Journal of Applied Mathematics and Physics, 2017, 5, 449-452 http://www.scirp.org/journal/jamp ISSN Online: 2327-4379 ISSN Print: 2327-4352
Revisiting the Evaluation of a Multidimensional Gaussian Integral R. P. Mondaini, S. C. de Albuquerque Neto Federal University of Rio de Janeiro, Centre of Technology, COPPE, Rio de Janeiro, Brazil
How to cite this paper: Mondaini, R.P. and de Albuquerque Neto, S.C. (2017) Revisiting the Evaluation of a Multidimensional Gaussian Integral. Journal of Applied Mathematics and Physics, 5, 449-452. https://doi.org/10.4236/jamp.2017.52039 Received: January 4, 2017 Accepted: February 18, 2017 Published: February 21, 2017 Copyright © 2017 by authors and Scientific Research Publishing Inc. This work is licensed under the Creative Commons Attribution International License (CC BY 4.0). http://creativecommons.org/licenses/by/4.0/ Open Access
Abstract The evaluation of Gaussian functional integrals is essential on the application to statistical physics and the general calculation of path integrals of stochastic processes. In this work, we present an elementary extension of an usual result of the literature as well as an alternative new derivation.
Keywords Gaussian Integral, Determinants, Spectral Theorem, Sylvester’s Criterion
1. Introduction In the present work, we apply theorems of Linear Algebra to derive and extend an usual result of the literature on evaluation of multidimensional Gaussian integrals of the form [1]: ∞
∫−∞e
− xT Ax
dx1 dxn
where x T is the transpose of every non-zero column vector x ∈ IR n and x T Ax is a real positive definite quadratic form of n variables. In order to guarantee the convergence of the integrals, we should have (1)
x T Ax > 0
We can also write A as a sum of its symmetric and skew-symmetric com-
A + AT = A ponents, 2
A − AT + and we have 2
A + AT = x T Ax x T 2 A − AT since x T 2 DOI: 10.4236/jamp.2017.52039 February 21, 2017
x ≡ 0.
x >0
(2)
R. P. Mondaini, S. C. de Albuquerque Neto
2. Application of the Spectral Theorem of Linear Algebra From the Spectral Theorem of Linear Algebra [2], a real matrix will be diagonalized by an orthogonal transformation if and only if this matrix is symmetric. We then apply an orthogonal transformation to the quadratic form
A + AT xT 2
x: ; θ Tθ 1l = x θ= y ; x T y Tθ T=
(3)
where the columns of the matrix θ are the orthonormal eigenvectors of the
A + AT matrix . 2 We then have
A + AT 2
θT
A + AT θ = 2 d
(4)
A + AT where is the corresponding diagonal form. 2 d From Equation (3) and Equation (4) we have:
A + AT A + AT = = λ1n1 λ2n2 λlnl det det 2 2 d
(5)
where λ1 , , λl are the eigenvalues and n1 , , nl their algebraic multiplicities [2] with (6)
n1 + n2 + + nl = n The transformation of the volume element is
dx1 dxn = det θ dy1 dyn
(7)
det θ = 1
(8)
and we can choose from Equation (3) and the adequate organization of the orthonormal eigenvectors as the columns of the matrix θ . The quadratic form can then be written as
A + AT x T Ax = y T y 2 d n1
= λ1 ∑ y + + λl j =1
2 j
(9)
n
∑
y
j =n − nl +1
2 j
From Equation (8) and Equation (9), the multidimensional integral will result ∞
∫−∞e
− xT Ax
∞
dx1 dxn = ∫−∞e n1
A + AT − yT 2
∞
= ∏ ∫−∞e j =1
− λ1 y 2j
y d
dy j
since each unidimensional integral is given by 450
dy1 dyn n
∏
j =n − nl +1
∞
∫−∞e
(10) − λl y 2j
dy j
R. P. Mondaini, S. C. de Albuquerque Neto 12
π ∞ − λk y 2j dy j = e , k 1, , l. ∫−∞= λk
(11)
We finally write, from Equations ((5), (10), (11)),
12 n ∞ − xT Ax π πn dx1 dxn = e = ∫−∞ nl n1 n2 A + AT λ1 λ2 λl det 2
12
(12)
and we see from Equation (12) that the original matrix A does not need to be diagonalizable [1]. The usual result of the literature will follows if A = AT , i.e., if A is itself a symmetric matrix.
