The Inverse Eigenvalue Problem For Several Classes Of Nonnegative Matrices

The theory of nonnegative matrices has always been one of the most activeresearch areas in the matrix theory and has been widely applied in mathematics andother branches of natural and social sciences. There are, for example, game theory,Markov chains (stochastic martices), theory of probability, probabilistic algorithms,numerical analysis, discrete distribution, group theory, matrix scaling, theory of smallosillations of elastic systems (oscillation marrices), economics and so on. In recentyears, the inverse eigenvalue problem comes to be the focus of the matrix theory. Thisthesis will study the inverse eigenvalue problem for nonnegative matrices (NIEP). Themajor researches of this theisis focus on the inverse eigenvalue problem for severalspecial classes of nonnegative matrices, the necessary and sufficient conditions andsome sufficient conditions of which are derived. Moreover, the numerical algorithms ofthe inverse eigenvalue problem for these special classes of nonnegative matrices aregiven, the accuracy of which together with the correcteness of related theories istestified by several numerical examples. The main procedures of this theisis are asfollows:In the first chapter, the significance and the development of the inverse eigenvalueproblem for nonnegative matrices are addressed, and the research situation home andabroad is introduced.In the second chapter, the inverse eigenvalue problem for nonnegative tridiagonalmatrices is studied. First, the inverse eigenvalue problem for33nonnegativetridiagonal matrices is solved by discussion of a variety of situations. Moveover，thenecessary and sufficient conditions of the solutions of the inverse eigenvalue problemfor33nonnegative tridiagonal matrices are derived. Then, the properties ofeigenvalue of n nnonnegative tridiagonal matrices are derived by characteristicpolynomial of truncated matrices of nonnegative tridiagonal matrices, with thecombination of the relationship between eigenvalues of Jacobi matrix. Finally, theinverse eigenvalue problem for nonnegative tridiagonal matrices is solved.In the third chapter, the inverse eigenvalue problem for nonnegative five-diagonalmatrices is studied.33nonnegative five-diagonal matrices is also33nonnegative matrices, the necessary and sufficient conditions of the solutions of theinverse eigenvalue problem for which are given in this thesis. For the inverse eigenvalue problem for n nnonnegative five-diagonal matrices, only some sufficientconditions are given because of its complexity.In the fourth chapter, the inverse eigenvalue problem for nonnegative circulantmatrices is studied. First, some remarkable conclusions of the inverse eigenvalueproblem for nonnegative matrices in recent years are summarized. Then, the inverseeigenvalue problem for real circulant matrices is advanced and successfully solved, thenecessary and sufficient conditions of which are given also. Finally, the inverseeigenvalue problem for nonnegative circulant matrices is advanced based on the inverseeigenvalue problem for real circulant matrices, whose sufficient conditions and somerelevant conclusions are given.In the fifth chapter, some algorithms and numerical examples are given based onthe conclusions derived in the previous three chapters.In the sixth chapter, the summary of the paper is given and the future researchwork is put forward.