On Idempotent-Hermite Matrices

Matrix theory plays an indispensable rule in many fields such as modern natural science,engineering technique and social science.Idempotent Matrix and Hermite Matrix are two kinds of particular matrix,which were investigated and analysed in many dissertations.In this paper,the author tries to combine these two kinds of matrix into one kind of more particular matrix - Idempotent-Hermite Matrix.The study of Idempotent-Hermite Matrix is an important part of matrix theory, and it is getting to be an active and important direction in the field of application mathematics.This kind of matrix has good qualities and constructions,so it is necessary to extend it and to discuss its special charaters,constructions,forms, decomposition of matrices,and profound contacts with space, Moore-Penrose inverse.The following is the study contents in this paper:1.Offering the condition that two Idempotent - Hermite Matrices' um,difference and product is Idempotent - Hermite Matrix,Posing the conception of anti-Idempotent - Hermite Matrices,Offering the condition that two anti-Idempotent -Hermite Matrices' um,difference and product is anti-Idempotent - Hermite Matrix,and the bound of Idempotent - Hermite Matrix' s Rayleigh-Ritz-quotient2.Posing some special characters,computing methods and expressions of Idempotent - Hermite Matrix in space,and studing how does space or subspace show by Idempotent - Hermite Matrix.3.Offering the full rank decomposition and spectral decomposition of Idempotent - Hermite Matrix and their applications.The formula of combination and meeting of two subspaces expressing by Idempotent - Hermite Matrix.4.Studying the relatioship between Idempotent - Hermite Matrix and other generalized inverse,offering the formula of Idempotent - Hermite Matrix expressing by Moore-Penrose inverse and the formular of sum,difference,product of two Idempotent - Hermite Matrixes expressing by Moore-Penrose inverse.