Real-symmetric matrices

The Jacobi method is probably the simplest diagonalization method that is well adapted to computers. It is limited to real symmetric matrices, but that is the only kind we will get by the formula for generating simple Huckel molecular orbital method (HMO) matrices just described. A rotation matrix is defined, for example. 

Without losing generality, this review will concentrate on real-symmetric matrices, whereas their Hermitian counterparts can be handled in a similar way. In some special cases, solutions of complex-symmetric matrices are required. This situation will be discussed separately. 

Smith and Yu have used MINRES to construct filtered vectors in Lanczos-based FD calculations for Hermitian/real-symmetric matrices.76,77,181,182 To this end, these authors demonstrated that the filtered vectors can be written as a linear combination of the Lanczos vectors ... 

The aforementioned applications of recursive methods in reaction dynamics do not involve diagonalization explicitly. In some quantum mechanical formulations of reactive scattering problems, however, diagonalization of sub-Hamiltonian matrices is needed. Recursive diagonalizers for Hermitian and real-symmetric matrices described earlier in this chapter have been used by several authors.73,81... 

Let us denote by S the space of block-diagonal real symmetric matrices (i.e., multiple symmetric matrices arranged diagonally in a unique large matrix) with prescribed dimensions, and by U " the m-dimensional real space. Given the constants C,Ai,A2,. .., A e S, and b e IR , an SDP problem is usually defined either as the primal SDP problem. 

The self-adjoint ( Hermitian ) matrices often play a particularly important role in representing physical phenomena (e.g., as the observables in quantum theory), and they also include the real symmetric matrices to be encountered in the metric geometry of equilibrium thermodynamics. 

For real symmetric matrices of dimension n, a triangular pattern (referred to as T) is used with the location of i,j computed as L 3 i+j(j-l)/2 for ij]. The Cl hamiltonian matrix is a large real symmetric matrix with mostly zero entries (provided orthonormal configurations constructed from orthonormal orbitals are used). If more than half the entries are zero it is more efficient to omit zero entries and include the index as a label (if the word length is long and the matrix is small enough, this label may be packed into the insignificant bits of the matrix element). 

We will confine ourselves to the case when A and S are real symmetric matrices and S is positive definite, since this simplifies the problem considerably, while excluding more troublesome cases which may be of more interest to specialists. 

E. Davidson, The iterative calculation of a few of the lowest eigenvalues and corresponding eigenvectors of large real-symmetric matrices, J. Comp. Phys. 17 (1975) 87. 

Lowest Eigenvalues and Corresponding Eigenvectors of Large Real-Symmetric Matrices. 

Eds., Lawrence Berkeley Laboratory, CA, 1978, Vol. LBL-8158, pp. 49-53. The Simultaneous Expansion Method for the Iterative Solution of Several of the Lowest Eigenvalues and Corresponding Eigenvectors of Large Real-Symmetric Matrices. 

This is a mathematical property of real symmetric matrices. 

Standard numerical methods exist for diagonalizing real symmetric matrices such as the Hamiltonian H.1 However, such methods usually require the stor-... 

Figure 5 The Davidson-Liu Iterative Method for the Lowest Few Eigenvectors and Eigenvalues of Real, Symmetric Matrices (Ref. 174)...

As a prelude to the proof as such it will be found convenient to cast the Huckel equations in matrix form as we have continually emphasised, much of Huckel theory is essentially the linear algebra associated with real-symmetric matrices, albeit in a somewhat disguised form. Let us begin by considering again (in the original matrix-notation rather than the a, / notation we have lately adopted) the rth secular equation, satisfied by e, and c/r, r = 1,2,..., n. This is... 

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] =... 

Clint, M. and Jennings, A. (1970) The evaluation of eigenvectors of real symmetric matrices by simultaneous iteration. The Computer Journal, 13, 76-80. 

Excel can find eigenvalues and eigenvectors of real, symmetric matrices as follows. If the... 

The large amplitude of the continuum resonance states is a direct result of the non Hermitian properties of Hamiltonian (i.e. the resonance eigenfunctions which are associated with complex eigenvalues are not in the Hermitian domain of the molecular Hamiltonian). Let us explain this point in some more detail. As was mentioned above Moiseyev and Priedland [7] have proved that if two N x N real symmetric matrices H and H2 do not commute, there exists at least one value of parameter A = such that matrix H + XH2 possesses incomplete spectrum. That is at A A there are at least two specific eigenstates i and j for which ]imx j ei — ej) =0 and also lim ( i j) — 0- Since and ipj are orthogonal (within the general inner product definition i.e., i/ il i/ j) = = 0 not in the... 

The correct normalization of c can be obtained without calculating Cb but by using equation (4.125) and the fact that V and G° (for E in the band gaps) are real symmetric matrices ... 

Two real symmetric matrices and have the same index of inertia if and only if there exists a real nonsingular matrix C with M =C M2C. 

An important category of transformation matrices is that of unitary matrices. These are employed for diagonalizing real, symmetric matrices, and we will use U to designate a unitary matrix in the following discussion. A unitary matrix is one whose inverse is the same as or equal to its transpose. Or, when the elements of U are complex, unitarity means the inverse is equal to the adjoint ... 

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