Diagonal elements of a matrix

Finally, using the eigenvalues there are some further subdivision possible If the product of eigenvalues of a unitary matrix or operator is equal to +1, it is called a special unitary (SU) matrix or operator. Similar for real orthogonal matrices, where the only possible choice is +1 or -1 the former case is called special orthogonal (SO) matrices. For a matrix, this product equals the determinant of the matrix. For both matrices and operators, the sum of eigenvalues is called the trace of the matrix or operator. This equals the sum of the diagonal elements of a matrix representation. 

The sum of the diagonal elements of a matrix is called the character (%) of the matrix. Hereafter, we use the term character rather than the representation since there is a one-to-one correspondence between them and since mathematical manipulation with x is simpler than with the representation. The characters of the reducible representations for the E, and o operations are 3, 0 and 1, respectively. The characters for C3 (counterclockwise rotation by 120°) is the same as that of C, and those for cr2 and cr3 are the same as that of crj. By grouping symmetry operations of the same character ( class ), we obtain... 

The expansions in even powers of normal frequencies are of special interest, because they provide means for obtaining explicit relations between the equations of motion and the thermodynamic quantities, through the use of the method of moments The sum of over all the normal vibrations can be expressed as the trace, or the sum of all the diagonal elements, of a matrix H" obtained by multiplying the Hamiltonian matrix H of the system by itself (n — 1) times. Such expansions thus enable us to estimate the thermodynamic functions and their isotope effects from known force fields and structures without solving the secular equations, or alternatively, to estimate the force fields from experimental data on the thermodynamic quantities and their isotope effects. The expansions explicitly correlate the motions of particles with the thermodynamic quantities. They can also be used to evaluate analytically a characteristic temperature associated with the system, such as the cross-over temperature of an isotope exchange equilibrium. Such possible applications, however, are useful only if the expansion yields a sufficiently close approximation. The precision of results obtainable with orthogonal polynomial expansions will be explored later. 

For the calculation of stationary mutant distributions we restrict attention to a uniform error rate per digit (1 — ) and assume equal degradation rate coefficients Dy = D2= =D = D. Since the addition of a constant to all diagonal elements of a matrix just shifts the spectrum of eigenvalues and has no influence on the eigenvectors, we need only consider the case D = 0 without loss of generality. Then the elements of the matrix W are determined by the replication rate coefficients (as in Section III.2) and are of the form... 

Finally, there is one further source of information on the harmonic force field that has been used occasionally, namely mean square amplitudes of vibration in the various intemuclear distances, as observed by gas-phase, electron-diffraction techniques. These can be measured experimentally from the widths of the peaks observed in the radial distribution function obtained from the Fourier transform of the observed diffraction pattern. They are related to the harmonic force field as follows.23 If < n > denotes the mean square displacement in the distance between atoms m and /t, then the mean amplitudes <2 > are given as the diagonal elements of a matrix 2, where... 

The sum of the diagonal elements of a matrix is called the character of the matrix and is denoted by %. It is to be noted in Table 1.5 that the character of each of the elements belonging to the same class is the same. Thus, using the character. Table 1.5 can be simplified to Table 1.6. Such a table is called the character table of the point group 3. ... 

The partition function Z is the trace (sum of diagonal elements) of a matrix for which the diagonal elements are... 

It should be mentioned that the Jacobi method for diagonalizing N x N matrices is a generalization of the above procedure. The basic idea of this method is to eliminate iteratively the off-diagonal elements of a matrix by repeated applications of orthogonal transformations, such as the ones we have considered here. 

The copolymer composition can be obtained using the trace of the matrix defined in Equation (10.36). Trace is a sum of all the diagonal elements of a matrix. Thus ... 

The character we assign for a particular basis vector has been linked to the diagonal element in the operation s matrix. This can be generalized to say that the sum of the diagonal elements of a matrix representing an operation on a particular basis is the sum of the characters for the basis under that operation. In matrix algebra, the sum of the diagonal elements of a matrix A is known as the trace of the matrix, Tr(A), i.e. 

The diagonal elements of this matrix approximate the variances of the corresponding parameters. The square roots of these variances are estimates of the standard errors in the parameters and, in effect, are a measure of the uncertainties of those parameters. 

It is important to realize that while the uppennost diagonal elements of these matrices are numbers, the other diagonal element is a matrix of dimension N. Specifically, these are the matrix representations of Hq and Fin the basis q which consists of all the original set, apart from i.e. 

