Powers And Products Of Some Special Matrices

In the first chapter,we give some basic concepts and conclusions involvedin this dissertation.In the second chapter,we give a new proof of ProfessorJiayu Shao's result on a necessary and sufficient condition for a nonnegativematrix to be decomposed into a product of irreducible nonnegative matrices,and determine the least possible number of factors.In the third chapter,wediscuss the possible numbers of ones in the powers of 0-1 matrices with agiven rank.In the fourth chapter,we discuss the possible numbers of onesin 0-1 matrices whose powers are still 0-1 matrices.In the fifth chapter,westudy the powers of Toeplitz matrices and Hankel matrices,and give lowerbounds for Frobenius norms and spectral norms of Cauchy-Toeplitz matricesand Cauchy-Hankel matrices respectively.In the sixth chapter,we give anexplicit expression for any power of a certain type of anti-tridiagonal matricesof even order.This dissertation discusses five kinds of problems as follows.1.Decomposition of nonnegative matricesMatrix decomposition problems are important in both matrix theory andmatrix computations.This kind of problems discusses the problems of decom-posing a matrix into products or sums of several matrices of some special kinds.Professor Jiayu Shao gave a necessary and sufficient condition for a nonnegativematrix to be decomposed into a product of irreducible nonnegative matricesin 1985 ([38]).We reprove his result using a graph-theoretic method,and alsoshow that when such a decomposition is possible,the number of factors canbe required to be at most three.The methods used here are constructive,and they give an algorithm to produce the factors.2.Powers of 0-1 matrices with a given rankA 0-1 matrix is a matrix whose entries are 0 or 1.Such matrices arisefrequently in combinatorics and graph theory.In 2005,Qi Hu,Yaqin Li andXingzhi Zhan determined the possible numbers of ones in a 0-1 matrix witha given rank in the generic case and in the symmetric case ([24]).ProfessorXingzhi Zhan asked later:what are the possible numbers of ones in the powersof a 0-1 matrix with a given rank? We answer this question partially.3.0-1 matrices whose powers are still 0-1 matricesIntuitively,if a 0-1 matrix has too many ones,its powers can not be still0-1 matrices.In 2007 Professor Xingzhi Zhan posed the following problem ata seminar:Given positive integers n and k,if the kth power of a 0-1 matrix A oforder n is still a 0-1 matrix,then what is the maximum number of ones in A?Furthermore,how to characterize those matrices which attain the maximumnumber? We solve the case k=2 of Zhan's problem.That is,we determinethe maximum number of ones in the 0-1 matrices whose squares are still 0-1matrices,and the maximizing matrices are also specified.Furthermore,westudy the case k＞2 partially.4.Powers and norms of special matricesThere are some matrices with special structures which have importantapplications.On the one hand,we discuss the powers of Toeplitz matrices and Han-kel matrices.Tamir Shalom gave a necessary and sufficient condition for thepowers of Toeplitz matrices to be still Toeplitz matrices ([37]).We give a newproof of his result.Moreover,we give a sufficient condition for the powers ofHankel matrices to be still Hankel matrices.On the other hand,we give lower bounds for the Frobenius norms andspectral norms of Cauchy-Toeplitz matrices and Cauchy-Hankel matrices, using the polygamma function and assistant matrices.5.Powers of one kind of anti-tridiagonal matricesIn solving some difference equations and differential equations we need tocompute the arbitrary powers of square matrices ([1],[25],[33]).We derive anexplicit expression for the powers of some kind of anti-tridiagonal matrices ofeven order.