The Generalized Idempotency(Involutivity) Of The Linear Combination Of M-1Times Generalized Idempotent Matrices Of The Power Of K And Any Matrix

The research on the idempotency of the linear combination of idempotent matrices has become an active field. The research has attracted many scholars in domestic and overseas in recent years, yet, there is seldom research on the problem of the linear combination of generalized idempotent matrices.The dissertation researches about the generalized idempotency(involutivity) of the linear combination of m-1times generalized idempotent matrices of the power of k and any matrix, the main results are as follows:1. Let A be an generalized idempotent matrix of the power of k and Am an arbitrary matrix, which satisfy the condition AiAj=AiAj,Then the characteristics of all cases in which the linear combination∑mciAj is an generalized idempotent matrix of the power of k are given, where ci=Ci=1/{0}.2. Let A be an generalized idempotent matrix of the power of k and Am an arbitrary matrix, which satisfy the condition AiAj=AiAj,Then the characteristics of all cases in which the linear combination∑mciAj is an generalized involutive matrix of the power of k are given, where ci=C/i-j{0}.Lastly the dissertation gets the generalized involutive properties of the linear combination of m times generalized idempotent matrices of the power of k in the same way.