3. Application of Sylvester’s Criterion Theorem We now present an alternative derivation of the result obtained above. We will show that there is no need to apply an orthogonal transformation to diagonalize a quadratic form in order to derive formula (12). Let us write the IR n vectors:
= x
n
n
x j eˆ j , b j ∑b jk eˆk ∑=
(13)
k 1 =j 1 =
where eˆ j , j = 1, , n is an orthonormal basis, eˆ j ⋅ eˆk = δ jk
(14)
We now define the matrices
b11 b1 j −1 b1 j b b b j j jj −1 j1
b1 ⋅ eˆ1 b1 ⋅ eˆ j −1 b1 ⋅ eˆ j B j × j = = b ⋅ eˆ b ⋅ eˆ b j ⋅ eˆ j j j −1 j 1 b1 ⋅ eˆ1 b1 ⋅ eˆ j −1 b1 ⋅ x Bx = = j× j b ⋅ eˆ b ⋅ eˆ b j ⋅ x j j −1 j 1 The first
( j − 1)
nants of the B x
j× j
b11 b1 j −1 b b jj −1 j1
b1 ⋅ x b j ⋅ x
(15)
(16)
terms of the expansion of x will produce null determimatrix. The j th term will correspond to the determinant
det B j × j times x j . The
( j + 1)
th
term will lead to a determinant of a B j +1 j× j
matrix which is obtained by replacement of j column of the matrix B j × j by a column whose elements are b1 j +1 b jj +1 , times x j +1 . The nth term will corth
respond to the determinant of a Bn matrix which is obtained by replacement j× j of the j th column of the matrix B j × j by a column whose elements are b1n b j n , times xn . We can then write,
= det B x x j det B j × j + j× j
n
∑ xk det Bkj× j
k = j +1
It should be noted that if Bn×n = B is a symmetric matrix= like B
(17)
A + AT , ∀A , 2 451
R. P. Mondaini, S. C. de Albuquerque Neto
the quadratic form x T Bx can be written as 2
det B xj × j n x T Ax ≡ x T Bx = ∑ det B j =1 j −1× j −1 det B j × j
(18)
where
det B0×0= 1, det B1×1= b11 , det B x = b 1⋅ x 1×1
From Equation (17), we can write Equation (18) as
det B j× j 1 = x Ax ∑ x j + det B j× j j= 1 det B j −1× j −1 n
T
n
∑ xk det Bkj× j
k = j +1
2
(19)
From Sylvester’s Criterion [3], the quadratic form x T Bx is positive definite if and only if all upper left determinants ( det B j × j , j = 1, , n ) of the symmetric matrix B are positive. We should note [4] that for each variable x j : n 1 − xj + xk det B k det B j −1× j −1 det B j× j j× j k = j +1
∑
det B j× j
∞
∫−∞e
2
π dx j = det B j× j det B j −1× j −1
12
(20)
since the other variables x j +1 , , xn which are contained on the term 1 det B j × j
n
∑ xk det Bkj× j
k = j +1
do not contribute to unidimensional integrals of the
form ∞
∫−∞e
(
(
−α x j + f x j +1 ,, xn
))
2
π dx j = α
12
where α is a real constant and f a generic function of its arguments. We then have from Equation (19) and Equation (20): 12
12 n πn ∞ − x T Ax π = = e x x d d ∏ det B det B n 1 ∫−∞ j× j j =1 det B j −1× j −1 πn = A + AT det 2
12
(21)
q.e.d .
References [1]
Chaichian, M. and Demichev, A. (2001) Path Integrals in Physics. Vol. 1, Stochastic Processes and Quantum Mechanics, IOP Publishing, Bristol.
[2]
Lay, D.C. (2012) Linear Algebra and its Applications. Addison-Wesley, Boston.
[3] [4]
452
Gilbert, G.T. (1991) Positive Definite Matrices and Sylvester’s Criterion. American
Mathematical Monthly, 98, 44-46.
Yu, B., Rumer, M. and Rivkin, Sh. (1980) Thermodynamics, Statistical Physics and Kinetics. Mir Publishers, Moscow.
Submit or recommend next manuscript to SCIRP and we will provide best service for you: Accepting pre-submission inquiries through Email, Facebook, LinkedIn, Twitter, etc. A wide selection of journals (inclusive of 9 subjects, more than 200 journals) Providing 24-hour high-quality service User-friendly online submission system Fair and swift peer-review system Efficient typesetting and proofreading procedure Display of the result of downloads and visits, as well as the number of cited articles Maximum dissemination of your research work
Submit your manuscript at: http://papersubmission.scirp.org/ Or contact
[email protected] 
