Negative definite matrix

The status of H can be used to identify the character of extrema. A quadratic form <2(x) = xrHx is said to be positive-definite if Q(x) > 0 for all x = 0, and said to be positive-semidefinite if Q(x) > 0 for all x = 0. Negative-definite and negative-semidefinite are analogous except the inequality sign is reversed. If Q(x) is positive-definite (semidefinite), H(x) is said to be a positive-definite (semidefinite) matrix. These concepts can be summarized as follows ... 

It can be shown from a Taylor series expansion that if/(x) has continuous second partial derivatives, /(x) is concave if and only if its Hessian matrix is negative-semidefinite. For/(x) to be strictly concave, H must be negative-definite. For /(x) to be convex H(x) must be positive-semidefinite and for/(x) to be strictly convex, H(x) must be positive-definite. 

Determine whether the following matrix is positive-definite, positive-semidefinite, negative-definite, negative-semidefinite, or none of the above. Show all calculations. 

Solution to the above Mth order equation always exists if the M + 1 by M + 1 Toeplitz matrix has a non-negative definite. In a typical experimental situation, lm f i-6-. the nonresonant contribution, dominates the CARS... 

Exercise. Define jump moments for the case that Y has more components. Show that the matrix a (ye) must be negative definite, or at least semi-definite. 

A symmetric matrix A is said to be positive-definite if the quadratic form uTAu > 0 for all nonzero vectors u. Similarly, the symmetric matrix A is positive-semidefinite if uTAu 2 0 for all nonzero vectors u. Positive-definite matrices have strictly positive eigenvalues. We classify A as negative-definite if u Au < 0 for all nonzero vectors u. A is indefinite if uTAu is positive for some u and negative for others. 

Figure 5 illustrates more generally various cases that can occur for simple quadratic functions of form q x) — JxTHx, for n = 2, where H is a constant matrix. The contour plots display different characteristics when H is (a) positive-definite (elliptical contours with lowest function value at the center) and q is said to be a convex quadratic, (b) positive-semidefinite, (c) indefinite, or (d) negative-definite (elliptical contours with highest function value at the center), and q is a concave quadratic. For this figure, the following matrices are used for those different functions ... 

Equations (12.27) and (12.64) show the stability of the nonequilibrium stationary states in light of the fluctuations Sev The linear regime requires P > 0 and dP/dt < 0, which are Lyapunov conditions, as the matrix (dAJdej) is negative definite at near equilibrium. 

In accordance with the said criterion, the stationary state will only be stable at the negative definiteness of aU intrinsic values of matrix M p of eigenvalue equation ... 

Another concept that is important is that of positive (or negative) definite. For this, it is required that>l be a symmetric matrix, that is a,j = Cy,. An important theorem is the following. 

A symmetric matrix A is said to be positive definite if all of the eigenvalues are positive it is said to be negative definite if all of the eigenvalues are negative. Semidefinite is similarly defined. There is a simple test to determine if a symmetric matrix is positive or negative definite. 

Theorem A. 10 allows us to conclude stability if the matrix is the variational matrix evaluated at a rest point. An important result is that the test in Theorem A.10 will work for some matrices that are not symmetric, not in the sense of being negative definite but in the sense of yielding stability based on the sign of the real parts of the eigenvalues. The type of matrix is closely associated with the orderings and monotone flow discussed in Appendices B and C. 

It is assumed that the diagonal matrix W is negative definite and that k and the square root in the transformation matrix are chosen positive definite. The elements of GW(E) are closely related to the original basis since they are obtained by a process which is similar to Lowdin s with the property of being optimally close to initial ones. Accordingly, it is useful to derive populations and bond orders from its elements. 

It is difficult to determine if a particular minimum is the global minimum, which is the lowest energy point where force is zero and second derivative matrix is definitely positive. Local minimum results from the net zero forces and positive definite second derivative matrix and saddle point results from the net zero forces and at least one negative eigenvalue of the second derivative matrix. 

Positive and negative definiteness can also be checked using the leading principal minor test, although semidefiniteness cannot be verified in this manner. Using standard matrix notation, let... 

This means that the poles of the Green s function, i.e., the molecular orbital energies, are obtained as eigenvalues of a negative definite matrix with elements -Kr Srs + Srs)Ks. The molecular orbital coeflficients can straightforwardly be inferred from the residues at the poles of this spectral representation. 

If S S is to be negative for all possible variations 8li and 5V, then the matrix S must be negative definite or equivalently, (-S) must be positive definite. The conditions under which S is negative definite are given by a theorem from linear algebra it is necessary and sufficient that the principal minors of S satisfy the following inequalities [5] ... 

The above procedure can be repeated to obtain the stability criteria for multicomponent mixtures. For a mixture of C components, the criterion is still (8.3.4) in which S is the (C + 2)2 matrix of second derivatives analogous to (8.3.5). The fluid is stable to small disturbances when S is negative definite that is, when odd-order principal minors of S are negative and simultaneously those of even order are positive. The reduction of those minors to economical forms is a tedious exercise that can often be alleviated by posing the criteria in terms of G or A rather than S. 

Stable node is a fixed point for which the gradient of / evaluated at x is zero and the Hessian matrix at x" is negative definite (Fien and Liu, 1994), namely. 

Theorem 6. Suppose the strategy space of the game is convex and all equilibria are interior. Then if the determinant H is negative quasi-definite (i.e., if the matrix H + is negative definite) on the players strategy set, there is a unique NE. 

Thus, since each is a positive matrix from Theorem 1, we can inductively show that each Lt, I i m, is also a positive matrix. By definition, each matrix Vt is a non-negative diagonal matrix. We now assume that at least one of the sub-regions Bk contains some fissionable material. By the same permutation of rows and columns of the matrices F, we can write... 

Theorem 5. If H and V are symmetric and positive definite matrices, and 2 is a non-negative definite matrix, then the Peaceman-Rachford matrix Tp is convergent for any fixed p > 0. 

Since 5 / can be positive or negative, condition (17.3.6) for stability of the equilibrium state implies that the matrix (5A,/S )gq must be negative definite. In a neighborhood of the equilibrium state, (0A,/6 ) would retain its negative definiteness. In this neighborhood, expression (17.3.5) must also be negative definite. Hence in the neighborhood of equilibrium we have the inequalities... 

The matrices B and C are both positive definite and so is their product, the matrix A, When (X — X ) is not zero, the rhs of (A5) is negative definite and when it is zero, that is at the stationary state, the rhs of (A5) is zero. Therefore the rhs of (A5) is negative semi-definite. Thus we have shown that... 

---