Matrix Determinant And Algebraic Polynomial Root Calculation

The problems—the determent of a matrix and the polynomial in one indeterminate evaluation—are in fact one of complex and classical Mathematical problems. Many famous mathematicians do lots of investigation on them since a long long ago. Therefore, it is very important and useful to estimate them. In this paper, we give lots of evaluations for these problems, basing on some new investigations. Our results generalize and modify some classical ones.This paper mainly includes two parts:1. Bounds for determinants with dominant principal diagonal: In this part, some new upper and lower bounds for determinants with dominant principal diagonal are presented. These bounds are some improvements of some similar bounds. For example:(1) In Chapter two, we firstly give thatTheorem 2.1. If A = [ ai j]∈D, then, for an arbitrary index k∈N, we haveThe equality holds if and only ifTheorem 2.2. If A = [ ai j]∈D, then, for an arbitrary index k∈N, whereυj and w j( j∈N)are the same as those in Theorem 2.1. The equalities hold if and only ifTheorem 2.3. If A = [ ai j]∈D, then A ?1 = [b ij]exists, and, for an arbitrary index k∈N, whereυij and wi j(i , j∈N)as in Theorem 2.1. The equalities hold if and only when(2) For doubly (row) diagonally dominant, we also show thatTheorem 2.4. If A = [ ai j ]∈DD , and N 0= { j || a jj |≤R j( A) , j∈N }≠0,then2. Evaluation for the polynomial in one indeterminate: We give some simple bounds, which generalize and modify some classical ones. For example:Theorem 3.1. For the polynomial (3-2),its any real eigenvalueλsatisfies that:Theorem 3.3. For the polynomial (3-2) and any of 1≤s≤n ,σ> 0,its any eigenvalueλsatisfies that: where { }...