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# Accession Number:

## ADA080216

# Title:

## Solutions of the Matrix Equation AXB + CXD = E.

# Descriptive Note:

## Master's thesis,

# Corporate Author:

## AIR FORCE INST OF TECH WRIGHT-PATTERSON AFB OH SCHOOL OF ENGINEERING

# Report Date:

## 1979-12-01

# Pagination or Media Count:

##
106.0

# Abstract:

## Solutions of the general linear matrix equation summation A sub i XB sub i c where i 1 and N is the limit are obtained and presented in this thesis. Some special cases do arise like the Lyapunov matrix equation. Necessary conditions and sufficient conditions are established for the solution of the general linear matrix equation. Other forms of solutions than those obtained through the use of similarity transformations that have been considered make use of the spectral decomposition of matrices and tensor products of matrices or Kronecker products. In considering the general linear matrix equation, linear matrix equations in which two different variables appear are also studied. Conditions for the existence of a solution for this type of equation are given. The theory of the generalized inverses of a matrix was used in obtaining a solution to the general linear matrix equation. More general forms of the solution are given and conditions under which these solutions exist have been established. Solutions to systems of matrix equations were also considered. As a by-product of this investigation, some aspects of the model reduction problem may be treated from the point of view of matrix equations.

# Distribution Statement:

## APPROVED FOR PUBLIC RELEASE

#