The Inverse And Generalized Inverse Of Special Combinations Of Idempotent Matrices

Any square matrix over complex number field can be represented by linear combinations of three idempotent matrices and any bounded linear operator of a Hilbert space can be written as a combination of at most five idempotent operators.These show that linear combination and combination of idempotents play an important role on various algebraic systems.By utilizing the properties of idempotent elements and various concepts of inverse,linear combination and combination of two and three idempotents are studied.The relationship between inverse,Drazin inverse,Groups of the combination and the combinations coefficients are characterized.The formulars for these inverses are also presented.The main contents are the following:The relation between the invertibility of the linear combinations of two and three idempotents over skew field and the coefficients of the combination are investigated.The relation between the inverse of combinations of two idempotent matrices over skew field and the combination coefficients are also studied.The Drazin inverse and its index of a special combination of two idempotents matrices over complex number field are discussed under four conditions.The necessary and sufficient condition for the existence of Group inverse of a special combination of two idempotent matrices over complex number field is proved under a condition.The formular for this combination is also presented.