[線代] 對角化性質的證明
我想請問在Frideberg的Thm5.9的證明問題:
A liner operator T on a finite-dimensional VS V is diagonalizable iff
the mutiplicity of λ(i) is equal to dim(E(i)) for all i.
For each i,E(i) is the eigenspace corresponding to λ(i),
where λ(i) are distinct eigenvalue, 1≦ｉ≦ｋ
而證明的過程是先假設這個linear operator T是diagonalizable的,
並令一個β是由T的eigenvector所組成,一個V的basis,
再來定義一個β(i)=β∩E(i),並且令n(i)表示β(i)中的向量數目.
接下來的這個不等式就是我的問題:
n(i)≦dim(E(i)) for each i because β(i) is a LID subset of a subspace
with dimension dim(E(i))
後面我先省略不打了,如果覺得這樣不完整沒辦法判斷我再補打~
我想請問的是,β是由T的eigenvector所組成,那E(i)的basis中的向量難道不會通通包含在
β之中嗎?如果包含在其中,那麼β(i)就應該等於E(i),則n(i)≦dim(E(i))中的等式就成
立了,因此我想知道,＂n(i)＜dim(E(i))＂是在什麼情形下發生~
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