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Given an m × m m \times m symmetric nonnegative matrix A A and a positive vector R = ( r 1 , ⋯ , r m ) R = ({r_1}, \cdots ,{r_m}) , necessary and sufficient conditions are obtained in order that there exist a diagonal matrix D D with positive main diagonal such that DAD has row sum vector R R .

New lower bounds on the minimal condition numbers of a matrix with respect to both one-sided and two-sided scaling by diagonal matrices are obtained. These bounds improve certain results obtained by F. L. Bauer.

Let be a linear model with independently - not necessary normally – distribused error components ϵ j and where V(i=1, … p) are known diagonal matrices and the Θ i are unknown scalars (veriance components). Starting from prior distributions with respect to β and Θ BAYES solutions for four elasses of quedratie unblased estimaters for linear functions of the vaciance components are given. They result from solutions of linear equation systems and is general they depend - beside on the experimental design (X,U,V 1,…V p ) -– only on skewness and kurtosis of the ϵ,j 's and on the first two moments of the prior distribution. For special models there oxist solutions depending neither on the prior distribution nor on the distribution of the ϵj 's.

A new method of finding the eigenvalues and eigenvectors of an arbitrary complex matrix is presented. The new method is a similarity transformation method which transforms an arbitrary N × N matrix to a Jordan canonical form in N-1 or less transformations. Each transformation matrix is a matrix function-the matrix sign function with a ± added to the main diagonal elements. Using this matrix function as a similarity transformation gives a block diagonal form which is a reduced form of the transformed matrix. As the Jordan canonical form is found, the eigenvectors are simultaneously found since the product of transformation matrices must be a matrix of eigenvectors. The theoretical development of the new method and a computational scheme with examples are given. In the examples, the computational scheme is applied successfully to matrices which have characteristics that cause problems for most numerical techniques.

Thermodynamic coupling due to thermal diffusion, diffusion-thermo, and ob aliam diffusion effects in boundary layers is considered. Decoupling of the set of conservation equations is achieved by diagonalisation of the phenomenological coefficients matrix. A new Peclet number for coupled transfer is defined in terms of the eigenvalues of the diagonal matrix. The specific and overall degrees of thermodynamic coupling are shown to depend both on the values of the coupling coefficients and on the magnitude and direction of the gradients across the boundary layers. The application of the theory is illustrated for coupled transfer in two-phase particulate systems.

We study the properties of the collision operator, as defined by the Brussels school, for systems whose hamiltonian is a cyclic matrix (finite case) or a Laurent matrix (infinite case). Explicit calculations in particular examples permit us to illustrate the difference in the behaviour of this operator when long-range interactions are present.

Variational principles are considered for the approximate evaluation of the diagonal matrix elements of an arbitrary known linear Hermitian operator. A method is derived that is immediately applicable to the variational determination of both the off-diagonal and diagonal matrix elements of normal and modified Green's functions.

An algorithm to choose the initial multipliers optimally for quasilinearization solution of optimal control problems with bounded controls and constant multipliers is proposed. It uses diagonal matrices of weighting coefficients in the performance index of an auxiliary minimization problem. This auxiliary performance index comprises the cumulative error in the system constraints and the optimum conditions in the original extremization problem. The auxiliary performance index is quadratically dependent on the multipliers for given state and control functions. The resulting variational problem leads to linear Euler equations. The computational characteristics of the proposed method are demonstrated with two numerical examples.

It is proved that each matrix over a principal ideal ring is equivalent to some diagonal matrix. Partial results are obtained on the uniqueness of the diagonal form obtained. These results are obtained by specializing some general properties about simultaneous decompositions of a projective module and a homomorphic image of finite (composition) length over any ring. These general results are also specialized to obtain results about matrices and projective modules over hereditary prime rings.

An expression for a time-reversal-invariant projection operator which can be used to generate the columns of the unitary matrix which block diagonalizes the dynamical matrix of the lattice-dynamics problem in the harmonic approximation is given. The use of this operator and some of its properties is summarized. One interesting result is that the eigenvectors of the dynamical matrix can be expanded in terms of the columns of the unitary matrix which block diagonalizes the dynamical matrix and that the coefficients in the expansion are real numbers.

