On characterizations of linear combination of an arbitrary matrix and a quadratic matrix

In the first chapter, it is given some information about the subject and scope of the study. Also, it is mentioned from some studies in the literature related to matrix classes discussed in the study. Some fundamental concepts and properties which will be used throughout the study are given in the second chapter. In the Chapter 3, it has been established necessary and sufficient conditions for the idempotency of the linear combination matrix and implying certain conditions imposed on the matrices and where is an – quadratic matrix, is an arbitrary matrix, , and . In the Chapter 4, it has been established necessary and sufficient conditions for the involutiveness of the same linear combination matrix and implying the same conditions. Examples explaining the results which are obtained in Chapter 3 and 4 are given in Chapter 5. Keywords: idempotent matrix, involutive matrix, quadratic matrix, linear combination, direct sum, diagonalization