Revisiting the Evaluation of a Multidimensional Gaussian Integral ()
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] :
where
is the transpose of every non-zero column vector
and
is a real positive definite quadratic form of
variables. In order to guarantee the convergence of the integrals, we should have
(1)
We can also write
as a sum of its symmetric and skew-symmetric components,
and we have
(2)
since
.
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
:
(3)
where the columns of the matrix
are the orthonormal eigenvectors of the matrix
.
We then have
(4)
where
is the corresponding diagonal form.
From Equation (3) and Equation (4) we have:
(5)
where
are the eigenvalues and
their algebraic multiplicities [2] with
(6)
The transformation of the volume element is
(7)
and we can choose
(8)
from Equation (3) and the adequate organization of the orthonormal eigenvectors as the columns of the matrix
.
The quadratic form can then be written as
(9)
From Equation (8) and Equation (9), the multidimensional integral will result
(10)
since each unidimensional integral is given by
(11)
We finally write, from Equations ((5), (10), (11)),
(12)
and we see from Equation (12) that the original matrix
does not need to be diagonalizable [1] . The usual result of the literature will follows if
, i.e., if
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
vectors:
(13)
where
is an orthonormal basis,
(14)
We now define the matrices
(15)
(16)
The first
terms of the expansion of
will produce null determinants of the
matrix. The
term will correspond to the determinant
times
. The
term will lead to a determinant of a
matrix which is obtained by replacement of
column of the matrix
by a column whose elements are
, times
. The
term will correspond to the determinant of a
matrix which is obtained by replacement of the
column of the matrix
by a column whose elements are
, times
. We can then write,
(17)
It should be noted that if
is a symmetric matrix like
,
the quadratic form
can be written as
(18)
where
From Equation (17), we can write Equation (18) as
(19)
From Sylvester’s Criterion [3] , the quadratic form
is positive definite if and only if all upper left determinants
of the symmetric matrix
are positive. We should note [4] that for each variable
:
(20)
since the other variables
which are contained on the term
do not contribute to unidimensional integrals of the form
where
is a real constant and
a generic function of its arguments.
We then have from Equation (19) and Equation (20):
(21)