w.l.o.g. assume Im{eig(A)} ≠ 0 (o.w. B=A, C=I)
∵ A is real
=> A is not scalar matrix
∴ exist real vector x s.t. R^2 = span{x,Ax}
=> ax + bAx + A^2x = 0 for some real number a and b
represent A using bases {x,Ax}
A|_{x,Ax} = [ 0 -a ] := R
[ 1 -b ]
=> A = P R P^{-1} , where P = [x Ax]
= P [ 0 a ] P^{-1} P [ 1 0 ] P^{-1}
[ 1 b ] [ 0 -1 ]
= BC
where B := P [ 0 a ] P^{-1} and C := P [ 1 0 ] P^{-1}
[ 1 b ] [ 0 -1 ]
are real matrices both with real eigenvalues
( poly(A) = poly(R) = x^2 + bx + a => b^2 - 4a < 0
poly(B) = poly([0 a]) = x^2 -bx -a , D = (-b)^2 + 4a > 0 )
[1 b]
※ 引述《cholauda (cholauda)》之銘言：
: 請教大大~
: For any real 2x2 matrix, A,
: there exists two real 2x2 matrices with real eigenvalues, B & C, such that
: A=BC
: 請問這該怎麼證明？
: 感激!!!
: hint: 可能可以從Jordan form來看?!
: 信站: 批踢踢實業坊(ptt.cc)
: ◆ From: 140.113.150.197
: ※ 編輯: cholauda 來自: 140.113.150.197 (10/25 15:51)
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※ 發信站: 批踢踢實業坊(ptt.cc)
◆ From: 218.168.28.154
※ 編輯: GSXSP 來自: 218.168.28.154 (10/25 22:20)