Diagonal damping matrices were computed for three systems which have non-proportional damping matrices. These diagonal damping matrices were computed on three bases, as follows: 1. After normalizing the equations of motion by the modal matrix, the diagonal terms are retained ignoring the non-diagonal terms. 2. Diagonal damping matrix is established by the optimization algorithm which minimizes the mean square error of the frequency response. 3. Diagonal damping is determined from the normalized differential equation by matching the peaks of the coupled and uncoupled system. The frequency responses for the three cases of one of the three systems are presented together with a comparison of the energy dissipation.

The purpose of this paper is to provide a comprehensive study of the generalized impact formalism as applied to overlapping neutral lines, especially to the 4471.48 A HeI line emitted in a plasma. First, we will discuss the specific algebraic modifications needed to adapt the impact theory to partially degenerate lines. Secondly, we will review briefly the time-averaged dipole width and shift functions. The corresponding impact parameter averages and other properties are given and discussed at length. Thirdly, we will show that the various proposals made for the long range cutoff of the electron impact parameter do not lead to significant discrepancies in the final line shape. The short range cutoff is shown to be dependent on the static splitting and it is the smallest for the nondiagonal elements of the collision matrix. Moreover, the nonmarkovian corrections to the frequency-independent impact theory appear to be of only marginal importance for the 4471.48 A line in the line core, in contradistinction to the wings where more significant discrepancies are exhibited. These results strongly suggest that the non-diagonal part of the collision operator is dependent on the close perturbers collisions (weak collisions with the smaller impact parameter), so that the extension of the impact formalism to off diagonal matrix elements appears as an acceptable approximation, as long as the impact theory retains its validity. The impact formulation is seen to be self-consistent with respect to slow electron collisions which are automatically excluded when the static splitting is introduced in the collision operator. Finally, our profile calculations are compared with the previous ones and additional results with 2 × 1016 ⩽ Ne ⩽ 6 × 1016e-cm−3 and Te = 2 × 104 K are presented. The dip between the two peaks is seen to be sensitively dependent on the perturber temperature. This effect combined with ion dynamic shielding and a self-consistent choice of the short impact parameter cutoff could allow for an important enhancement of the line intensity between the two peaks, and also a significent reduction of the theory-experiment discrepancy.

A system is considered for which the diagonal matrix elements of the hamiltonian with respect to a basis of localized orbitals are fixed, while the nearest-neighbour off-diagonal elements are a random variable with a given continuous probability distribution. The spatial average of the system's Green's function is calculated by means of a two-site extension of the coherent potential approximation in which the self-energy has both diagonal and off-diagonal elements. The density of states and spectral density are calculated for various amounts of disorder, and the mean lifetimes of the quasi-particle Bloch states are derived from the latter.

In this paper we consider the trace identities satisfied in a full matrix algebra of order n. For the case of a field of characteristic zero we prove that all trace identities are consequences of one obtained from the Hamilton-Cayley theorem.

In a previous study of bound-state matrix elements of a Hermitian operator $W$, it was possible to obtain at most an upper (or lower) stationary bound. The possibility arose only for the diagonal matrix element case, and only for $W$ nonpositive (or nonnegative). In the present treatment, both upper and lower stationary bounds are obtained, for diagonal and off-diagonal matrix elements, and, though some restrictions on $W$ remain, the requirement that $W$ be of well-defined sign can be dropped. The derivation also improves upon that given previously in that the possibility of any difficulty with near singularities in the equation defining the trial auxiliary (or Lagrange) function is unambiguously avoided. As an example, the method is applied to the problem of the zero-energy scattering of positrons by atoms or ions, and an expression is derived which provides a rigorous stationary upper bound on the scattering length; the target ground-state wave function need not be known exactly. Crude but rigorous numerical results are obtained quite simply in the Born approximation.

