1. Use similarity. If A is similar to B, there is a invertible matrix P such that P (- 1) AP = B, and A N = Pb NP (- 1). In order to simplify the operation, the matrix B similar to A is generally the diagonal matrix of A or Jordan standard form:

(1) diagonal matrix: that is, B=diag{λ 1, λ2, ..., λm}, the multiplication of two diagonal matrices is still a diagonal matrix, and each element on the diagonal is the product of the corresponding position elements of two corresponding matrices;

(2)Jordan canonical form: then B is a block diagonal matrix, and each block on the main diagonal is a Jordan block, which can be expressed as aE and shape.

[0 1 0 ...0 0]

[0 0 1 ...0 0]

[.........]

[0 0 0 ...0 1]

[000 ... 000] (denoted as C), if Jordan block M=aE+C, then m n = (AE+C) N, which is expanded according to binomial theorem, because C (if C is of order S) is a nilpotent matrix with nilpotent exponent of S (that is, C = 0, C (.) Calculate the n power of each Jordan block respectively, and then.

2. Direct expansion with binomial theorem. Similar to the above method, if A can be directly expressed as the sum of a diagonal matrix and C, then it can be directly expanded into a n = (AE+c) n by binomial theorem.

3. Use mathematical induction. If the order of a is indefinite, the elements in a are not constant, and a is abstract, mathematical induction is usually used. Write down the first few items A, A 2, A 3 ..., try to find the law, and then prove your conclusion by mathematical induction.