Eigenvalue 2-positivity conditions

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]. 

A careful reader will observe that this algebraic transformation will produce a dual SDP problem that does not have y G IR such that the matrix in Eq. (16) has all of its eigenvalues positive and, therefore, will not satisfy the Slater conditions. However, numerical experiments have shown that practical algorithms stUl can solve these problems efhciently [16]. 

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 ... 

The real and positive condition on the eigenvalues places physical limits on the interdiffusivities. For the ternary case,... 

If ti satisfies necessaiy conditions [Eq. (3-80)], the second term disappears in this last line. Sufficient conditions for the point to be a local minimum are that the matrix of second partial derivatives F is positive definite. This matrix is symmetric, so all of its eigenvalues are real to be positive definite, they must all be greater than zero. 

The condition number is always greater than one and it represents the maximum amplification of the errors in the right hand side in the solution vector. The condition number is also equal to the square root of the ratio of the largest to the smallest singular value of A. In parameter estimation applications. A is a positive definite symmetric matrix and hence, the cond ) is also equal to the ratio of the largest to the smallest eigenvalue of A, i.e.,... 

These necessary conditions for local optimality can be strengthened to sufficient conditions by making the inequality in (3-87) strict (i.e., positive curvature in all directions). Equivalently, the sufficient (necessary) curvature conditions can be stated as follows V /(x ) has all positive (nonnegative) eigenvalues and is therefore defined as a positive (semidefinite) definite matrix. 

Consider the space state model R deflned by Eq.(52), showing an equilibrium point such that the matrix of the linearized system at this point has a real negative eigenvalue A and a pair of complex eigenvalues a j/3, j = /—1) with positive real parts 0.. In this situation, the equilibrium point has onedimensional stable manifold and two-dimensional unstable manifold. If the condition A < a is verified, it is possible that an homoclinic orbit appears, which tends to the equilibrium point. This orbit is very singular, and then the Shilnikov theorem asserts that every neighborhood of the homoclinic orbit contains infinite number of unstable periodic orbits. 

When N = p, the set B simply contains the p-particle reduced Hamiltonians, which are positive semidefinite, but when N = p + 1, because the lifting process raises the lowest eigenvalue of the reduced Hamiltonian, the set also contains p-particle reduced Hamiltonians that are lifted to positive semidefinite matrices. Consequently, the number of Wrepresentability constraints must increase with N, that is, B C B. To constrain the p-RDMs, we do not actually need to consider all pB in B, but only the members of the convex set B, which are extreme A member of a convex set is extreme if and only if it cannot be expressed as a positively weighted ensemble of other members of the set (i.e., the extreme points of a square are the four corners while every point on the boundary of a circle is extreme). These extreme constraints form a necessary and sufficient set of A-representability conditions for the p-RDM [18, 41, 42], which we can formally express as... 

For the next step, we show how we consider the N-representability conditions for the 1-RDM y for a system with N particles that is all of its eigenvalues should be between zero and one [17]. In other words, this condition is equivalent to saying that y and / — y are positive semidefinite, where / is the identity matrix. Assuming that H is the one-body Hamiltonian, we have... 

Therefore the 4-MCSE is not only determinate but, when solved, its solution is exact. As already mentioned, the price one has to pay is the fact of working in a four-electron space and the difficulty, as in the 2-CSE case, is that the matrices involved must be A-representable. Indeed, in order to ensure the convergence of the iterative process, the 4-RDM should be purified at each iteration, since the need for its A-representability is crucial. In practice, the optimizing procedure used is to antisymmetrize the at each iteration. This operation would not be needed if aU the matrices were A-representable but, if they are not, this condition is not satisfied. In order to impose that the 4-RDM, from which all the lower-order matrices are obtained, be positive semidefinite, the procedure followed by Alcoba has been to diagonalize this matrix and to apply to the eigenvalues the same purification as that applied to the diagonal elements in the 2-CSE case, by forcing the trace to also have a correct value. 

Figure 6 shows how the 5-representability is attained. Thus it can be seen from this hgure that the ofi Gaa fifi/spin-block converges very satisfactorily on a positive/negative semidehnite matrix. After twenty iterations the lowest/highest eigenvalue of these two matrices is —0.00010 and 0.00022, respectively. As was mentioned in Section II, these conditions are much more exacting than the well-known G-condition. 

In the previous sections we have implied that the loss of local stability which occurs for a stationary-state solution as the real part of the eigenvalues changes from negative to positive is closely linked to the conditions under which sustained oscillatory responses are born. 

With 0SS > 1, the lower root in (4.46) describes the small lower loop, which corresponds to the conditions for which the stationary state regains nodal character. Inside this region, the eigenvalues Al 2 are real and are positive, so we have an unstable node. This curve has a maximum at k = (3 — y/S) exp [ — i(3 + v/5)] w 0.0279, so this response is not to be found over a wide range of experimental conditions. 

We have specified that the conditions of interest here are those lying within the Hopf bifurcation locus, so tr(U) will be greater than zero. This means that the eigenvalues appropriate to this uniform state must have positive real parts. The system is unstable to uniform perturbations. We must exclude this n = 0 mode from any perturbations in the remainder of this section. 

For case (3), with /i in the range between ji and fi2, the condition tr(J) = 0, corresponding to eqn (10.80), is of less importance (and hence the locus is shown as a broken curve in Fig. 10.10). As n increases from zero, so the trace does become less positive, but now the determinant can also change sign. The latter occurs first, in fact, so the eigenvalues to change from two, real, and... 

The pressure-gradient term Ar requires either a further boundary condition or a determination of the domain size. In the semi-infinite cases, Ar is a constant that is specified in terms of the outer potential-flow characteristics. However, the extent of the domain end must be determined in such a way that the viscous boundary layer is entirely contained within the domain. In the finite-gap cases, the inlet velocity n(zend) is specified at a specified inlet position. Since the continuity equation is first order, another degree of freedom must be introduced to accommodate the two boundary conditions on u, namely u — 0 at the stagnation surface and u specified at the inlet manifold. The value of the constant Ar is taken as a variable (an eigenvalue) that must be determined in such a way that the two velocity boundary conditions for u are satisfied. 

However, the necessary and sufficient conditions for positive-definiteness of M can be expressed most concisely in terms of the positivity of its eigenvalues , ... 

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