Positive semidefinite matrix

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 ... 

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-... 

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

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,... 

As may easily be confirmed from (13.42), the Onsager matrix L is a positive-semidefinite matrix satisfying the eigenvalue equation... 

W is a positive definite matrix and R is a positive semidefinite matrix. State and input constraints are... 

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

Higham, N. J., Computing a nearest symmetric positive semidefinite matrix. Linear Algebra Applic., 103, 103 (1988). 

Finally, the matrix Jjj, is also positive definite, as the sum of a positive definite and a positive semidefinite matrix. 

Updated values of the Lagrange multiplier X are calculated from the stationary GENERIC equation(7.23) by inverting the symmetric, positive semidefinite matrix M. 

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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