Retentivity Of Linear Combinations Of K-Generalized Idempotent(Involutory) Matrix And Any Matrix

In recent years, the relative problems about of linear combination retentivity of theidempotent matrix have become a hot topic. Firstly, this paper gives the concepts of thefour class special matrixes. Based on this, the quality and the form of diagonalizationhave been studied. Finally using the diagonalization form of the four class specialmatrixes, the following study has been conducted.1. WhenA、B C^n×nandA^k lA, AB BA,the linear system c₁A＋c2_B （c1、c2C/{0}） is the necessary and sufficient condition of t-generalized idempotentmatrix, t-idempotent matrix, t-generalized involutory matrix and t-involutory matrix.2. WhenA、B C^n×nandA^k A, AB BA, the linear system c₁A＋c2_B （c1、c2C/{0}） is the necessary and sufficient condition of t-generalized idempotentmatrix, t-idempotent matrix, t-generalized involutory matrix and t-involutory matrix.3. WhenA、B C^n×nandA^k lI, AB BA, the linear system c₁A＋c2_B （c1、c2C/{0}） is the necessary and sufficient condition of t-generalized idempotentmatrix, t-idempotent matrix, t-generalized involutory matrix and t-involutory matrix.4. WhenA、B C^n×nandA^k I, AB BA, the linear system c₁A＋c2_B （c1、c2C/{0}） is the necessary and sufficient condition of t-generalized idempotentmatrix, t-idempotent matrix, t-generalized involutory matrix and t-involutory matrix.