Eigenvectors and eigenvalues

Eigenvectors arise from transformations and can only be found for square matrices, and if they are found, exactly n eigenvectors exist for an n times n matrix. All eigenvectors of a matrix are orthogonal that is, data can be represented in terms of 

We can combine all of them by using the single matrix equation 

The diagonal eigenvalues in this matrix make up the eigenvalue spectrum. The decomposition 

It is useful to characterize a set of values that has a tendency to cluster around a centered value by its centered moments of distribution, (the sums of the n integer powers of 

Consider a square matrix [/ll of order m. Let (x) be a column matrix of the same order, that is, with m rows and 1 column. From the definition of matrix multiplication it is known that the premultiplication of the matrix (x) by the matrix [ 4] generates a new column matrix (y) so that 

The matrix (y) can be considered to be a transformation of the original matrix (x). The question that will now be asked is Is it possible for the matrix (y) to be proportional to (x), that is, (y) = A(x), where A is a scalar multiplier That is, for (y) to have the same direction as the matrix (x). For the case of a collinear transformation we have 

This result is, however, trivial for (x) = (0) represents a trivial solution. But we may ask the question Are there values of A that will produce nontrivial solutions The necessary and sufficient condition that there be nontrivial solutions is that the determinant of the coefficients vanishes, that is. 

The that satisfy this Eq. A.4.8 may be complex, since an algebraic equation with real coefficients may have complex conjugate pairs of roots, or they may be complex since the polynomial may have complex coefficients if the matrix [yl] has nonreal elements. 

8 SOME MATHEMATICAL ASPECTS OF CPMs 3.8.1 Eigenvalues and Eigenvectors 

Due to the unity summation relationship we only need ( c 1) DPE equations to fully define the system. The resulting Jacobian system of equations is a matrix of size ( c— 1) X ( c— 1) where entry ij is the first partial derivative of ft with respect to component xf. 

The Jacobian matrix is a square matrix from which one may compute eigenvalues. To solve the eigenvalue problem it is necessary to compute the scalar values, 2, that satisfy the equation 

Equation 3.33 is a simple quadratic equation, known as the characteristic equation, with two unique solutions (Xi and I2), where li and I2 are known as the eigenvalues. It is the nature of these eigenvalues that allow us to identify nodes, according to Lyapunov s theorem. These nodes can be classified as follows [10]  

The roots of the characteristic equation have one zero value  

We assume now that Ais an N x N square matrix. In this case, there is at least one non-zero N vector v such that 

Any real square matrix has N (complex) eigenvalues (not necessarily different), and at most N linearly independent eigenvectors. It can be proved that all eigenvalues of a symmetric matrix A are real numbers, and A has exactly N different eigenvectors. The normalized eigenvectors of a symmetric matrix form an orthonormal set of vectors 

In other words, the normalized eigenvectors of a symmetric matrix form an orthonormal basis in Euclidean space Ej.  

The set of eigenvalues is called the spectrum and is denoted as A (A). The maximum eigenvalue of matrix A is called the spectral radius of A  

The symmetric matrix A is called positive definite, if all the eigenvalues of A are positive. In this case the following inequality holds for any non-zero vector x  

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Note that the Liouville matrix, iL+R+K may not be Hennitian, but it can still be diagonalized. Its eigenvalues and eigenvectors are not necessarily real, however, and the inverse of U may not be its complex-conjugate transpose. If complex numbers are allowed in it, equation (B2.4.33) is a general result. Since A is a diagonal matrix it can be expanded in tenns of the individual eigenvalues, X. . The inverse matrix can be applied... 

To determine the vibrational motions of the system, the eigenvalues and eigenvectors of a mass-weighted matrix of the second derivatives of potential function has to be calculated. Using the standard normal mode procedure, the secular equation... 

Figure 7-20. The Hiickel matrix (above) and the eigenvalues and eigenvectors for 1,3-butadiene.

The eigenvalues and eigenvectors of the mass-weighted force matrix can be obtained by diagonalizing equation (21.5). Then each eigenvalue corresponds to its normal coordinates, Qj,... 

We next solve the secular equation F — I = 0 to obtain the eigenvalues and eigenvectors o the matrix F. This step is usually performed using matrix diagonalisation, as outlined ii Section 1.10.3. If the Hessian is defined in terms of Cartesian coordinates then six of thes( eigenvalues will be zero as they correspond to translational and rotational motion of th( entire system. The frequency of each normal mode is then calculated from the eigenvalue using the relationship ... 

In the Mathcad calculation of eigenvalues and eigenvectors of the Huckel matrix for ethylene (j g), the eigenvalues are given in the order upper followed by lower. The matr ix E for this order is... 

The principal topics in linear algebra involve systems of linear equations, matrices, vec tor spaces, hnear transformations, eigenvalues and eigenvectors, and least-squares problems. The calculations are routinely done on a computer. 

The determination of some of the eigenvalues and eigenvectors of a large real symmetric matrix has a long history in numerical science. Of particular interest in the normal mode... 

In order to find the normal modes of vibration, I am going to write the above equations in matrix form, and then find the eigenvalues and eigenvectors of a certain matrix. In matrix form, we write... 

A short calculation will show you that S can be written in terms of its eigenvalues and eigenvectors as... 

Next we consider the eigenvalues and eigenvectors of p. For this purpose we observe that... 

This has all ground-state CC information in it via the coupled-clusterT amplitudes. Its eigenvalues and eigenvectors are given for A > 0 by... 

Let us dwell on the properties of eigenvalues and eigenvectors of a linear self-adjoint operator A. A number A such that there exists a vector 0 with = A is called an eigenvalue of the operator A. This vector... 

The eigenvalues and eigenvectors of the unperturbed hamiltonian are assumed to be known ... 

The explicit solution of Eq. (27), which uses a Fourier transform or a bilateral Laplace transform, is described in detail in Ref. 38. Its eigenvalues and eigenvectors are determined by the nonlinear eigenvalue equation... 

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