[理工] [工數] [矩陣] 證明特徵值

[iamwwj (阿Ken)]

Show that the eigenvalues of [αβ] are real.
                             [βγ]
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這題的λ是α和γ沒錯吧?(目測)

請問該怎麼證明它們是實數呢??

帶數字進去嗎??還是?

題目短短一行卻不知怎麼下筆

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◆ From: 220.139.149.158
→ kagato:A^T X = AX , AX=λX下去證
  [αβ]^T  [X1]      [αX1 βX2]     [αβ][X1]
  [βγ]    [X2]  =   [βX1 γX2]  =  [βγ][X2]

  [αβ][X1]      [λ1  0][X1]
  [βγ][X2]  =   [0  λ2][X2]

故λ1=α   λ2=γ   β=0    特徵值為實數     (寫對了?)





※ 編輯: iamwwj          來自: 220.139.149.158      (03/22 21:13)
→ iamwwj:謝謝kagato大指點^^
推 kagato:(AX)^T=(λX)^T,X^T*A^T=λ*X^T,X^T(A^T*X)=λ(X^T*X)
→ kagato:X^T(AX)=λ'(X^T*X),X^T(λX)=λ'(X^T*X)
→ kagato:(λ-λ')X^T*X=0 , λ=λ' , '表示共厄
→ iamwwj:我懂了,寫的很清楚,謝謝kagato大^^