Diagonal form

Simulations of the adaptive reconstruction have been performed for a single slice of a porosity in ferritic weld as shown in Fig. 2a [11]. The image matrix has the dimensions 230x120 pixels. The number of beams in each projection is M=131. The total number of projections K was chosen to be 50. For the projections the usual CT setup was used restricted to angels between 0° and 180° with the uniform step size of about 3.7°. The diagonal form of the quadratic criteria F(a,a) and f(a,a) were used for the reconstruction algorithms (5) and (6). 

If we start with an t -dimensional representation of A consisting of the matrices M, M2, M3,. .., it may be that we can find a matrix V such that when it is used with ( equation A1.4.34) it produces an equivalent representation M, M 2, M 3,. .. each of whose matrices is in the same block diagonal form. For example, the nonvanishing elements of each of the matrices could fonn an upper-left-comer ... 

Finally, the unitary transformation W t) was shown to have a near-diagonal form ([252], Eq. XVII.97)... 

This scheme requires the exponential only of matrices that are diagonal or transformed to diagonal form by fast Fourier transforms. Unfortunately, this matrix splitting leads to time step restrictions of the order of the inverse of the largest eigenvalue of T/fi. A simple, Verlet-like scheme that uses no matrix splitting, is the following ... 

Note that every matrix in the four dimensional group representation labeled DN) has the so-called block diagonal form... 

The distance matrix A, which holds the relative distances (by whatever similarity measure) between the individual confonnations, is rarely informative by itself. For example, when sampling along a molecular dynamics trajectory, the A matrix can have a block diagonal form, indicating that the trajectory has moved from one conformational basin to another. Nonetheless, even in this case, the matrix in itself does not give reliable information about the size and shape of the respective basins. In general, the distance matrix requires further processing. 

The uniform phase becomes unstable if there exist fluctuations for which 0,2, given by Eq. (56), vanishes. Since a2 > 2 for all temperatures (see (48) and (49)), the instability of the uniform phase is related to fluctuations of the fields (f) and j li. O2 can be written in a diagonal form... 

Just like the electric quadrupole moment, the electric field gradient matrix can be written in diagonal form for a suitable choice of coordinate axes. 

Some coordinate transformations are non-linear, like transforming Cartesian to polar coordinates, where the polar coordinates are given in terms of square root and trigonometric functions of the Cartesian coordinates. This for example allows the Schrodinger equation for the hydrogen atom to be solved. Other transformations are linear, i.e. the new coordinate axes are linear combinations of the old coordinates. Such transfonnations can be used for reducing a matrix representation of an operator to a diagonal form. In the new coordinate system, the many-dimensional operator can be written as a sum of one-dimensional operators. 

Finally, we consider the contribution from the last term, j d>g A j. To find the eigenvalues in this case, we note that we can relabel the submatrix d>g so that it appears in block diagonal form, with each block corresponding to a particular cycle, Ci, i = 1,2,..., / ... 

In order to try to approach the HF scheme as much as possible, we will now introduce the basic orthonormal set fc which has maximum occupation numbers. Let U be the unitary matrix which brings the hermitean matrix (ylk) to diagonal form ... 

Our theorem permits the following inference. The statistical matrix of every pure case in quantum mechanics is equivalent to an elementary matrix and can be transformed into it by a similarity transformation. Because p is hermitian, the transforming matrix is unitary. A mixture can, therefore, always be written in the diagonal form Eq. (7-92). 

We have now achieved one objective, that of expressing the hamiltonian in essentially diagonal form. In order to deal with operators that satisfy S-function commutation rules rather than SJy commutation rules, where... 

Haag, R., 388,517 Hadamard theorem, 58 Hahn, W., 343 Hamiltonian in diagonal form, 567... 

Considering the diagonalized form (5) of the image formation equation, a very tempting solution is to perform straightforward direct inversion in the Fourier space and then Fourier transform back to get the deconvolved image. 

In addition one can always find a transformation leading to a symmetry adapted basis [4] e, so that T is brought to the block diagonal form T via the associated similarity transformation. The T matrix can be written as a direct sum... 

S is the state space system Default is the diagonal form This is the observable companion... 

The third alternative to generate the diagonalized form is to use the state space to state space transformation function. The transformation is based on the modal matrix that we obtained earlier. 

As indicated above there may be many equivalent matrix representations for a given operation in a point group. Although the form depends on the choice of basis coordinates, the character is Independent of such a choice. However, for each application there exists a particular set of basis coordinates in terms of which the representation matrix is reduced to block-diagonal form. This result is shown symbolically in Fig. 4. ft can be expressed mathematically by the relation... 

Rfl. 4 Block-diagonal form of a representation matrix The tedueed representation. 

It is in general possible to find a Cartesian coordinate system attached to the molecule such that the tensor / takes a diagonal form. In terms of these so-called principal axes Equation (10) is simplified in that all cross terms are eliminated. Namely,... 

Coordinates such as these, which have the symmetry properties of the point group are known as symmetry coordinates. As they transform in the same manner as the IRs when used as basis coordinates, they factor the secular determinant into block-diagonal form. Thus, while normal coordinates most be found to diagonalize the secular determinant, the factorization resulting horn the use of symmetry coordinates often provides considerable simplification of the vibrational problem. Furthermore, symmetry coordinates can be chosen a priori by a simple analysis of the molecular structure. 

In general, the g- and nuclear hyperline coupling matrices, g and A can be written in diagonal form with three principal values, i.e., gx, gy, g. and A,x, Aiy, Aiz. In textbooks on ESR6a 30,33 35 it is usually assumed that the same set of principal axes diagonalizes all the relevant matrices. While this is sometimes true, there are many instances where the principal axes are non-coincident.36... 

This matrix describes the transformation from x y z to xyz as a rotation about the z axis over angle a, followed by a rotation about the new y" axis over angle /), followed by a final rotation over the new z " axis over angle y (Watanabe 1966 148). Formally, the low-symmetry situation is even a bit more complicated because the nondiagonal g-matrix in Equation 8.11 is not necessarily skew symmetric (gt] -g. Only the square g x g is symmetric and can be transformed into diagonal form by rotation. In mathematical terms, g x g is a second-rank tensor, and g is not. 

The elements of these vectors can be evaluated using an off-diagonal form of the Hellmann-Feynmann theorem, which in Cartesian coordinates, xa, is... 

Third, writing the discretized equations in matrix form results in sparse matrices, often of a tri-diagonal form, which traditionally are solved by successive under- or over-relaxation methods using the tri-diagonal matrix algorithm... 

Suppose that the density matrix has been brought to diagonal form, whereby it is immediately apparent that all its eigenvalues should be either zero or one. More precisely, since trg=l, only one of the eigenvalues can be equal to one and all the others are zero. 

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