Matrix elements of the residual interaction in 208 Pb are derived from the wavefunctions and the energies of states and configurations in 208 Pb. From the experimentally determined wavefunctions of 20 low-lying states in 208 Pb with spins I π = 2 − , 3 − , 4 − , 5 − , 6 − , 7 − one obtains matrix elements of the effective residual interaction among particlehole configurations defined with respect to the physical ground state of 208 Pb. The average value of these matrix elements is about 100 keV. No significant difference in the average value of the off-diagonal matrix elements is found for the different spin values. Also many nondiagonal matrix elements have values approaching the diagonal matrix elements.

This paper describes a method of solving the Liapounov equation (1)HM+M * H=2D, M in upper Hessenberg form,D diagonal. Initialising the first row of the matrixA arbitrarily, one can find (by solving equations with one unknown) the unknown elements ofA such that (2)AM+M * A * =2F, whereA differs from a Hermitian matrix only in that its diagonal elements need not be real.F is a diagonal matrix which is uniquely determined by the first row ofA. By solving Eq. (2) for several initial values one may generate several matricesA andF (in the most unfavourable case 2n?1A's andF's are needed) and superpose them to getn linearly independent Hermitian matricesH j andD j respectively for whichH j M+M * H j =2D j is valid. Then one can solve the real system $$\sum\limits_{j = 1}^n {p_j D_j } = D$$ to obtain the solution $$H: = \sum\limits_{j = 1}^n {p_j ,} H_j$$ of Eq. (1).

The number of multiplications required for matrix multiplication, for the triangular decomposition of a matrix with partial pivoting, and for the Cholesky decomposition of a positive definite symmetric matrix, can be roughly halved if Winograd's identity is used to compute the inner products involved. Floating-point error bounds for these algorithms are shown to be comparable to those for the normal methods provided that care is taken with scaling.

The method of analytical calculation of off-diagonal matrix elements for the hydrogen atom which involve the continuum wave functions is discussed in some detail. The method is then applied to the calculation of the continuum contribution of some infinite sums which represent known observables. It is demonstrated that the exact integral over all energies of the continuum is easy to obtain by analytical methods, involving approximately as much labor as the calculation of one matrix element involving a discrete state. Moreover, the continuum contribution to the sum is sometimes quite large, accounting, in extreme cases, for all but a few percent of the total.

The correspondence between Self-Consistent Huckel MO methods and Differential Ionization Energies methods is discussed in terms of the approximations used for the diagonal matrix elements. The two methods are shown to be equivalent if electronic correlation is neglected. Ground-state properties of the hydrogen halides are calculated by these simple methods and shown to be in good overall agreement with experimental data.

Necessary and sufficient conditions for a matrix to be realizable as the A-matrix of an RLC network are developed. The RLC network is assumed to have no cut-set of inductors, no circuit of capacitors, and is assumed to have a connected resistive part. It is shown that if there exists a realization then the given matrix A can be factored into two matrices: one, a diagonal matrix of positive entries, and the other a symmetric-skew-symmetric (hybrid) matrix. The former determines directly the values of capacitances and inductances in the network. A technique is given by which the terminal matrix of the resistive part and the fundamental circuit matrix of the reactive part can be obtained from the other factor. It is shown that the given matrix A has a realization with a half-degenerate RLC network which has a connected resistive part if and only if the factorization exists and both the terminal matrix and the circuit matrix are realizable with the same terminal tree.

In this paper, we consider the homogeneous difference equation \[ \sum _{j = 0}^k {\alpha _j y_{n - j} } = 0,\quad n = k,k + 1,k + 2, \cdots ,\] with initial values \[ y_j = q_j,\quad j = 0(1)k - 1 .\] The $y_j$ are d-component column vectors, the $\alpha _j $ are $d \times d$ matrices independent of n. We derive algebraic criteria for numerical stability of the difference equation, which is understood in the sense that the solution $\{ y_j \} $ and its difference quotients up to order $s \in \{ {0,1,2,3, \ldots } \}$ depend continuously on the initial values $\{ q_j \} $. This generalizes the well-known case where $s = 0$ and the $\alpha _j $ are diagonal matrices.

