Eigenvalues and eigenvectors of a symmetric matrix

Example 1.6 Eigenvalues and eigenvectors of a symmetric matrix The eigenanalysis of the matrix (ref. 2)... 

Furthermore, this development is sufficiently general as to be considered a way to obtain the minimal eigenvalue and eigenvector of a symmetric matrix. 

The simplest and most easily implemented method of computing the eigenvalues and eigenvectors of a symmetric matrix is developed and implemented here. 

Once the matrix SCF equations have been set up and transformed to an orthogonal basis, the only numericad problem in their solution is the calculation of the eigenvalues and eigenvectors of a symmetric matrix. This is a problem which occurs in many branches of science, particularly those involving optimisation of some kind and has, consequently, received much attention. 

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... 

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... 

This procedure transforms the ill-conditioned problem of finding the roots of a polynomial into the well-conditioned problem of finding the eigenvalues and eigenvectors of a tridiagonal symmetric matrix. As shown by Wilf (1962), the N weights can then be calculated as Wa = OToV ai where tpai is the first component of the ath eigenvector (pa of the Jacobi matrix. 

The method which has been implemented to generate the eigenvalues and eigenvectors of a real symmetric matrix is not, in fact, the fastest method there are methods which have asymptotic dependence on floating point operations, while the Jacobi method depends asymptotically on m. f77 implementations of these methods (the Givens and Householder methods) are available for most computers and calls to eigen may be simply replaced by corresponding calls to the other routine. 

Excel can find eigenvalues and eigenvectors of real, symmetric matrices, as follows. If the eigenvalues of the nth-order real, symmetric matrix H are arranged in increasing order Ai A2 A , an extension of a theorem due to Rayleigh and Ritz states that A] =... 

C. F. Bender and I. Shavitt, An Iterative Procedure for the Calculation of the Lowest Real Eigenvalue and Eigenvector of a Non-Symmetric Matrix, J. Comput. Phys. 6, 146—149 (1970). 

Greenstadt (1967) proposed evaluating the eigenvalues and eigenvectors of G. As the matrix is symmetric, there are techniques that allow these values to be obtained in a safe and accurate way (Buzzi-Ferraris and Manenti, 2010a). 

This polynomial, which is called the characteristic equation of matrix A, has n roots, which are the eigenvalues of A. These roots may be real distinct, real repeated, or complex, depending on matrix A (see Table 2.4). A nonsingular real symmetric matrix of order n has n real nonzero eigenvalues and n linearly independent eigenvectors. The eigenvectors of areal symmetric matrix are orthogonal to each other. The coefficients a, of the characteristic polynomial are functions of the matrix elements and must be delermined before the polynomial can be used. 

A symmetric matrix A, can usually be factored using the common-dimension expansion of the matrix product (Section 2.1.3). This is known as the singular value decomposition (SVD) of the matrix A. Let A, and u, be a pair of associated eigenvalues and eigenvectors. Then equation (2.3.9) can be rewritten, using equation (2.1.21)... 

If X is the matrix formed from the eigenvectors of a matrix A, then the similarity transformation X lAX will produce a diagonal matrix whose elements are the eigenvalues of A. Furthermore, if A is Hermitian, then X will be unitary and therefore we can see that a Hermitian matrix can always be diagonalized by a unitary transformation, and a symmetric matrix by an orthogonal transformation. 

The derivatives with respect to x sample the off-diagonal behaviour of F > and generate terms related to the current density j and the quantum stress tensor er. The first-order term is proportional to the current density, and this vector field is the x complement of the gradient vector field Vp. The second-order term is proportional to the stress tensor. Considered as a real symmetric matrix, its eigenvalues and eigenvectors will characterize the critical points in the vector field J and its trace determines the kinetic energy densities jK(r) and G(r). The cross-term in the expansion is a dyadic whose trace is the divergence of the current density. 

Inverselteration Given a symmetric matrix and its eigenvalues this operation returns the eigenvectors of the matrix. 

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