http://i.imgur.com/F5KiXt1.jpg
想請問第三題該如何證明
想了很久都想不出如何利用rank=3的條件
麻煩各位大大幫忙解惑

det(A-λI)=0, since A is singular, det(A)=0, we haveλ=0 is a eigenvalue of A. dim{N(A-λI),λ=0} = dim{N(A)} = nullity = n-r = 4-3 = 1 = GM(λ=0)<=AM(λ=0). since A is 4x4, and λ≠0 has multiplicity 3 , so AM(λ=0) = 4-3 = 1.GM(λ=0)應該是1 你算錯了 不過AM>=GM 要對角化每個λ一定要AM=GM(各eigenspace的基底總和=4) 題目沒說清楚 不過從答案來看應該是指AM=GM的情況

所以這題的λ不為0的eigenvalue不用去驗證am=gm囉？ 從題目好像沒辦法驗證