Matrices quadratic form

If A is a square matrix and AT is a column matrix, the product AX is a so a column. Therefore, the product XAX is a number. This matrix expression, which is known as a quadratic form, arises often in both classical and quantum mechanics (Section 7.13). In the particular case in which A is Hermitian, the product XxAX is called a Hermitian form, where the elements of X may now be complex. 

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 ... 

Quadrics associated with symmetric matrices Given A x a symmetric matrix, the quadratic form S = xTAx can be rewritten as... 

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. 

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. 

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 latter of the quadratic form may be easier to evaluate, because the inverse of the covariance matrix. Since e has n1elements (n1equations), h will have a chi-square distribution with n1 degrees of freedom. Thus at specified significance level a... 

With the dimensions of these quantities a dimensional matrix is formed. Their columns are assigned to the individual physical quantities and the rows to the exponents with which the base dimensions appear in the respective dimensions of these quantities (example Ap [M1 L-1 T"2]). This dimensional matrix is subdivided into a quadratic core matrix and a residual matrix, whereby the rank r of the matrix (here r = 3) in most cases corresponds to the number of the base dimensions appearing in the dimensions of the physical quantities. 

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. 

The Lyapunov function resembles the thermodynamic entropy production function and the asymptotic stability principle. If the eigenvalues of the coefficient matrix of the quadratic form of the entropy production are very large, then the convergence to equilibrium state will be rapid. 

The eigenvalues are obtained from the characteristic equation dct. / - AT] = 0, or for a 2 X 2 matrix we have the following quadratic form ... 

Using the vector form, eqn.(34), it is easy to show that fa is rotationally invariant that is, invariant to any orthogonal transformation. Let U be any 3x3 orthogonal matrix (any proper or improper rotation in 3 space) then the action of U on fa is to transform the quadratic forms in the pre-multiplying factors and exponential factor as (using the exponential factor as an example) ... 

Besides the measure of the dispersion of the one-dimensional projection i.e. the projective index, another distinction of PP PCA from the classical PCA is the procedure of computation. Since the projective index is the quadratic form of X as stated above, the extremal problem of Eqn. 1 can be turned into the problem of finding the eigenvalues and eigenvectors of the sample covariance matrix for which a lot of algorithms such as SVD, QR are available. Because of the adoption of the robust projective index in PP PCA, some nonlinear optimization approaches should be used. In order to guarantee the global optimum. Simulated annealing (SA) is adopted which is the main topic of this book. 

The representation of this equation for anything greater than two variates is difficult to visualize, but the bivariate form (m = 2) serves to illustrate the general case. The exponential term in Equation (26) is of the form x Ax and is known as a quadratic form of a matrix product (Appendix A). Although the mathematical details associated with the quadratic form are not important for us here, one important property is that they have a well known geometric interpretation. All quadratic forms that occur in chemometrics and statistical data analysis expand to produce a quadratic smface that is a closed ellipse. Just as the univariate normal distribution appears bell-shaped, so the bivariate normal distribution is elliptical. 

The second term in the right-hand side of Equation (12) defining the discriminant function is the quadratic form of a matrix expansion. Its relevance to our discussions here can be seen with reference to Figure 3 which illustrates the division of the sample space for two groups using a simple quadratic function. This Bayes classifier is able to separate groups with very differently shaped distributions, i.e. with differing covariance matrices, and it is commonly referred to as the quadratic discriminant function. 

Mapping displays 23 Matrix, confusion, 127 determinant, 212 dispersion, 82 identity, 206 inverse, 210 quadratic form, 212 singular, 211 square, 204 symmetric, 204 Matrix multiplication, 207 Mean centring, 17 Mean value, 2 Membership function, 117 Minkowski metrics, 99 Moving average, 36 Multiple correlation, 183 Multiple regression, backward elimination, 182... 

If A is a square matrix, the quadratic form x Ax is a scalar product. In Chapter 12 on response surface models was discussed how the stationary point on the response surface was determined as the roots of the systems of equations defined by setting all partial derivatives of the response surface model to zero. In matrix language this corresponds to determining for which values of the x variables, the vector d Mdx is the null vector. This derivative is computed as... 

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-... 

In order to write S2E as a quadratic form involving a self-adjoint matrix we introduce the 2N x 2N matrix... 

It is important to spell out the limitations on the derivation of the distribution (2.2S) of fluctuations. Consider the most general initial state, which is XPinitial state is pure if the rank of the matrix p is unity. Otherwise, it is a mixture. The transition intensity to the final state / is y = Y,ijx Pijxj where x = < T />. y = x px is then a quadratic form where the amplitudes x have a gaussian probability density... 

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