T：V→V is self-adjoint over F , dim(V) = n
f(v) = <Tv,v>
= <[T]β[v]β,[v]β> , if β is an orthonormal basis
= < A [v]α,[v]α> , if α is a basis , α={α1,...,αn}
where A = [ <T(α1),α1>‧‧‧<T(α1),αn>]
[ ‧ ‧ ]
[ ‧ ‧ ]
[ ‧ ‧ ]
[ <T(αn),α1>‧‧‧<T(αn),αn>]
so A*=A
T is positive-definite(T>0) is defined by f(v) > 0 for all v≠0
Hence by theorem
1.T>0 iff [T]β>0 iff A>0
2.T>0 iff all eigenvalues of T are positive , denoted by e1>=...>=en>0
Since [T]β and A are both self-adjoint
They can be diagonalized by an orthonormal basis
and the diagonal matrix of [T]β must be [e1 0 .. 0 ]
[0 e2.. 0 ]
[. . . . ]
[0 0 en]
But What about A ??
我們只知道A的所有eigenvalue是正的
而會與e1~en有關係嗎?? 謝謝
(A = E [T]α , E = gram determinant of {α1,...,αn}
這個條件會有幫助嗎??)
--
※ 發信站: 批踢踢實業坊(ptt.cc)
◆ From: 111.243.156.82
※ 編輯: znmkhxrw 來自: 111.243.156.82 (03/22 10:23)