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
A Doctoral Thesis. Submitted in partial fulfilment of the requirements for the award of Doctor of Philosophy of Loughborough University.