The five-dimensional quasi-spin formalism is used to factor out the n, T dependent parts of shell-model matrix elements in the seniority scheme and derive reduction formulae which make it possible to express matrix elements for states of definite isospin T in the configuration jn in terms of the corresponding matrix elements for the configuration jv. The n, T dependent factors for one- and two-nucleon c.f.p. and for the matrix elements of one-body operators and the two-body interaction are expressed in terms of generalized R(5) Wigner coefficients. The needed R(5) Wigner coefficients are calculated in the form of general algebraic expressions for the seniorities v and reduced isospins t corresponding to the simpler R(5) irreducible representations. In this first contribution, the R(5) representations (ω 1 t) = (j + 1 2 − 1 2 v, t) are restricted to (ω10), ( ω 1 1 2 ), (tt), and the states of (ω11) with n−v = 4k−2T, (k is an integer). Explicit expressions are given for the diagonal matrix elements of the general, charge-independent, two-body interaction and the isovector and isotensor parts of the Coulomb interaction for seniorities v = 0 and 1, and the v = 2 states with n = 4k +2−2T.

The model (or pseudo) Schrodinger equation describing free-electron-like metals is solved by a modified perturbation method in which the diagonal matrix element of the potential is accounted for exactly by defining two κ-dependent effective masses which incorporate it from the outset. The new perturbation approach is used to derive mass-renormalized expressions for the energy of electronic states, the screened form factor, and the energy-wave-number characteristic. Numerical results for the effective masses are obtained using the optimized model potential, and a constant-mass approximation is used to estimate corrections to the form factors and energy-wave-number characteristics calculated from ordinary perturbation theory. The magnitude of these corrections suggests that mass-renormalized results may be required to predict metallic properties.

The activity of a neural net is represented in terms of a matrix vector equation with a normalizing operator in which the matrix represents only the complete structure of the net, and the normalized vector-matrix product represents the activity of all the non-afferent neurons. The activity vectors are functions of a quantized time variable whose elements are zero (no activity) or one (activity). Certain properties of the structure matrix are discussed and the computational procedure which results from the matrix vector equation is illustrated by a specific example.

Let Y=Xβ+e be a Gauss-Markoff linear model such that E(e) = 0 and D(e), the dispersion matrix of the error vector, is a diagonal matrix δ whose ith diagonal element is σi 2, the variance of the ith observation yi. Some of the σi 2 may be equal. The problem is to estimate all the different variances. In this article, a new method known as MINQUE (Minimum Norm Quadratic Unbiased Estimation) is introduced for the estimation of the heteroscedastic variances. This method satisfies some intuitive properties: (i) if S 1 is the MINQUE of Σ piσi 2 and S 2 that of Σqiσi 2 then S 1+S 2 is the MINQUE of σ(pi + qi )σi 2, (ii) it is invariant under orthogonal transformation, etc. Some sufficient conditions for the estimation of all linear functions of the σi 2 are given. The use of estimated variances in problems of inference on the β parameters is briefly indicated.

This article concerns dominance conditions for an n × m n \times m matrix. In the simplest kind of dominance, the (absolute) value of the diagonal element exceeds the sum of the absolute values of the nondiagonal elements on the same row. This condition has been generalized in the literature in several ways, of which we consider ways in which the rows of the matrix cooperate. Our work amounts to a sorting out of certain dominance conditions that belong to a class C \mathcal {C} of dominance conditions. We prove a theorem characterizing all true statements of the form \[ C 1 , C 2 , ⋯ , C s ⇒ C 0 {C_1},{C_2}, \cdots ,{C_s} \Rightarrow {C_0} \] where C i ∈ C ( i = 0 , 1 , ⋯ , s ) {C_i} \in \mathcal {C}(i = 0,1, \cdots ,s) .

The principle of mass renormalization in the quantum electrodynamics is applied to derive the equations which the potential for the individual particle motions satisfies self-consistently in the nuclei with the doubly closed shells. The potential is to be such that the excitation energy of each single particle (hole) state is invariant when the residual interactions are taken into account. Since this condition is concerned only with the diagonal matrix elements of the potential, the other matrix elements are so determined that the wave functions of the unperturbed ground and single particle (hole) states can best simulate the corresponding perturbed ones. The perturbed ground state is supposed not to have the components with one (particle and hole) pair excitation. Each perturbed single particle (hole) state should have the components of the different unperturbed single particle (hole) states as small as possible. The equations for the potential are given in a form of power series expansion with respect to the two-body interactions and are explicitly written up to the second order. The result turns out to be practically identical with that for the real part of the optical potential. The relation with Sawada's theory in an infinite system is pointed out.

