On the proof of matrix complex field, we will add 1-2 times points Let a be a matrix of order n over a complex field 1) A is similar to a matrix in the form of: λ1 c12 c13 ... c1n 0 λ2 c23 ... c2n 0 0 λ3 ... c3n ... ... ... ... ... 0 0 0 ... λn 2) A has n eigenvalues (multiple roots count multiple numbers) in complex field, and if λ 1, λ 2,..., λ n are all eigenvalues and f (x) is any polynomial in complex field, then f (λ 1), f (λ 2),..., f (λ n) are all eigenvalues of F (a)

On the proof of matrix complex field, we will add 1-2 times points Let a be a matrix of order n over a complex field 1) A is similar to a matrix in the form of: λ1 c12 c13 ... c1n 0 λ2 c23 ... c2n 0 0 λ3 ... c3n ... ... ... ... ... 0 0 0 ... λn 2) A has n eigenvalues (multiple roots count multiple numbers) in complex field, and if λ 1, λ 2,..., λ n are all eigenvalues and f (x) is any polynomial in complex field, then f (λ 1), f (λ 2),..., f (λ n) are all eigenvalues of F (a)

1. A can be transformed into a Jordan matrix, and then the order of the bases of each radical subspace can be reversed
2. From the basic theorem of algebra, we know that a has n characteristic roots. On the other hand, if a is transformed into a Jordan matrix, then f (a) is a lower triangular matrix, and its diagonal elements are f (λ 1), f (λ 2),..., f (λ n), so f (λ 1), f (λ 2),..., f (λ n) are all the characteristic roots of F (a)