Matrix Polynomial Equation And The Canonical Decomposition Of Invertible System

This thesis mainly studies the following problems:1. Restricted solutions and the maximal rank solutions of linear equations with applicationsWe present a new method, the base method, to solve general linear matrix equations in any finite dimensional space, and especially we give an effective way to obtain the restricted solutions of linear equations, such as (skew)symmetric solutions, (skew) Her-mitian solutions, circulant solutions etc. On the basis of the structure theory of solutions of linear matrix equations, we give a method for finding the invertible(or maximal rank) solutions of linear equations and discuss their applications.2. Solutions of polynomial matrix equations with applications On the basis of invertible solutions of linear equations, we propose a method to solvepolynomial matrix equations which has not been dealed with before, and give the specific steps to find solutions and diagonalizable solutions of polynomial matrix equations on the real (complex)field. The method can also be used to solve many other non-linear matrix equations.3. Quaternion polynomialsWe put forward the concept of the irreducible quaternion polynomial and discuss its properties such as divisibility, factorization, division algorithm and the structures of root. On the basis of the diagonalizable solutions of polynomial matrix equations and the complex representation of quaternion, we establish the relation between quaternion polynomial and polynomial on the complex field, and give the specific steps to find solutions of the quaternion polynomial equations.4. Solutions of matrix function equationsWe give the definition of the solvable matrix function equation f(X) = A,where f(x) is a kind of general analytic functions. By the concept and properties of matrix function and the Jordan canonical form, we discuss the necessary and sufficient conditions about the consistency of f(X) = A on the real and complex field, and give the specific steps to find such solutions.5. Canonical decomposition of the invertible system with applicationsThe decoupling problem of linear control system is one of the most concerned problems in the control system. With the elementary operations, we give a new decomposition...