Decomposition Of Several Types Of Special Symmetric Matrices

In matrix theory,related symmetric matrix theory has wide applications in engineering calculation,numerical analysis,automatic control,optimization theory and system identification.Matrix decomposition has a wide range of applications and rich theoretical research values in the fields of signal processing,neural networks,statistics,cybernetics,and systems science.Therefore,the decomposition of special symmetric matrices has attracted more and more scholars'interest,and has become a hot topic in the field of contemporary mathematics.On the basis of reading a large number of references,this paper further explores the decomposition of several special symmetric matrices,their generalized inverses and group inverses,and then generalizes some useful conclusions.The specific contents include the following aspects:In the first chapter,the research background and current situation in this field were introduced.The definitions of singular value decomposition,full rank decomposition,Moore-Penrose inverse and group inverse and their related properties were given.Finally,the basic structure of this paper was given.In the second chapter,the quasi-row(column)symmetric matrices were mainly be studied.By using the singular value decomposition formula and Moore-Penrose inverse of the mother matrix,the conclusion about the singular value decomposition formula of the quasi-row(column)symmetric matrices and the calculation formula of Moore-Penrose inverse were deduced.The singular value decomposition and Moore-Penrose inverse of quasi-row(column)symmetric matr:ix have a quantitative relationship with singular value decomposition of matrix and Moore-Penrose inverse of matrix was found.This theorem simplifies the calculation of singular value decomposition and Moore-Penrose inverse of quasi-row(column)symmetric matrix.In the third chapter,the quasi-row(colxumn)symmetric matrix was also studied,but in this chapter,the full rank decomposition of the quasi-row(column)symmetric matrix was mainly studied.And the Moore-Penrose inverse of the(column)symmetric matrix was obtained by using the way which the full rank decomposition formula of the quasi-row(column)symmetric matrix,and the relevant theorem conclusions were obtained.The second and third chapters are summarized.The calculation method and steps of the Moore-Penrose inverse of the symmetric(column)symmetric matrix are given.In the fourth chapter,the special central symmetric matrix was mainly studied,and the definition of special central symmetric matrix was given.The full rank decomposition theorem of special central symmetry was derived by the full rank decomposition method of the known mother matrix.On this basis,the further study the group inverse of the special central symmetric matrix was further studied and the correlation theorem was given.The theorem is greatly reduced,but without losing the accuracy.In the fifth chapter,the row(column)conjugate symmetric matrix was mainly studied.Based on the definition of row(column)conjugate symmetric matrix,the singular value decomposition of such row(column)conjugate symmetric matrix was studied.It was found that the singular value decomposition of the mother matrix can be calculated first when calculating the singular value decomposition of the row(column)conjugate symmetric matrix,and then the singularity of the row(column)conjugate symmetric matrix can be calculated simply and quickly according to the given theorem formula.The method promotes the relevant literature conclusions.