Positive semi definite

We assume that A is a symmetric and positive semi-definite matrix. The case of interest is when the largest eigenvalue of A is significantly larger than the norm of the derivative of the nonlinear force f. A may be a constant matrix, or else A = A(y) is assumed to be slowly changing along solution trajectories, in which case A will be evaluated at the current averaged position in the numerical schemes below. In the standard Verlet scheme, which yields approximations y to y nAt) via... 

Here C = aa ) is the covariance of the basis functions used to model the turbulence. Covariance matrices are positive semi-definite by dehnition which implies a C a > 0, and thus a dehned maximum of Pr a exists. 

It can be shown that all symmetric matrices of the form X X and XX are positive semi-definite [2]. These cross-product matrices include the widely used dispersion matrices which can take the form of a variance-covariance or correlation matrix, among others (see Section 29.7). 

By way of example we construct a positive semi-definite matrix A of dimensions 2x2 from which we propose to determine the characteristic roots. The square matrix A is derived as the product of a rectangular matrix X with its transpose in order to ensure symmetry and positive semi-definitiveness ... 

If A is a symmetric positive definite matrix then we obtain that all eigenvalues are positive. As we have seen, this occurs when all columns (or rows) of the matrix A are linearly independent. Conversely, a linear dependence in the columns (or rows) of A will produce a zero eigenvalue. More generally, if A is symmetric and positive semi-definite of rank r[Pg.32]

Thus far we have considered the eigenvalue decomposition of a symmetric matrix which is of full rank, i.e. which is positive definite. In the more general case of a symmetric positive semi-definite pxp matrix A we will obtain r positive eigenvalues where r general case we obtain a pxr matrix of eigenvectors V such that ... 

Newton, the limit h —> 0 is singular. The symmetries underlying quantum and classical dynamics - unitarity and symplecticity, respectively - are fundamentally incompatible with the opposing theory s notion of a physical state quantum-mechanically, a positive semi-definite density matrix classically, a positive phase-space distribution function. 

The covariance matrix is positive semi-definite and symmetric. Thus, it can be written in terms of eigenvalues and eigenvectors as... 

To gain an understanding of this mechanism, consider the Hamiltonian operator (H — Egl) with only two-body interactions, where Eg is the lowest energy for an A -particle system with Hamiltonian H and the identity operator I. Because Eg is the lowest (or ground-state) energy, the Hamiltonian operator is positive semi-definite on the A -electron space that is, the expectation values of H with respect to all A -particle functions are nonnegative. Assume that the Hamiltonian may be expanded as a sum of operators G,G,... 

Because all of the components of J are Hermitian, and because the scalar product of any function with itself is positive semi-definite, the following identity holds ... 

Here x stands for the set x,, and summation over repeated indices is implied. The matrix By(x) is symmetric and positive semi-definite. Alternatively one may write... 

Exercise. Prove the following lemma If H is a positive semi-definite Hermitian matrix, and F anti-Hermitian then the eigenvalues of A = H + F have nonnegative real parts. Moreover, if the real part is zero the corresponding eigenvector is an eigenvector of H and F separately. Use this lemma to show that (5.12) is the solution of (5.10). 

Note that since the matrix T is positive (semi)definite, the first K eigenvalues Tk, k - 1,. .., K, are all nonnegative. The other N - K eigenvectors of T all have zero as an eigenvalue. (Note that the nonvanishing eigenvalues of T must equal those of the K x K Hermitian matrix ft s V V.)... 

In theorem 12 of P it was shown that the first term is that of a positive definite Gramian matrix. The second is clearly the general term of a positive semi-definite matrix we cannot assert that it is definite since one or more of the A Hi might be zero. But the sum of a positive definite and a positive semi-definite matrix is definite and so the Jacobian nowhere vanishes. 

The square matrix of second-order partial derivatives of a potential energy over the nuclear displacements, Hessian, H, is positive semi-definite if Q HQ > 0 for any arbitrary vector Q. 

In both the dual solution and decision function, only the inner product in the attribute space and the kernel function based on attributes appear, but not the elements of the very high dimensional feature space. The constraints in the dual solution imply that only the attributes closest to the hyperplane, the so-called SVs, are involved in the expressions for weights w. Data points that are not SVs have no influence and slight variations in them (for example caused by noise) will not affect the solution, provides a more quantitative leverage against noise in data that may prevent linear separation in feature space [42]. Imposing the requirement that the kernel satisfies Mercer s conditions (K(xj, must be positive semi-definite)... 

Equation (11.10) arises from inserting equation (11.52) into a = Uo ria/N and then expanding c(a) in powers of fl. The matrix (gij) coincides with the matrix of the linearized macroscopic equations (11.5). The fluctuations enter through the matrix (hij). The Hurwitz criterion [15] assures that this matrix is positive semi definite, which means that equation (11.7) is a linear multivariate Fokker-Planck equation. 

Hybird kernel can be expressed in K = where K, form a family of various kernel functions. The hybird coefficients A, are either simply required to be nonnegative or determined in the such way that the hybird kernel is positive semi-definite. So the hybird kernel function can be described as following ormula ... 

Q is symmetric positive semi-definite and R is symmetric positive definite. The desired output can either be a constant (regulator problem) or varying (tracking problem). The existence of weights on the control moves alleviates the problem of requiring large sampling times when nonminimum phase zeros exist in plants even in linear unconstrained optimization [17]. 

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