The Inverse Eigenvalue Problem Of Symmetric Double Stochastic Matrices

The inverse eigenvalue problem of matrices refers to the problem of constructing a corresponding matrix from given eigenvalues or eigenvectors.The inverse eigenvalue problem is widely used in practical engineering technology,and the mathematical charm of the problem itself makes its research have a very broad prospect and become a very active topic increasingly in mathematics.A stochastic matrix is a kind of special and widely used nonnegative matrix.It has important applications in the finite homogeneous Markov chain theory,combinatorial mathematics,evolutionary system model and discrete economic model in biological and social sciences.In this thesis,the inverse eigenvalue problem of low-order symmetric double stochastic matrices is discussed,and a construction method for the inverse eigenvalue problem of high-order symmetric double stochastic matrices is proposed.On the basis of predecessors,some results of inverse eigenvalue problem of symmetric double stochastic matrices are improved,and the conclusion is convenient for application.The sufficient conditions for the existence of symmetric double stochastic matrices are obtained.The construction of a solution matrix of the inverse eigenvalue problem of low order double stochastic matrices is given.It makes the judgment of the inverse eigenvalue problem of low order symmetric double stochastic matrices simpler and the application more convenient.Examples are given to illustrate the feasibility and simplicity of the application.For the inverse eigenvalue problem of higher-order symmetric double stochastic matrices,the conclusion that two symmetric double stochastic matrices are combined into higher-order symmetric double stochastic matrices is proved.In other words,the method of constructing a new symmetric double random matrix by using the smaller matrix of known spectrum is applied to the inverse eigenvalue problem of higher-order symmetric double stochastic matrix.This method avoids discussing the parity of the number of eigenvalues.The results are applied to practical examples.