The changes in the unitary polar factor under both multiplicative and additive perturbation are studied. A multiplicative perturbation bound and a new additive perturbation bound, in which a different measure of perturbation is introduced, are presented. AMS subject classification (2000): 15A18, 15A23, 65F35.

In $1994,$ degree distance  of a graph was introduced by Dobrynin, Kochetova and Gutman. And Gutman proposed the Gutman index of a graph in $1994.$ In this paper, we introduce the concepts of  multiplicative version of degree distance and the multiplicative version of Gutman index of a graph. We find the sharp upper bound for the  multiplicative version of degree distance and multiplicative ver...

In this paper we obtain an upper bound and also a lower bound for maximum edges of strongly 2 multiplicative graphs of order n. Also we prove that triangular ladder the graph obtained by duplication of an arbitrary edge by a new vertex in path and the graphobtained by duplicating all vertices by new edges in a path and some other graphs are strongly 2 multiplicative

Let complex matrices $A$ and $B$ have the same sizes. Using the singular value decomposition, we characterize the $g$-inverse $B^{(1)}$ of $B$ such that the distance between a given $g$-inverse of $A$ and the set of all $g$-inverses of the matrix $B$ reaches minimum under the unitarily invariant norm. With this result, we derive additive and multiplicative perturbation bounds of the nearest per...

This paper is concerned with how the QR factors change when a real matrix A suffers from a left or right multiplicative perturbation, where A is assumed to have full column rank. It is proved that for a left multiplicative perturbation the relative changes in the QR factors in norm are no bigger than a small constant multiple of the norm of the difference between the perturbation and the identi...

In this paper, we propose an algorithm base on decomposition technique for solvingthe mixed integer linear multiplicative-linear bilevel problems. In actuality, this al-gorithm is an application of the algorithm given by G. K. Saharidis et al for casethat the rst level objective function is linear multiplicative. We use properties ofquasi-concave of bilevel programming problems and decompose th...

We use a simple matrix splitting technique to give an elementary new proof of the Lidskii-Mirsky-Wielandt Theorem and to obtain a multiplicative analog of the Lidskii-Mirsky-Wielandt Theorem, which we argue is the fundamental bound in the study of relative perturbation theory for eigenvalues of Hermitian matrices and singular values of general matrices. We apply our bound to obtain numerous bou...

The major challenge of data perturbation is to achieve the desired balance between the level of privacy guarantee and the level of data utility. Data privacy and data utility are commonly considered as a pair of conflicting requirements in privacy-preserving data mining systems and applications. Multiplicative perturbation algorithms aim at improving data privacy while maintaining the desired l...

The double angle theorems of Davis and Kahan bound the change in an invariant subspace when a Hermitian matrix A is subject to an additive perturbation A ! e A = A+A. This paper supplies analogous results when A is subject to a congruential, or multiplicative, perturbation A ! e A = D AD. The relative gaps that appear in the bounds involve the spectrum of only one matrix, either A or e A, in co...