A study of matrix equations

Matrix equations have been studied by Mathematicians for many years. Interest in them has grown due to the fact that these equations arise in many different fields such as vibration analysis, optimal control, stability theory etc. This thesis is concerned with methods of solution of various matrix equations with particular emphasis on quadratic matrix equations. Large scale numerical techniques are not investigated but algebraic aspects of matrix equations are considered. Many established methods are described and the solution of a matrix equation by consideration of an equivalent system of multivariable polynomial equations is investigated. Matrix equations are also solved by a method which combines the given equation with the characteristic equation of the unknown matrix. Several iterative processes used for the solution of scalar equations are applied directly to the matrix equation. A new iterative process based on elimination methods is also described and examples given. The solutions of the equation x2 = P are obtained by a method which derives a set of polynomial equations connecting the characteristic coefficients of X and P. It is also shown that the equation X2 = P has an infinite number of solutions if P is a derogatory matrix. Acknowledgements