Multidimensional Normal (Gaussian)
Distribution
Centralized (unnormalized) multidimensional Normal (Gaussian)
distribution is denoted by: N(0, K) where the mean is 0
(zero) and the
covariance matrix is K = ( K_{i,j}
) = Cov(X_{i}, X_{j}) . K is symmetric by the identity, i.e., Cov(X_{i},
X_{j}) = Cov(X_{j} , X_{i}) . Let X = [X_{1},
X_{2}, X_{3}, ...X_{n}]^{T} and ndimensional derivatives are
written as
.
The density function of N(0, K)
is exp ( 1/2 X^{T} K^{1}
X ),
normalizing constant of the N(0, K)
density is:

(1) 
where K = det( K
) is the determinant. In general, if multidimensional Gaussian is not
centered
then the (possibly offset) density has the form: exp [ 1/2 (Xμ)^{T} K^{1} (Xμ) ].
 Because K^{1} is real and symmetric (since (K^{1})^{T}
= (K^{T})^{1} ), if A = K
we can decompose
A into A
= T Λ T^{1},
where T is an orthonormal matrix ( T^{T}
T = I ) of the eigenvectors of A and Λ is a diagonal
matrix of
the eigenvalues of A.
Then the multidimensional Gaussian density can be expressed as:

(2) 
However, T
is
orthonormal and we have T^{1} = T^{T}
. Now define a new vector variable Y = T^{T}
X, and substitute in (2):
where  J  is the determinant of the Jacobian
matrix J = ( J_{m,n} ) = ( ∂X_{m}
/ ∂Y_{n}). As X = (T^{T})^{1}Y
= Y , J ≡ T and thus  J 
= 1.
 Because Λ
is diagonal, the integral (4) may be separated into the product of n
independent Gaussians, each of which we can integrate separately using
the 1D Gaussian:

(5) 
Summarizing:
Orthonormal matrix multiplication does not change the determinant,
so we
have
 A  =  T Λ T^{1} =  T   Λ   T^{1} =
 Λ 
And therefore:

(10) 
Substituting back in for K^{1} ( A = K and  K^{1}  = 1 / K^{} ), we get

(11) 
as we expected.
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