There exist several multivariable control systems with the characteristic of having equal inputs and equal outputs; equality of the outputs is usually achieved by means of high performance components in each of the control system loops. Such a high performance is required in order to reduce the output sensitivity to disturbances and parametric variations within prescribed amounts. In practice we try to make the system transfer matrix a diagonal matrix.

It is proven that a necessary and sufficient condition for an $n$-dimensional Hermitian matrix $\ensuremath{\rho}$ to be positive definite is that it be expressible in the form $\ensuremath{\rho}=OEO\ifmmode\dagger\else\textdagger\fi{}$, where $O$ is a complex orthogonal matrix and $E$ is a diagonal matrix with positive elements. This accomplishes a parametrization since $O$ has ${n}^{2}\ensuremath{-}n$ real parameters and $E$ has $n$ of them. The proof is constructive, giving $O$ and $E$. It is further shown that the limit forms of this expression yield all the non-negative definite matrices. The parametrization for the polarization matrix of a spin-one particle is given explicitly.

The general functional matrix has the computational disadvantages of increasing rapidly with system order and of having large numbers of zero elements. If a spectral analysis of the linear functional matrix is available, these disadvantages may be overcome by making use of relationships between the dyadic expansion of a matrix and the dyadic expansions of its inverse powers. A set of tables is given which shows the eigenvectors and reciprocal eigenvectors of the general functional matrix in terms of those of the linear functional matrix. These tables thus provide an alternative means of computing the functionals considered in the two previous papers.

A shell-model calculation of 42Ca, 92Zr, and 18O is described in which the nuclear states are assumed to belong to configurations involving two nucleons outside a closed shell and in which the residual interaction is taken to be Green's velocity-dependent potential and Tabakin's separable potential, both of which reproduce the free two-nucleon scattering data. The two potential yield similar two-particle matrix elements. Some of the J = 0 diagonal matrix elements are small and even positive; some off-diagonal ones are large. The energy spectra of the nuclei are similar; configuration mixing is important. The agreement with experiment is very poor. This may be due to using these interactions as residual shell-model interactions. It is more probably due to not including core excitation in the calculation.

The transition energies and transition probabilities for the family of dye ions Me2N–(CH=CH)n−1–CH=N lim +Me2 are calculated using the method of Pariser and Parr. The nitrogens are taken into account by a perturbation method. Configuration interaction considering single excitations from the ground configuration is employed, though in the comparison with experiment it is judged better to use the diagonal matrix elements for the transition energies.

A general method is given for the determination of diagonal matrix elements of relative state operators in shell model states of a given supermultiplet and SU3 symmetry, for states in s, p and sd shells, and for cross-terms between these shells, enabling the excitation energies of particle–hole configurations to be evaluated.

A formulation of analysis of variance based on a model for the subclass means is presented. The deficiency of rank in the model matrix is handled, not by restricting the parameters, but by factoring the matrix as a product of two matrices, one providing a column basis for the model and the other representing linear functions of the parameters. In terms of the column basis and a diagonal matrix of subclass or incidence numbers, a compact matrix solution is derived which provides for testing a hierarchy of hypotheses in the non-orthogonal case. Two theorems are given showing that a column basis for crossed and/or nested designs can be constructed from Kronecker products of equi-angular vectors, contrast matrices, and identity matrices. This construction can be controlled in machine computation by a symbolic representation of each degree of freedom for hypothesis in the analysis. Provision for a multivariate analysis of variance procedure for multiple response data is described. Analysis of covariance, both ...

