The Research On The Idempotency Of The Linear Combinations Of Three Matrices

The idempotency of the linear combinations of the matrices has important applications in statistics and coding theory. In statistics, the problem that the linear combination of several quadratics obeyχ2-distribution whether obeysχ2-distribution may converse to the idempotency of the linear combination of several idempotent matrices in algebra. In coding theory, the certification of information transmission is an important problem. For the information, its mathematics model is an idempotent matrix, many authors all utilize the standards of idempotent matrices in constructing authentication. Hence, idempotent matrices have important effects in constructing authentication, and the research on the idempotency of the linear combinations of the matrices has important significance.First, this paper gives the research background, the research significance and the research status of the idempotency of the linear combinations of the matrices. Next, we introduce some basic knowledge of matrix theory. In the last two Chapter, we gives the main conclusions in this paper, the main results are follouing:1. Let P1 and P2 be idempotent matrices of order n, and P3 be an arbitrary complex matrix of n-order, which satisfy P1, P2 and P3 are commutative each other. Then the necessary and sufficient conditions of the idempotency of the linear combination P= c1P1+c2P2+c3P3 are given, where c1,c2,c3 are nonzero complex numbers.2. Let P1 and P2 be involutive matrices of order n, and P3 be an arbitrary complex matrix of n-order, which satisfy P1, P2 and P3 are commutative each other. Then the necessary and sufficient conditions of the idempotency of the linear combination P= c1P1+c2P2+c3P3 are given, where c1,c2,c3 are nonzero complex numbers.3. Let P1 be an idempotent matrix of order n, P2 be an involutive matrix of n-order, and P3 be an arbitrary complex matrix of order n, which satisfy P1, P2 and P3 are commutative each other. Then the necessary and sufficient conditions of the idempotency of the linear combination P=c1P1+c2P2+c3P3 are given, where c1,c2,c3 are nonzero complex numbers.4. Let P1,P2 and P3 be are tripotant matrices of order n, which satisfy P1, P2 and P3 are commutative each other. Then some sufficient conditions of the idempotency of the linear combination P= c1P1+c2P2+c3P3 are given, where c1,c2,c3 are nonzero complex numbers.