of n (H Egl) operator, consistent with Eq. (70), as a function of the 2-positive 2-RDM cannot produce an energy less than zero. For this class of Hamiltonians, we conclude, the 2-positivity conditions on the 2-RDM are sufficient to compute the exact ground-state A-particle energy on the two-particle space. 

D. A. Mazziotti, Solution of the 1,3-contracted Schrodinger equation through positivity conditions on the 2-particle reduced density matrix. Phys. Rev. A 66, 062503 (2002). 

G. Gidofalvi and D. A. Mazziotti, Variational reduced-density-matrix theory strength of Hamiltonian-dependent positivity conditions. Chem. Phys. Lett. 398, 434 (2004). 

J. R. Hammond and D. A. Mazziotti, Variational two-electron reduced-density-matrix theory partial 3-positivity conditions for A-representability. Phys. Rev. A 71, 062503 (2005). 

If the subspace S contains a positive definite element S, neither the energy problem nor the spectral optimization problem has an optimal solution since there is no positive semidefinite matrix P satisfying both the conditions (P, = 0, (P, I)j = 1, the convex set Pq n 5 is empty, and the energy pro-... 

In Table V we check the A-representability of the CSE 2-RDMs through three well-known positivity conditions, the D-, the Q-, and the G-conditions [4, 5, 63], The D- and the Q-conditions are given in Eqs. (86) and (87), while the G-condition states that the following matrix (known as the G-matrix)... 

The above mentioned positivity conditions state that the 2-RDM D, the electron-hole density matrix G, and the two-hole density matrix Q must be positive semidefinite. A matrix is positive semidefinite if and only if all of its eigenvalues are nonnegative. The solution of the corresponding eigenproblems is readily carried out [73]. For D, it yields the following set of eigenvalues ... 

Because the diagonal elements of a positive semidefinite matrix are never negative, this implies the first (2, 2) condition, Tr[%n,rAr] > 0. 

Mass spectrometric studies were carried out as a first qualitative means of checking for dicarboxylate anion binding (see also Section 3). Here, mixtures of sapphyrin dimer 15 and several representative dicarboxylate anions, such as oxalate, 4-nitrophthalate, 5-nitroisophthalate and nitroterephthalate in methanol, were subjected to high resolution FAB mass spectrometric (HR FAB MS) analysis using FAB positive NBA matrix. In general, peaks for the putative complexes were seen, lending credence to the hypothesis that the dicarboxylate substrates in question were, in fact, being bound by 15 under the matrix desorption/gas phase conditions used to effect these mass spectrometric analyses. 

However, (S 10.2-7) is incompatible with the vanishing of any diagonal element (i = j) of the determinant, because it is well known from general matrix theory that a positive-semidefinite matrix with a vanishing diagonal element is necessarily singular, and therefore has vanishing determinant, contrary to (S 10.2-7). Thus, the only condition under which =o is the trivial case where dR[ is stationary,... 

Riley, J.D., Solving systems of linear equations with a positive definite symmetric but possibly ill-conditioned matrix, Math. Table Aids Comput., 9, 96-101, 1955. 

Preconditioning is a technique which improves the condition number of a matrix and thereby increases the convergence rate of Krylov subspace methods. Thus, if the preconditioner A4 is a symmetric, positive definite matrix, the original problem Ax = b can be solved indirectly by solving M Ax = M h. The the residual can then be written as ... 

Physically, the semidefinite conditions on the D, Q, and matrices restrict the probabilities of finding particle-particle, hole-hole, and particle-hole pairs to be nonnegative, respectively. Even though the nonnegativity constraints in Eqs. (6-8) are non-redundant, these matrices contain equivalent information as each matrix can be expressed in a one-to-one mapping of another by the fermionic anticommutation relations. The (2,2)-positivity conditions are often denoted as DQG. Contraction of the positive semidefinite D, Q, and matrices generates one-particle D and one-hole 2 matrices that are also positive semidefinite. 

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