In [7] Sinkhorn proved that if A is a positive square matrix, then there exist two diagonal matrices D, = {@r,..., d$) and D, = {dj2),..., di2r) with positive entries such that D,AD, is doubly stochastic. This problem was studied also by Marcus and Newman [3], Maxfield and Mint [4] and Menon [5]. Later Sinkhorn and Knopp [8] considered the same problem for A nonnegative. Using a limit process of alternately normalizing the rows and columns sums of A, they obtained a necessary and sufficient condition for the existence of D, and D, such that D,AD, is doubly stochastic. Brualdi, Parter and Schneider [l] obtained the same theorem by a quite different method using spectral properties of some nonlinear operators. In this note we give a new proof of the same theorem. We introduce an extremal problem, and from the existence of a solution to this problem we derive the existence of D, and D, . This method yields also a variational characterization for JJTC1 (din dj2’), which can be applied to obtain bounds for this quantity. We note that bounds for I7y-r (d,!‘) dj”)) may be of interest in connection with inequalities for the permanent of doubly stochastic matrices [3].

In multigroup transport theory the nonnegative n x n matrix C = (c/sub ij/) is encountered, in which the c/sub ij/ are transfer coefficients. The weighted probability that a particle in group j will transfer into group i is given by c/sub ij/. Considering the condition on C that there exists a positive diagonal matrix (D) such that DCD/sup -1/ is a nonnegative symmetric matrix,'' the necessary and sufficient conditions for the existence of D are found. (T.F.H.)

Energy spectra up to Ex = 20 MeV and angular distributions of τ-particles from the 15N(d, τ)14C reaction have been measured at an incident energy of 52 MeV and have been analysed by the DWBA. Five strong l = 1 transitions to 1p proton hole states occur up to Ex = 11.29 MeV, which is 3 MeV above the particle emission threshold. The absence of direct l = 0 and l = 2 transitions demonstrates that 15N is a good shell-model nucleus with respect to protons. The isobaric analogue 1p hole states in 14C and 14N are identified by a comparison with T = 1 states excited in the 15N(p, d) reaction. This implies the following assignments for 14C levels: Jπ = 2+ for the 8.32 MeV level; Jπ = 1+ for a new level at 11.29±0.04 MeV with a width of 150±50 keV; Jπ = (0, 1, 2)+ for one of the 10.433 and 10.453 MeV levels. This state corresponds to the formerly unknown T = 1, Tz = 0 isobaric analogue at 12.52 MeV in 14N. The Jπ = 0+ and 1+, T = 1 states of mass 14 are rather pure 1p hole states, while the (2+, 1) states are strongly mixed with (s, d)2 configurations. The complete 1p−2 spectrum could be observed. Combined with neutron pick-up data the six diagonal matrix elements of the 1 p 1 2 −2 and the 1 p 1 2 −1 1 p 3 2 −1 interaction have been determined and compared to theoretical results. The eight multipole coefficients ακτ(j1j2) have been calculated by the use of the energy weighted sum rules. The 1 p 1 2 −1 1 p 3 2 −1 spectrum is nearly identical to one produced by a δ-force without exchange terms.

Several new results are presented concerning matrices having nonnegative inverses. We consider the problem of determining conditions on the pair of real square matrices $(A,B)$ such that for every diagonal matrix D with positive elements on the main diagonal, the inverse of the matrix $AD + B$ has only nonnegative elements. Of primary concern are the two special cases which occur frequently in the applications, (a) the case in which A is the identity matrix, and (b) the case in which both of the matrices $A,B$ have only nonpositive off-diagonal elements.

It is shown that a stationary, ergodic stochastic process satisfies conditions analogous to Van Hove's1) well known diagonality conditions. In this context Van Hove's irreducible diagonal matrix elements are shown to be simply cumulant averages. An illustrative application to the theory of isotopically disordered crystal lattices is briefly discussed.

The author (2) has shown that corresponding to each positive square matrix A (i.e. every aij > 0) is a unique doubly stochastic matrix of the form D1AD2, where the Di are diagonal matrices with positive diagonals. This doubly stochastic matrix can be obtained as the limit of the iteration defined by alternately normalizing the rows and columns of A.In this paper, it is shown that with a sacrifice of one diagonal D it is still possible to obtain a stochastic matrix. Of course, it is necessary to modify the iteration somewhat. More precisely, it is shown that corresponding to each positive square matrix A is a unique stochastic matrix of the form DAD where D is a diagonal matrix with a positive diagonal. It is shown further how this stochastic matrix can be obtained as a limit to an iteration on A.