Re: [線代] 試證 此矩陣為hermitian
※ 引述《pennyleo (落日黃花)》之銘言：
: 考慮一有限維度n維矩陣
: 若此矩陣存在n個互為正交的eigenvector
: 且這些eigenvector對應到的eigenvale皆為實數
: 試證明
: 此矩陣為hermitian矩陣
: 誰能幫忙解答
: 謝謝
Assume that {v_i} is a set of orthonormal eigenvectors
of A, where A:V->V is the given linear operator.
Since V has dimension n, {v_i} is orthonormal, it forms
a basis for V. For every vector v in V, v can be expressed
in terms of linear combination of {v_i}:
v=Σ <v,v_i>v_i.
Then Av=Σ<v,v_i>Av_i==Σλ_i<v,v_i>v_i, where λ_i
is the corresponding eigenvalue of A w.r.t. v_i.
Hence
_____
<Av,w>=Σλ_i<v,v_i><w,i_i>
= <v,Aw>
Hence A is Hermitian.
--
※ 發信站: 批踢踢實業坊(ptt.cc)
◆ From: 128.120.178.219
討論串 (同標題文章)
本文引述了以下文章的的內容：
完整討論串 (本文為第 2 之 3 篇)：