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  :
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
We can also write as a sum of its symmetric and skew-symmetric components, and we have
2. Application of the Spectral Theorem of Linear Algebra
From the Spectral Theorem of Linear Algebra  , 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 :
where the columns of the matrix are the orthonormal eigenvectors of the matrix .
We then have
where is the corresponding diagonal form.
From Equation (3) and Equation (4) we have:
where are the eigenvalues and their algebraic multiplicities  with
The transformation of the volume element is
and we can choose
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
From Equation (8) and Equation (9), the multidimensional integral will result
since each unidimensional integral is given by
We finally write, from Equations ((5), (10), (11)),
and we see from Equation (12) that the original matrix does not need to be diagonalizable  . 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:
where is an orthonormal basis,
We now define the matrices
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,
It should be noted that if is a symmetric matrix like ,
the quadratic form can be written as
From Equation (17), we can write Equation (18) as
From Sylvester’s Criterion  , the quadratic form is positive definite if and only if all upper left determinants of the symmetric matrix are positive. We should note  that for each variable :
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):