Nonpositive-definite matrix

Nonpositive-definite matrix A symmetric matrix M is positive definite if its determinant, each diagonal element, and each term MuMkt MikMti is positive. If these conditions for a synunetric matrix do not hold, the matrix is said to be nonpositive definite. 

It is also straightforward to generalize the off-diagonal interaction to incorporate the previously mentioned resonance picture of unstable states by using a complex symmetric operator. For general discussions on this issue, we refer to the proceedings of the Uppsala-, Lertorpet- and the Nobel-Satellite workshops and references therein [13-15]. Thus one may arrive at a complex symmetric secular problem (note that the same matrix construction may be derived from a suitable hermitean matrix in combination with a nonpositive definite metric [9] see also below), which surprisingly leads to a comparable secular equation as the one obtained from Eq. (1). To be more specific we write... 

The symmetric matrix A is nonpositive definite. Saddle points are present in this case certain eigenvalues are positive and some others are negative. In the two-dimensional case, function contours are hyperboles. 

If the Hessian matrix is nonpositive definite during the search, d may not be a direction of function decrease. 

The Levenberg-Marquardt method is able to move between Newton s method and the gradient method. This feature will be discussed later conversely, it is now important to consider this method for removing the problem of a nonpositive definite Hessian matrix. 

W is also nonpositive definite when the Hessian G is positive definite and the matrix A is m-rank. 

In Sect. 3.2, we have seen that a dynamical system with unilateral or bilateral frictional contact can possess a peculiar characteristic, namely the inertia matrix may be asymmetric and nonpositive definite. Painleve was the first to point out the difficulties that may arise in such cases [53, 95]. As we will see in this section through examples, the presence of a kinematic constraint with friction could lead to situations where the equations of motion of the system do not have a bounded solution (inconsistency) or the solution is not unique (indeterminacy). These situations where the existence and uniqueness properties of the solution of the equations of motion are violated are known as the Painleve s paradoxes. There is a vast literature on the general theory of the rigid body dynamics with frictional constraints... 

So far, lumping has been defined, but nothing has been said concerning the dynamic behavior of the" system. Now we come to the definition of exact lumping A system is said to be exactly lumpable by the matrix L if there exists an N X N matrix K, enjoying the same properties as K does (i.e., off-diagonal elements of K are nonpositive, = 0, and there exists at least an m" =... 

Here E is the energy of the degenerate bound states and I is the unit matrix. The second term describes the loss of population of the bound space due to dissociation. We know it is a loss term because Eq. (Ill) implies that the rate matrix Y is (semi)positive definite, i.e., the eigenvalues of the effective Hamiltonian must have a nonpositive imaginary part or that exp(—i nt/h) = exp(—/( - iY )t/h) = exp( - iEj/h)cxp(- Y t/h) so that the states do decay in time because T 0. To determine the eigenvalues we need to diagonalize Eq. (113). The assumption of Eq. (112) implies that there are K eigenvectors which can be explicitly written down ... 

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