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[Abstract]
The properties of class I matrix equation solution were studied in this paper. Firstly it proposed a new concept of idempotent matrix by defining the idempotent matrix (idemfactor), then it studied the relevant properties by making reference to properties of some idempotent matrixes, thereafter it summed up some principles. Discussions were focused on issues such as the determinants, orders, traces and eigenvalues of idempotent matrixes as well as the diagonalizability. Firstly the values of matrix determinants were calculated, then the formulae for idempotent matrix traces were developed and the eigenvalues of idempotent matrixes were studied, thereafter such matrixes were demonstrated to be diagonalizable and worked out the sufficient conditions for idempotent matrix invertibility, and finally solutions to some idempotent matrix equation were investigated.

Properties for a class of matrix equations in this paper, first of all, through the definition of idempotent matrix, put forward the new concept - idempotent matrix, and then studied on the nature of the reference some of the properties of idempotent matrix, the induction summarizes some theorems. Focus on - idempotent matrix characteristic value, determinant, rank, trace, and may the diagonalization problem, we first calculated the value of the determinant of a matrix, then wrote - idempotent matrix trace formulas, and studied - idempotent matrix characteristic value, and then proves that this kind of matrix diagonalization matrix is also developed, the sufficient condition of idempotent matrix is reversible. Finally some - idempotent matrix equation is studied to solve the problem.