Positive-semidefinite Hermitian matrices

This means that L must be positive-semidefinite. A matrix L is a positive-semidefinite matrix if and only if L is Hermitian and... 

Proposition 3 Let P be the k-matrix for the k-density p. Then P is Hermitian, positive semidefinite, and has unit trace. ... 

The G-matrix is also Hermitian and positive semidefinite. The condition... 

The 2-RDM, the 2-HRDM, and the G-matrix are the only three second-order matrices which (by themselves) are Hermitian and positive semidefinite thus they are at the center of the research in this field. Recently, a formally exact solution of the A -representability problem was published [12] but this solution is unfeasable in practice [40]. 

The preceding is a rather comprehensive—but not exhaustive— review of N-representability constraints for diagonal elements of reduced density matrices. The most general and most powerful V-representability conditions seem to take the form of linear inequalities, wherein one states that the expectation value of some positive semidefinite linear Hermitian operator is greater than or equal to zero, Tr [PnTn] > 0. If Pn depends only on 2-body operators, then it can be reduced into a g-electron reduced operator, Pq, and Tr[Pg vrg] > 0 provides a constraint for the V-representability of the g-electron reduced density matrix, or 2-matrix. Requiring that Tr[Pg Arrg] > 0 for every 2-body positive semidefinite linear operator is necessary and sufficient for the V-representability of the 2-matrix [22]. 

A < 1. Some of these coefficients can be close to 1 (strongly occupied NSO) and some close to 0 (weakly occupied NSO), but in any case the post-HF density matrix loses its property = p. Turning to the basis of natural orbitals (NOs) results in the appearance of two spin components of the density matrix, p° and p. For RHF, ROHF, and UHF cases, these spin components are idempotent, being projectors on the subspace of the full MO space spanned by the occupied density operator is the Hermitian positive-semidefinite operator with a spur equal to the number of electrons. At the same time, in general, this operator does not possess any other specific properties such as, for example, idempontency. After the convolution over the spin variables, the density operator breaks down into two components whose matrix representation in the basis set of atomic orbitals (AOs) has the form... 

Secondly, we note that the density matrix may become singular. is a Hermitian, positive semidefinite matrix and - as mentioned above - its eigenvalues, called natural weights, characterize the importance of the corresponding natural orbital. A zero eigenvalue occurs if there is a natural orbital (i.e., a linear combination of the single-particle functions) that does not contribute to the MCTDH wave function. Its time evolution may thus be modified by replacing with p( )... 

We have assumed that Xi > IX2I. What happens if ki = k2Y> la. general, the power method oscillates and the sequence ut l, 1,. . . fails to converge. However, for the special case of a Hermitian, positive-semidefinite matrix, all eigenvalues must be real and non-negative. The only way that Xi = IX2I is if A.i = Xj. In the limit k00,... 

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