Algebraic Matrix Roots

The problem related to algebraic matrix equations is one of the important branches, and the non-linear matrix equations play an essential role in many fields. The purpose of the thesis is to survey the very important area of Matrix Theory: Non-linear Algebraic Matrix Equations. We consider mainly the equationX~m =A and the quadratic matrix equation X~2 -2AX+B = 0(where m is positiveinteger, and X , A are square matrices of the same size).Throught the results of this paper, many differences between the scalarequation x~m = a,x2 - 2ax + b = 0 and the matrix equationX~m=A, X~2 -2AX + B = 0 (III.1)will become apparent.In contrast to the properties of the solutions of a scalar equation,the solutions of the matrix equation are not necessarily polynomials in A,they do not necessarily commute, and they may be nilpotent. Moreover,the matrix equation ,such as (III.2), may have infinitely many solutions when the corresponding scalar equations have only a finite number of solutions. For example, the solutions of the matrix equationand all matrix of the formThe thesis first gives a survey of earlier known results in this field, then obtains some conditions for the solvability of the equation X~m = A . Some algorithms for the solutions are presented. Relationship between singular values of a complex matrix andits mth order roots is also investigated.In chapter 1, we explain the corresponding backgrounds of the thesis, the purpose of this thesis,the methods of research,the results of this thesis,etc.In chapter 2,we investigate the equation Xm = A for any given matrix AE.Cnxn.Our purpose here is by discussing the matrix mth roots of Jordan normal form matrices, then discussing the usual matrices. It shows that nonsingular Jordan normal form matrices have mth roots. We prove the existence of matrix mlh roots of nonsingular complex matrices. Eventually, we give the sufficient and necessary conditions for the existence of matrix mth root of a general complex matrix.we use numerical method to study the equation Xm - A . We present an algorithm to find the roots, by using the Schur decomposition of A . We also point out some errors appear in [25],[33],[37].In particular,we consider the roots of a Jordan normal form,for msil,and analysis the existence of the solutions,the number of solutions,and the general forms of its solutions.In chapter 3, based on the analysis of the 2th order roots of a matrix, we establishsome further results on I2-2AX + B = 0. Specifically, we get the condition for theuniqueness of its solution.In the end, we state some interesting results of the thesis1. Any unitarily invarianf norm of a non-unitary algebraic root of unitary matrix is greater than 1.2. If m is an even number, then the solutions of X'" = A span whole C"x" if and only if A = A/ , A * 0.3. spanO"2 = {AECnxn :tr(A) = 0}...