Some Researches On The Generalized Quadratic Matrices

The research on the matrix classes with power conditions and their linear com-binations has been one of important subjects since last century. As research pro-gressed, the generalized quadratic matrix class with a wide sense has been introduced by R.W.Farebrother and G.Trenkler in 2005, generalizes and unifies involutory ma-trices, idempotent matrices and operators and so on, which already have rich con-clusions. Due to the basic work of R.W.Farebrother and G.Trenkler, many scholars pay attentions to generalized quadratic matrices. Since then the study about gener-alized quadratic matrix and its related issues attract more and more people. It has important applications in probability and statistical theory, cryptography, control theory, quantum mechanics and many other fields of mathematics and physical.In this thesis, taking the equivalent definition of generalized quadratic matrix, we make further investigations on the fundamental properties of generalized quadrat-ic matrices, and obtain the solutions of generalized quadratic matrices to the matrix equation aA+bX=AX. We also discuss the relationships between scalar tripo-tent matrices and generalized quadratic matrices, and give out rank equalities for generalized quadratic matrices under some operations and show their many appli-cations and so on. Our work will enrich and develop the quadratic matrix theory and quadratic operator theory, and make a powerful tool for the further discussion.The concrete structure of the thesis is as follows:At the beginning of this thesis, we make a review on the related study of the common matrix classes with power conditions, such as idempotent matrices and involutory matrices and so on, then introduce the development history and research situation of generalized quadratic matrices, and show the research content and framework of this thesis at last.In the first chapter, based on the work of R.W.Farebrother and G.Trenkler. we further study the fundamental properties of generalized quadratic matrices, from the representations of generalized quadratic matrices, to the similar canonical for-m, matrix power, inverse and the generalized inverse and so on, and obtain some profound results. For example, it shows the explicit expressions of the power of generalized quadratic matrices, illustrates that the inverse of generalized quadratic matrices discussed in [39] is incorrect, then gets the all inverse matrices of invert-ible generalized quadratic matrices, proves that the inverse and power of generalized quadratic matrices are also generalized quadratic, and shows the relationship of gen-eralized quadratic characteristics between A^k?A^-1 and A as well. Finally, we get the all{1},{1,2}-generalized inverses and group inverses of generalized quadratic matrices.Notice that the research on matrix equation is one of the basic topics in the matrix theory. In order to derive the sufficient and necessary condition for the so-lution of generalized quadratic matrices to the matrix equation aA+bX=AX, in the second chapter, we introduce current situation of the matrix pairs with linear combinations being equal to the product firstly, then discuss the relationships be-tween the matrix pairs on the eigenvalue, invertibility. and generalized quadraticity, which provides another sufficient, condition for the sum and product of generalized quadratic matrices keeping generalized quadraticity as well.In the third chapter, we consider when the scalar tripotent matrices are gen-eralized quadratic, then make a classification of scalar tripotent matrices. We also give out rank equalities for the sum of any finite scalar tripotent matrices. Since idempotent matrices is scalar tripotent, then as an application, it solves the open problem proposed by Y.Tian and G.P.H.Styan about rank equalities for the sum of any finite idempotent matrices. Finally, we obtain rank equalities for the sum of any finite generalized quadratic matrices as well.Based on the close relationships between generalized quadratic matrices and idempotent matrices, in the fourth chapter, we focus on studying rank equalities for generalized quadratic matrices under some operations. We obtain the invariance of rank and nullity of linear combinations of generalized quadratic matrices, and the one of rank of generalized Jordan product of generalized quadratic matrices. Rank equalities for commutator of generalized quadratic matrices are given out as well. All of these have many applications, and generalize the corresponding results of J.Gr????, G. Trenkler, J.J.Koliha, Y.Tian and G.P.H.Styan and so on about idempotent matrices and involutory matrices. Finally, we give out a rank equality, which unifies various rank equalities about the invariance of rank of matrices.Finally, a brief conclusion and a prospect are made in the last chapter, it points out that many researches maybe extended to quadratic operators.