Symmetric stochastic inverse eigenvalue problem

Quanbing Zhang,Changqing Xu,Shangjun Yang

doi:10.1186/s13660-015-0698-0

Abstract

Firstly we study necessary and sufficient conditions for the constant row sums symmetric inverse eigenvalue problem to have a solution and sufficient conditions for the symmetric stochastic inverse eigenvalue problem to have a solution. Then we introduce the concept of general solutions for the symmetric stochastic inverse eigenvalue problem and the concept of totally general solutions for the $3\times3$ symmetric stochastic inverse eigenvalue problem. Finally we study the necessary and sufficient conditions for the symmetric stochastic inverse eigenvalue problems of order 3 to have general solutions, and the necessary and sufficient conditions for the symmetric stochastic inverse eigenvalue problems of order 3 to have a totally general solution.

Highlights

For a square matrix A, let σ (A) denote the spectrum of A
Given an n-tuple = (λ, . . . , λn) of numbers, the problem of deciding the existence of an n × n matrix A with σ (A) = is called the inverse eigenvalue problem which has for a long time been one of the problems of main interest in the theory of matrices
Since stochastic matrices are important nonnegative matrices, it is surely important to investigate the existence of stochastic matrices with a prescribed spectrum under certain conditions

Summary

Introduction

For a square matrix A, let σ (A) denote the spectrum of A. Proof ( ) If S, S are equivalent, S = PSQD for some permutation matrices P, Q and diagonal unipotent matrix D. That two solutions of the SCRSIEP for associated with equivalent typical orthogonal matrices are permutationally similar to each other. Sn∗ is a non-symmetric typical orthogonal matrix for any integer n >. Sn∗ are the typical orthogonal matrices given by Any × typical orthogonal matrix which having no entry equal to zero must be equivalent to. Proof Let S be a × typical orthogonal matrix which has no zero entry. Any × typical orthogonal matrix must be equivalent to the matrix S (x) given by Proof Let S be a × typical orthogonal matrix. The following are some important × irreducible typical orthogonal matrices:

Note that is equivalent to both

Proof We have

Now condition implies