Quadratic form, positive definite

To estimate Apa, we choose some values of T, py and therefore some values of rio, Apy, fay are fixed. For simplicity, we assume that quadratic form (4.180) is positive definite with elements fay of symmetrized matrix and denote by t y the elements of its inversion. Taking first derivative of fli (4.169) (in arbitrary real trDy at chosen T, py) as zero we obtain the extremal values trDy (in fact in minimum because second derivatives of (4.169) form positive definite matrix of (4.180), cf. [134, Sect. 11.3-3]). Inserting this values into (4.169) (for which this inequality is valid too) we obtain the following minimal values of n i... 

The Hamiltonian function for a system of bound harmonic oscillators is, in the most general form, a sum of two positively definite quadratic forms composed of the particle momentum vectors and the Cartesian projections of particle displacements about equilibrium positions ... 

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 ... 

To describe bound stationary states of the system, the cji s have to be square-normalizable functions. The square-integrability of these functions may be achieved using the following general form of an n-particle correlated Gaussian with the negative exponential of a positive definite quadratic form in 3n variables ... 

As you learned in the previous sections, LU decomposition with built-in partial pivoting, followed by backsubstitution is a good method to solve the matrix equation Ax = b. You can use, however, considerable simpler technics if the matrix A has some special structure. In this section we assume that A is symmetric (i.e., AT = A), and positive definite (i.e., x Ax > 0 for all x 0 you will encounter the expression x Ax many times in this book, and hence we note that it is called quadratic form.) The problem considered here is special, but very important. In particular, estimating parameters in Chapter 3 you will have to invert matrices of the form A = X X many times, where X is an nxm matrix. The matrix X X is clearly symmetric, and it is positive definite if the columns of X are linearly independent. Indeed, x (x" X)x = (Xx) (Xx) > 0 for every x since it is a sum of squares. Thus (Xx) (Xx) = 0 implies Xx = 0 and also x = 0 if the columns of X are linearly independent. 

If J is a positive definite quadratic form with a closed extension, its closure J and hence the self-adjoint operator If belonging to J are also positive definite. 

Let Ju J2 be two positive-definite quadratic forms such that... 

Case ii). C) is automatically satisfied if B), B0) hold and, moreover, the spectrum of IIt) at least belotu Aq consists of finite discrete eigenvalues with finite multiplicities. To show this we may assume U0 and i cl) to be positive definite (see the remark of 7.1). Then the quadratic form ((i70 + /) is an increas-... 

The variance of the restricted least squares estimator is given in the second equation in the previous exercise. We know that this matrix is positive definite, since it is derived in the form B positive definite. Therefore, it remains to show only that the matrix subtracted from Var[b] to obtain Var[b ] is positive definite. Consider, then, a quadratic form in Var[b ]... 

It remains to show, therefore, that the inverse matrix in brackets is positive definite. This is obvious since its inverse is positive definite. This shows that every quadratic form in Var[b ] is less than a quadratic form in Var[b] in the same vector. 

Also this is a positive definite quadratic form, since (—dF/dt) must always be positive. (F tends to its minimum.)... 

At this stage of the derivation, emphasis must be stressed on the positive definiteness of both quadratic forms involved. As a consequence of this positive definiteness, the transformation matrix must be real and the values t, must all be real and positive. 

Furthermore, Eq. 2.42 guarantees that all the eigenvalues of Eq. 2.21 will be real numbers. Also, the quadratic form in Eq. 2.23 together with Eq. 2.16 implies that the kinetic matrix (La ) will be positive definite all the eigenvalues are nonnegative.8... 

The quadratic form d2f/d(kd t is positive definite since, for any set of values the quantity... 

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 ... 

The first iteration in a CG method is the same as in SD, with a step along the current negative gradient vector. Successive directions are constructed differently so that they form a set of mutually conjugate vectors with respect to the (positive-definite) Hessian A of a general convex quadratic function. 

There are n(n -1)/2 independent diffusivities Ay, which are also the coefficients in a positive definite quadratic form, since according to the second law of thermodynamics, the internal entropy of a single process never decreases. In terms of these symmetric diffusivities, the mass flow becomes... 

If the function f(xv. .., xn) is convex (concave) on squadratic form (Eq. (2)) in s variables is positive definite (negative definite). The quadratic form (Eq. (2)) in s variables for which is positive definite (negative definite) if [18]... 

The second-order change in the free energy AA . thus appears to be a quadratic form in these coefficients. If A A has to be positive for arbitrary variation coefficients ca P, the Hessian of this form has to be positive definite. By Fourier transforming the coefficients, the Hessian can be block-diagonalized with blocks... 

Show that V = ax +2bxy + cy is positive definite if and only if a>0 and ac- b > 0. (This is a useful criterion that allows us to test for positive definiteness when the quadratic form F includes a cross term 2tey.)... 

In a linear system Ax = b where the matrix A is symmetric and positive definite, the solution is obtained by minimizing the quadratic form (12.331). This implies that the gradient, / (x) = Ax — b, is zero. In the iteration procedure an approximate solution, x +i, can be expressed as a linear combination of the previous solution and a search direction, p, which is scaled by a scaling factor am-... 

To define the quadratic form, we can always choose symmetrical coefficients Ajt = Au which must be considered as the matrix elements of a definite positive operator A. In other words, there always exists a positive number rj having the property that for any non-zero set x = (x,.. ., xN),... 

Example Spatial Oscillator.—A massive particle is restrained by any set of forces in a position of stable equilibrium (t.g. a light atom in a molecule otherwise consisting of heavy, and therefore relatively immovable atoms). The potential eneigy is then, for small displacement, a positive definite quadratic function of the displacement components. The axes of the co-ordinate system (x, y, z) can always be chosen to lie along the principal axes of the ellipsoid corresponding to this quadratic form. The Hamiltonian function is then... 

The conditions for stability are analogous to those in the case of one degree of freedom. The particular motion fp=ijp=0 is stable when, and only when, the quadratic form i (17) is definite. The energy is a minimum if it is positive definite. 

The thermodynamics of irreversible processes begins with three basic microscopic transport equations for overall mass (i.e., the equation of continuity), species mass, and linear momentum, and develops a microscopic equation of change for specific entropy. The most important aspects of this development are the terms that represent the rate of generation of entropy and the linear transport laws that result from the fact that entropy generation conforms to a positive-definite quadratic form. The multicomponent mixture contains N components that participate in R independent chemical reactions. Without invoking any approximations, the three basic transport equations are summarized below. 

Step 8. Postulate Unear relations between these fluxes and forces that obey the Curie restriction, and demonstrate that entropy generation can be expressed as a positive-definite quadratic form. 

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