[問題] Gaussian integral

[chungweitw (.)]

更新: 問題(1) 已解 ( 0:10am 20100621 )
      問題(2) 已解 ( 0:39am 20100621 )

(1) Let M be a complex n*n matrix.

   (a) All of the eigenvalues of the Hermitian part of M are positive.
       ( Hermitian part of M = (M+M^{\dagger}) / 2 )

   (b) All of the real part of eigenvalues of M are positive.


 請問(a)和(b) 有存在任何的 sufficient or necessary conditions 嗎?

Ans:
(a) => (b). However, (b) can not imply (a).

(a)=>(b)
Decompose M into the form M = H + A,
where H is the hermitian part of M while A is its anti-hermitian part.

If the hermitian part of M is positive definitie,
it can be shown that ,for any given vector z with its conjugate denoted
by z*,

Re( z* M z )
= Re( z* H z + z* A z )
= Re( ci* xi* H cj xj + di* yi* A dj yj )
= Re( ci* xi* H cj xj )
> 0
where z = ci xi = di yi.
xi and yi are eigenvectors of H and A, respectively.

Therefore,
for any eigenvector z of M with the norm ||z||=1,
the real part of eigenvalue corresponding to z is

Re(z* M z) > 0.


(b) can not imply (a):

example

          [ -1     3I/2 ]
Let M =   [ 3I/2    2   ]

I : imaginary number

The eigenvalues are positive, but the hermitian part is not positive definite.






(2) For a Gaussian integral with multiple complex variable xi* and xi
    , where i = 1, ....,n,

   the integral turns out to be 1 / det M

   if the exponent of the integrand is - xi* Mij xj,

      and "the Hermitian part" of M is positive definite.


   ( Here, I have used the Einstein summation convention )


proof:

If M is positive definitive Hermitian, the proof is easy.

The problem is what it becomes if M is not hermitian.

Let M = H + A as I did in the problem (1).

If H is positive definite, the integral converges as the antihermitian
part only contributes the oscillation.

Therefore, the result 1/detM can be analytically continuated from the
real axis to the complex plane around the real axis where the integral
is convergent if H is positive definite.









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◆ From: 24.250.202.249
※ 編輯: chungweitw      來自: 24.250.202.249       (06/19 21:04)
推 xgcj:第二的你只要講矩陣對角化之後就出來了
推 sunev:第一的M是normal嗎?
→ chungweitw:第二個不是對角化就出來
→ chungweitw:第一個M 不是normal...就任意矩陣. 不過, 如果是normal
→ chungweitw:的話, 就會等價嗎?
→ chungweitw:再補充第二個...M 不是 Hermitian matrix
→ chungweitw:所以似乎不是那麼顯然
→ xgcj:原PO可以跟我們講講對解第二題的看法嗎?我想知道你的想法~
→ chungweitw:老實說, 想不出來.
→ xgcj:如果可以對角話答案就很明顯了~如果不行對角化(我猜答案是無
→ xgcj:限大)
→ chungweitw:M 可以對角化. 但是 Jacobian 怎麼辦?
→ chungweitw:M 不是Hermitian..
→ chungweitw:一般書上都直接拿 M is hermitian 來證明.
→ chungweitw:然後有些書在以M is hermitian 證完之後, 說其實只要
→ chungweitw:Hermitian part is positive definite 就可以. 但是沒
→ chungweitw:有證明.
推 xgcj:第二題的答案應該是π^0.5*(detM)^(-0.5)
→ xgcj:更正 是π^0.5n*(detM)^(-0.5)      (n是vector的dim)
→ xgcj:你題目有寫錯~(對角化矩陣的det就是Jacobian)
→ chungweitw:題目哪裡錯? 1/detM 該換成 1/detM^{.5} 嗎?
→ chungweitw:No. 應該是 1/detM 沒錯.
→ chungweitw:x* 和 x 共有 2n 個. 所以是 1/(detM^{.5})^2 =1/detM
推 youmehim:我這題目的答案也跟xgcj的一樣 有說M is hermitian
→ chungweitw:M 不需要是Hermitian. 就因為不需要, 我才來問的
→ chungweitw:只要求有positive Hermitian part 就可以了
→ youmehim:可是限定是hermitian答案應該也要一樣吧 XD
→ chungweitw:Hermitian 隨便一本書都有證明.
推 youmehim:所以這裡的Gaussian integral是積2n次嗎?
→ chungweitw:right...因為x* x 各有 n 個.
→ xgcj:這積分是在做路徑積分時會遇到的~
→ chungweitw:dx* dx 定義為 du dv  where x = u+iv
→ xgcj:難怪~跟我算的不一樣
※ 編輯: chungweitw      來自: 24.250.202.249       (06/21 00:10)
※ 編輯: chungweitw      來自: 24.250.202.249       (06/21 00:39)
※ 編輯: chungweitw      來自: 24.250.202.249       (06/21 00:43)
推 youmehim:這...只能推了 XD
→ sneak: 第一的M是normal https://noxiv.com
→ sneak: 老實說, 想不出來. https://daxiv.com
→ sneak: Hermitian p https://daxiv.com
→ sneak: M 不需要是Hermi https://muxiv.com
→ sneak: 所以似乎不是那麼顯然 http://yofuk.com
→ muxiv: 你題目有寫錯~(對角化 https://moxox.com