[理工] 線代
(1)假設T屬於L(V,V),其中V為複內積空間
若<T(u),v>=0,對所有u,v屬於V,則T=O
證:
因為<T(u),v>=0,對所有u,v屬於V
=><T(u),v>=0,對所有u屬於V
第一行到第二行用什麼觀念轉的.....?
(2)A保長度，即||Ax||=||x||，對所有x屬於R^n*1
則A為正交矩陣
證:
對所有x屬於R^n*1,
T T T 2 2 T T
x A Ax=(Ax) Ax=||Ax|| =||x|| =x x=x Ix
T
=>A A=I
所以A為正交矩陣
此題可以這樣證嗎?
(3)
hermatian matrix 跟 unitary matrix有一樣嗎?
我記得hermitian的定義是A^H=A
而unitary則是A^-1=A^H
但我剛在寫小黃的書卻說A is hermitian matrices,
所以AA^H=I,得A^-1=A^H
是我搞錯了，還是老師寫錯了
謝謝
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※ 編輯: KAINTS 來自: 111.70.42.106 (10/13 19:52)
※ 編輯: KAINTS 來自: 123.193.7.20 (10/13 20:51)
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可第二題小黃書中是寫
對所有x,y屬於R^n*1
T T T
y A Ax=(Ay)Ax=<Ax,Ay>=1/4||Ax+Ay||-1/4||Ax-Ay||
T T
=<x,y>=y x=y Ix
T
得A A=I,所以A is an orthogonal matrix
他也不知道A是否可逆ㄟ...
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Let A and B be Hermitian natrices.Please show that the
inverse of the matrix C=AB is also Hermitian.
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