How to prove that a real matrix is similar to an upper triangular matrix over a complex field, and give a proof (not Jordan matrix)

How to prove that a real matrix is similar to an upper triangular matrix over a complex field, and give a proof (not Jordan matrix)

The premise is the square array
If AX = λ x, X ≠ 0, then take an invertible matrix P = [x, *] with X as the first column to get
P^{-1}AP=
λ *
0 *
Just sum up the lower right corner

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