Note that the diagonal elements of the matrix, ap and hp, correspond to the populations in the energy levels, a and b, and contain no time dependence, while the off-diagonal elements, called the coherences, contain all the time dependence. 

The diagonal elements of the matrix [Eqs. (31) and (32)], actually being an effective operator that acts onto the basis functions Ro,i, are diagonal in the quantum number I as well. The factors exp( 2iAct)) [Eqs. (27)] determine the selection rule for the off-diagonal elements of this matrix in the vibrational basis—they couple the basis functions with different I values with one another (i.e., with I — l A). 

As can be seen in Figure 2-13, the diagonal elements of the matrix are always zero and it is symmetric around the diagonal elements (undirected, unlabeled graph). Thus, it is a redundant matrix and can be reduced to half of its entries (Figure 2-14b. For clarity, all zero entries are omitted in Figures 2-14b-d. 

The Wiener index was originally defined only for acyclic graphs and was initially called the path number [6]. "The path number, W, is defined as the sum of the distances between any two carbon atoms in the molecule in terms of carbon-carbon bonds". Hosoya extended the Wiener index and defined it as the half-sum of the off diagonal elements of a distance matrix D in the hydrogen-depleted molecular graph of Eq, (15), where dij is an element of the distance matrix D and gives the shortest path between atoms i and j. 

The force constants k 2 and k2 are the off-diagonal elements of the matrix. If they are zero, the oscillators are uncoupled, but even if they are not zero, the K matrix takes the simple fomi of a symmetrical matrix because ki2 = k2. The matrix is symmetrical even though may not be equal to k22-... 

A square matrix A is hermitian or self-adjoint if it is equal to its adjoint, i.e., if A = A or ay = a -. Thus, the diagonal elements of a hermitian matrix are real. 

The diagonal elements of the matrix A are af1 and the off-diagonal elements of Aij are Ty. Equation (9-21) determines how the dipoles are coupled to the static electric field. There are three major methods to determine the dipoles matrix inversion, iterative methods and predictive methods. 

In the ideal case, only the diagonal elements of the matrix at Eq. (7.17) are different from zero and all the off-diagonal elements Si = 0 (for I = /), sel(A,B,...,N) becomes 1. For real analytical systems, Eq. (7.24) expresses a certain degree of selectivity depending on the sensitivities of the analytes I, their contents Xi, and all the cross sensitivities % and contents xj of potentially disturbing species. 

The nonzero elements are those with both indices the same and are called the diagonal elements. The sum of the diagonal elements of a square matrix is called the trace... 

By changing the basis in the n-dimensional space L, the matrices D(R) will be replaced by their transforms by some matrix C. The matrices D (R) = CD R)C l also provide a representation of the group G, which is equivalent to the representation D(R). It should be clear that equivalent representations have the same structure, even though the matrices look different. What is needed to avoid any possible ambiguity are appropriate aspects of D(R) which remain invariant under a change of coordinate axes. One such invariant is easily defined in terms of the diagonal elements of the matrix, as... 

This matrix is the appropriate representation of an observable such as X. A Hermitian matrix is its own hermitian conjugate. The diagonal elements of a Hermitian matrix are real and each element is symmetry related to its complex conjugate across the main diagonal. 

Two alternative procedures have been suggested in the literature to solve the problem and they will be discussed next. Alternative 1 was proposed by Almasy and Mah (1984). They attempt to minimize the sum of the squares of the off-diagonal elements of a measurement error covariance matrix subject to the relation deduced from the... 

Remembering that the eigenvectors of a matrix are those vectors that, when multiplied by the matrix, become multiples of the vectors. As A=S2 is a diagonal matrix, each column of the product UA is a multiple of U and thus the columns of U are eigenvectors of YY. The diagonal elements of A=S2 are the eigenvalues for the corresponding columns of U. 

People rarely discuss the A -representabihty of the diagonal of a density matrix. The diagonal elements of a density matrix are A -representable if and only if there exists some A -electron ensemble that reproduces those elements. The preceding statement asserts that if the diagonal elements of the g-matrix, satisfy the (g, K) conditions, then there always exists an A -electron ensemble with these diagonal elements. Thus, if satisfies the (g, K) conditions, then there must be an ensemble-A -representable g-matrix with those... 

Because the diagonal elements of a positive semidefinite matrix are never negative, this implies the first (2, 2) condition, Tr[%n,rAr] > 0. 

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