In this note, a simple proof Jordan normal form and rational form of matrices over a field is given.
Let F be a field, M n (F ) the set of n × n matrices over F , λ i ∈ F, n i a natural number, i = 1, 2, • • • , r. A matrix of the form
is called a Jordan normal form, where for each i,
It is known that an n × n matrix over the complex field is similar to a Jordan normal form, see [1]. In this paper, a short proof of this result is given.
Theorem 1 (Jordan) Let F [λ] be the polynomial ring with one variable λ, A ∈ M n (F ) and λE -A the eigenmatrix. Suppose that there exist invertible λ-matrices P (λ), Q(λ) such that
Then in M n (F ), A is similar to (1).
Proof: Let V be a n-dimensional vector space over F with a basis e 1 , . . . , e n . Let A be a linear transformation of V such that
where
Then we have (Aλ 1 ) n 1 y 1 = 0, . . . , (Aλ r ) nr y r = 0. For any α ∈ V , suppose that
Noting that n 1 + • • • + n r = n, it is easy to see that y 1 , (Aλ 1 )y 1 , . . . , (Aλ 1 ) n 1 -1 y 1 , . . . . . . , y r , (Aλ r )y r , . . . , (Aλ r ) nr-1 y r forms a basis of V and the matrix of A under this basis is (1).
The proof is completed.
The following known theorem can be similarly proved.
Theorem 2 Let the notion be as in Theorem 1. Suppose that there exist invertible λmatrices P (λ), Q(λ) such that
where
where for each i,
Proof: By the proof as the same as Theorem 1, y 1 , Ay 1 , . . . , A n 1 -1 y 1 , . . . . . . , y r , Ay r , . . . , A nr-1 y r forms a basis of V and the matrix of A under this basis is (4). The proof is completed.
Remark: If F is algebraically closed, then one can find invertible λ-matrices P (λ), Q(λ) such that (2) holds. Generally, we have (3).
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