Approximate symmetric solutions of matrix equations by singular value decomposition

In the first chapter of the work, the concept of the SVD and its historical evolution are summarized.Some concepts and theorems that will be fundamental tools for the further chapters are introduced in the Chapter 2. In the next chapter, the SVD is discussed in detail and a numerical example is given.In the Chapter 4, first, a general theory about the linear equations systems is mentioned. Then, the best approximate solution to the problem of finding the vector x from among the least squares solutions set of the system Ax=g if it is inconsistent is presented.In the Chapter 5, firstly the results in the Chapter 4 are extended for the linear matrix equation AXB=C, secondly the necessary and sufficient conditions for consistency of the linear matrix equation AXB=C and the linear equations system Ax=g are given via the SVD. Finally, the general expression for the solutions of these equations is also established using the SVD again.The symmetric solutions of the consistent matrix equation AX=C which is a special case of the consistent matrix equation AXB=C are given in the next chapter.In the Chapter 7, the best approximate solution to the problem, studied in the literature recently, finding the symmetric matrix X from among the least squares symmetric solutions set of the linear matrix equations AXB=C if it is inconsistent is considered using the SVD. Moreover, a numerical example to explain the main theoretical results established is given as well.Keywords: Moore-Penrose inverse, least squares solution, singular value decomposition (SVD), inconsistent matrix equation, best approximate symmetric solution.