→ Johnfu:感謝~~~ 03/10 15:53
※ 引述《Johnfu (ㄟ...)》之銘言：
: a b
: A = ( ) be a real matrix.
: b c
: show that all eigenvalues or A are positive <=> a > 0 and ac - b^2 > 0
令eigenvalue為λ1, λ2
λ1 + λ2 = trace(A) = a + c
λ1 * λ2 = det(A) = ac - b^2
(=>)
λ1>0, λ2>0 => ac - b^2 = λ1 * λ2 > 0
ac > b^2 > 0 => a>0, c>0 or a<0, c<0
又 a + c = λ1 + λ2 > 0 故 a>0 得證
(<=)
a>0, ac - b^2 > 0
=> λ1 * λ2 = ac - b^2 >0, λ1>0, λ2>0 or λ1<0, λ2<0
ac > b^2 > 0, 又a>0 => c>0
λ1 + λ2 = a + c > 0 故λ1>0, λ2>0 得證
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