Characterization of quadraticity for linear combinations of quadratic matrices

In Chapter I, some brief information about the content of the work is given. In addition, some studies about the linear combinations of some special types of matrices in the literature are introduced. Chapter II is written as a preparation for the next chapters. In this chapter, some theorems without proofs and fundamental concepts, which will be used later, are given. In Chapter III, all the situations, where the linear combination of the form r_1Q_1+r_2Q_2 is a quadratic matrix when r_1 and r_2 are nonzero complex numbers and, Q_1 and Q_2 are nxn complex matrices, are characterized. The results obtained in this chapter consist of matrix equalities together with coefficient equalities which enable the linear combination to be a quadratic matrix. In Chapter IV, an alternate proof of the main result of Chapter III in the particular case that Q_1 and Q_2 in the linear combination r_1Q_1+r_2Q_2 are commuting matrices, is given. The results in this chapter are given in terms of diagonal forms of matrices which form the linear combination. At the end of Chapters III and IV, numerical examples are given to exemplify the results which are obtained in these chapters. Verification of these results was performed by using Mathematica software. The codes and outputs of associated programs are given in Appendix A and Appendix B. In Chapter V, the obtained results in this study are briefly introduced and some suggestions are given.