Study on some Classes of Square Matrices along with Operator Matrices and their Properties

Abstract

Abstract newlineI am very pleased to present my mathematical thesis and hope my work would be an effective source of information for the aspirants working in this field. There may be some discrepancies in the findings at the initial stage of my work but on the second review I made ceaseless efforts to solve them and this practice has empowered me with vision to complete the things with new perception. newlineFrom the beginning, I have keen interest in linear algebra and especially in matrix algebra. It is important to note its usage in wide range of real life problems. I was very curious about searching for some typical type of matrices. I started with the literature work, going through various theses and papers. Taking some examples as particular case for the solution of the problem, without considering a mathematical format but I began to find some ways to reach the perfect solution. newlineMy thinking process to find divisions of square matrices continued over a long time and finally I was satisfied. newlineIt, as I did, required detailed working on putting the square matrices under classification area. Matrices which followed a certain pattern in the flow of the elements of their columns and rows against which do not observe any pattern. newlineI called the previous falling in Class A and the latter falling in Class B. newlineThus, I have developed a group of square matrices of Class A which follow a certain pattern. A detailed study revealed that I need further subdivision of the matrices of Class A. newlineWe have defined basic algebraic structure of each class. The following is the wide order classification these classes are class -1,2,3, 4A, 4B, and finally the class5. newlineIn my work I have made detailed discussion on the algebraic structures and properties of class2 and class4A. newlineThe out-standing property of all such classes is its Libra value property that remains in accordance with the corresponding mathematical operation on the member matrices of the class. newlineThe important property is that the libra value of the resultant matrix obtained by either addition