Bounds on the diagonal elements of a unitary matrix

It was shown by A. Horn that the diagonal elements of a unitary n×n matrix satisfy a set of linear inequalities (Theorem I). We give a simple proof of this result, and we show that the diagonal elements satisfy additional linear inequalities (Theorem II) if the matrix is also symmetric.     

Upper bounds for nearly optimal diagonal scaling of matrices

Optimal diagonal scaling of an n×n matrix A consists in finding a diagonal matrix D that minimizes a condition number of AD. Often a nearly optimal scaling of A is achieved by taking a diagonal matrix D₁ such that all diagonal elements of D₁A^TAD₁ are equal to one. It is shown in this paper that the condition number of AD₁ can be at least (n/2)^1/2 times the minimal one. Some questions for a further research are posed.     

Counterexamples concerning the diagonal elements of normal matrices

An infinite class of conditions known to be satisfied by the diagonal elements of a normal matrix with prescribed spectrum is shown to be independent of other known conditions satisfied by the diagonal elements but nevertheless insufficient to characterize the diagonal.     

On the Square Root of an H-matrix with Positive Diagonal Elements

This paper proves that for an H-matrix with positive diagonal elements there exists one and only one square root which is also an H-matrix with positive diagonal elements. An algorithm approaching the square root is proposed.     

Estimates for the inverse of a matrix and bounds for eigenvalues

This paper gives new bounds for the relationship between the diagonal elements of a square matrix and the corresponding diagonal elements of the matrix inverse, as well as bounds for the eigenvalues of the matrix. The results given here generalize those of Ostrowski and Ky Fan, and have their origin in engineering application.     

Some improvements for two-sided bounds on the inverse of diagonally dominant tridiagonal matrices

In this paper, we obtain lower and upper bounds for the entries of the inverses of diagonally dominant tridiagonal matrices. First of all we derive the bounds for off-diagonal elements of the inverse as a function of the diagonal ones, then we improve the two-sided bounds for the diagonal entries obtaining sharper lower and upper bounds for all the elements of the inverse.     

Construction,properties and statistical applications of positive definite intraclass matrix

Given a positive definite (p.d.) matrix with real entries, it is possible to construct a p.d. intraclass matrix whose diagonal and off-diagonal elements are chosen as the averages of the diagonal elements and off-diagonal elements of the former matrix. Exploiting the very special structure of the latter matrix various interesting propositions are established. Statistical applications of such matrices are surveyed.     

The kernel of the bezoutian for operator Polynomials ∗

Based on two set partitions of the symmetric group S_n expansion theorems by diagonal elements for the permanent and the determinant are derived, for both the generic commuting and noncommuting cases. They are of the same type as the well-known Laplace expansions where either fixed rows or columns of a given matrix are chosen instead of diagonal elements.     

Invariant factors of integral quaternion matrices

Existence of a diagonal form under unimodular equivalence is proved for matrices with entries from the Hurwitz ring of integral quaternions. The diagonal elements satisfy certain divisibility relations with an unexpected character, and these force a degree of uniqueness to the diagonal form. Connections between the so obtained invariant factors of a full matrix and those of a submatrix are then established.     

A New Diagonal Quasi-Newton Updating Method With Scaled Forward Finite Differences Directional Derivative for Unconstrained Optimization

A new diagonal quasi-Newton updating algorithm for unconstrained optimization is presented. The elements of the diagonal matrix approximating the Hessian are determined as scaled forward finite differences directional derivatives of the components of the gradient. Under mild classical assumptions, the convergence of the algorithm is proved to be linear. Numerical experiments with 80 unconstrained optimization test problems, of different structures and complexities, as well as five applications from MINPACK-2 collection, prove that the suggested algorithm is more efficient and more robust than the quasi-Newton diagonal algorithm retaining only the diagonal elements of the BFGS update, than the weak quasi-Newton diagonal algorithm, than the quasi-Cauchy diagonal algorithm, than the diagonal approximation of the Hessian by the least-change secant updating strategy and minimizing the trace of the matrix, than the Cauchy with Oren and Luenberger scaling algorithm in its complementary form (i.e. the Barzilai-Borwein algorithm), than the steepest descent algorithm, and than the classical BFGS algorithm. However, our algorithm is inferior to the limited memory BFGS algorithm (L-BFGS).     

Invariant factors of integral quaternion matrices II

In the author's previous paper a diagonal form was found under unimodular equivalence for matrices over the Hurwitz ring of integral quaternions, and uniqueness was established for the norms of certain constituents of the diagonal elements. In the present paper it is shown that the odd, primitive, parts of the all but one of the diagonal elements may be freely chosen provided that the norm constraint is met.     

An inverse eigenvalue problem for totally nonnegative matrices

In a previous paper we proved that the diagonal elements of a totally nonnegative matrix are majorized by its eigenvalues. In this note we show that the majorization of a vector of nonnegative real numbers by another vector of nonnegative real numbers is not sufficient for the existence of a totally nonnegative matrix with diagonal elements taken from the entries of the majorized vector and eigenvalues taken from the entries of the majorizing vector.     

A diagonal quasi-Newton updating method for unconstrained optimization

Numerical Algorithms - A diagonal quasi-Newton updating algorithm is presented. The elements of the diagonal matrix approximating the Hessian are determined by minimizing both the size of the...     

On matrices stochastically similar to matrices with equal diagonal elements

A necessary and sufficient condition for a matrix to be stochastically similar to a matrix with equal diagonal elements is obtained Aand B are called Stochastically similar if B=SAS^{- 1} where S is quasi-stochastic i.e., all row sums of .S are I. An inverse elementary divisor problem for quasi-stochastic matrices is also considered.     

相容次序矩阵PSD迭代法的收敛性

本文是在求解大型线性方程组Ax=b的系数矩阵A为(1,1)相容次序矩阵且其Jacobi迭代矩阵的特征值均为纯虚数或零的条件下,得到PSD迭代法收敛的充分必要性定理,并在特殊情况下得到了相应的最优参数.     

Singular values and diagonal elements of complex symmetric matrices

Necessary and sufficient conditions are given for the existence of a complex symmetric matrix with prescribed diagonal elements and singular values.     

On functions which preserve the class of Stieltjes matrices

A positive definite symmetric matrix is called a Stieltjes matrix provided that all its off diagonal elements are nonpositive. We characterize functions which preserve the class of Stieltjes matrices.     

Condition number and diagonal preconditioning: comparison of the p-version and the spectral element methods

Summary. In the framework of adaptive methods, bases of hierarchical type are used in the -version of the finite element method. We study the matrices corresponding to the commonly used basis, introduced by Babuška and Szabo, in the case of -dimensional rectangular elements for 2 order elliptic boundary value problems. For the internal nodes, we show that the condition number is equivalent to and to for the stiffness and mass matrix, respectively. Moreover, we show that the usual diagonal preconditioning divides in the previous orders the exponents of by two. Finally, we compare these results with those obtained for spectral elements (nodal basis). Received November 24, 1994 / Revised version received March 20, 1995