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.     

Cyclic and diagonal products on a matrix

We unify the theory of cyclic and diagonal products of elements of matrices. We obtain some new results on diagonal similarity, diagonal equivalence, complete reducibility and total support.     

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.     

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.     

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.     

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.     

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).     

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.     

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...     

Two expansion theorems by diagonal elements for the permanent and the determinant

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.     

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.     

Restricted estimates of the fractional integral on the diagonal

For a function in a mixed-norm Lebesgue space, we study the restriction of its fractional integral on the diagonal. We show that the restriction is weakly bounded when the indices satisfy certain conditions. We give a complete characterization of such indices. To prove this result, we generalize a classical result on operators that commute with translations.     

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.     

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.     

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.     

DNA序列的特征数值及相似性分析

利用2维图表示DNA序列,计算与该图对应的距离矩阵,求出距离矩阵的不变量—距离矩阵主对角线以外的次对角线之和的平均值,进而得到了DNA序列的一种特征数值,利用这种新的特征数值,对DNA序列进行相似性比较,得到了与现有的资料符合很好的结果.     

对角方阵的变换群

在不改变对角方阵各行、各列、主对角线、次对角线的元素之集的条件下,其变换群是n次对称群S_n的直积S_n×S_n的子群,因对角拉丁方、对角拉丁方正交侣、幻方、高次幻方、加乘幻方均属此类方阵,本文对构作这类对象及研究它们的计数有重要意义.     

A note on generalizations of strict diagonal dominance for real matrices

We investigate classes of real square matrices possessing some weakened from of strict diagonal dominance of a real matrix whose diagonal entries are all positive. The intersection of each one of these classes with the set of all real matrices, with nonpositive off-diagonal elements, coincides with the set of all nonsingular M- matrices.     

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.