Re: [線代] 試證 此矩陣為hermitian

[herstein (翔爸)]

※ 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.

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