Symmetric matrices multiple

The application of the Chebyshev recursion to complex-symmetric problems is more restricted because Chebyshev polynomials may diverge outside the real axis. Nevertheless, eigenvalues of a complex-symmetric matrix that are close to the real energy axis can be obtained using the FD method based on the damped Chebyshev recursion.155,215 For broad and even overlapping resonances, it has been shown that the use of multiple cross-correlation functions may be beneficial.216... 

The expression is easily coded, since T<0>, Eq. 41, and the r 1 are known. It simplifies for substitution on a principal plane or axis and for symmetrically equivalent multiple substitution, because several of the elements of the matrix T will then vanish. It is clear from Eq. 45 that the derivatives are nonvanishing only for those atoms a that have actually been substitued in the particular isotopomer s. Therefore, the Jacobian matrix X generated from these derivatives is, in general, a sparse matrix. 

In fact the relative coefficients within a set of symmetrically equivalent atoms, such as Cl, C3, C5, and Cg in para-benzosemiquinone, can be determined by group theory alone. The appropriate set of symmetrized orbitals, also listed in Table 1, can be obtained by use of character tables and procedures described in Refs. 3 to 6. In matrix notation, the symmetrized combinations are <1> = Up, where <1) and p are column vectors and U is the transformation matrix giving the relations shown in part (c) of Table 1. The transformation U and its transpose U can be used to simplify the solution of the secular matrix X since the matrix multiplication UXU gives the block diagonal form shown in part (6) of Table 1. 

There are two template EXCEL files for the orthogonality tests using the Extended Htickel approximation in the BonusPack. The first is a similar template to the one used for the Htickel calculations. It is applied to check the 2-dimensional basis functions of the groups, which are taken in pairs, with the totally symmetric basis function included in the calculation to make up the 5 X 5 structure of the matrix multiplications. In the second template, for the 3-dimensional basis functions, a 9 x 9 structure is employed so that a full set of, for example, Itiu, 2tiu and 3tiu functions can be checked at the same time. 

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... 

A A has another important property, that can be shown by inspection using the rules of matrix multiplication outlined above - it is symmetrical about its principal diagonal. The terms in the top right half of the matrix are identical to the terms in the bottom left. 

Here we have expressed the eigenvectors as vectors of unit length any multiple of each eigenvector would also be a solution. A is a real, symmetric matrix. The eigenvalues of such matrices are always real and orthogonal (i.e. the scalar products of all parrs of eigenvectors are zero). This can be easily seen in our example. 

Note that we have added the superscript T to indicate a column vector, written as transpose. Equation (1.75) is a special case of matrix multiplication. A tensor is symmetric, when Txy = Tyx, Txz = T x, and Tyz — T y. The product of multiplication of a tensor with a vector reduces to a multiplication of the vector with a scalar, if the diagonal elements are equal, i.e., Txx = Tyy = and all other elements of the tensor are zero. Moreover, if Txx = Tyy = = I and all other elements are... 

The system (11.32) can be solved by either factorizing the symmetric matrix EV or using iterative methods such as the CG method or PCG method. In the latter case, it is not necessary to actually build the matrix EV since it appears in multiplications with vectors only (see Chapter 4). 

The direct covariance matrix multiplication of a homonuclear NMR spectrum yields a homonuclear covariance map whose square root is equivalent to the NMR spectrum obtained from Fourier transformation. As can be readily seen from Fig. 5.1 A, the higher resolution of the direct dimension is mapped onto the indirect dimension. Hence, a symmetric spectrum... 

The matrix R(j)(a,P) is a similar 3x3 block connecting jy, jz and jx. Carrying out the matrix multiplication yields a symmetric matrix which is diagonal except for the 6 X 6 block already mentioned. This has the structure... 

Therefore, the multiplication of the skew-symmetric matrix /3 x)) with the warping displacements Uo x,y,z) in Eq. (7.8) results in products of two angles or derivatives thereof respectively. In consequence of the above discussion on the orthogonality condition, these terms have to be neglected and the expression of the total displacements simplifies significantly ... 

This result can be verified by using matrix multiplication. Try it yourself. Follow eq. 1.2.14, switch rows with columns in T, and make fi a column vector on the other side. Of course, in the end this manipulation does not matter for T because it is symmetric, but we will find the operation useful later. 

The general case of a laminate with multiple anisotropic layers symmetrically disposed about the middle surface does not have any stiffness simplifications other than the elimination of the Bjj by virtue of symmetry. The Aig, A2g, Dig, and D2g stiffnesses all exist and do not necessarily go to zero as the number of layers is increased. That is, the Aig stiffness, for example, is derived from the Q matrix in Equation (2.84) for an anisotropic lamina which, of course, has more independent... 

Like the time propagation, the major computational task in Chebyshev propagation is repetitive matrix-vector multiplication, a task that is amenable to sparse matrix techniques with favorable scaling laws. The memory request is minimal because the Hamiltonian matrix need not be stored and its action on the recurring vector can be generated on the fly. Finally, the Chebyshev propagation can be performed in real space as long as a real initial wave packet and real-symmetric Hamiltonian are used. 

Because of round-off errors, symmetry contamination is often present even when the initial vector is properly symmetrized. To circumvent this problem, an effective scheme to reinforce the symmetry at every Lanczos recursion step has been proposed independently by Chen and Guo100 and by Wang and Carrington.195 Specifically, the Lanczos recursion is executed with symmetry-adapted vectors, but the matrix-vector multiplication is performed at every Lanczos step with the unsymmetrized vector. In other words, the symmetrized vectors are combined just before the operation Hq, and the resultant vector is symmetrized using the projection operators ... 

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. 

Note that and A r need not be the same matrix (although they are for the important special case of real symmetric A that we are most concerned with). Note also that AT1 need not exist, even if A 0. A matrix for which A"1 exists is called nonsingular (see below) and leads to many arithmetic extensions that are not permitted to singular matrices. The many varieties of singularity (not just A = 0) and the (potentially) noncommutative aspects of multiplication distinguish matrix algebra from its scalar counterpart in interesting ways. 

A unique synthetic molecule that can be used as a carrier is the so-called multiple antigenic peptide (MAP) (Tam, 1988 Bosnetteta/., 1988). The MAP core structure is composed of a scaffolding of sequential levels of poly-L-lysine. The matrix is constructed from a divalent lysine compound to which two additional levels of lysine are attached. The final MAP compound consists of a symmetrical, octavalent primary... 

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