I am awkward to calculate the matrix so I would like to get some help
$exp(y^{T}V^{T}\Sigma^{-1}S_{X}- \frac{1}{2}y^{T}ly)$ is proportional to
$exp(- \frac{1}{2}(y-a)^{T}l(y-a))$
and $a$ is $l^{-1}V^{T}\Sigma^{-1}S_{X}$
$ y = R \times 1 $ vector , $ V = CF \times R $ matrix, $ \Sigma = CF \times CF $ matrix $S_{X}=CF \times 1 $ vector, $ l = R \times R $ matrix.
and I know that symmetric matrix $ B $ is $CF \times CF $ matrix and $B=VV^{T}$
( acutally B and $ \Sigma$ are covariance matrix )
I don't know why abruptly proportional formula is comming.... I would like to know how the proprotional formula come from the $exp(y^{T}V^{T}\Sigma^{-1}S_{X}- \frac{1}{2}y^{T}ly)$
and I try to understand $exp(- \frac{1}{2}(y-a)^{T}l(y-a))$ and It seems that $l$ matrix should be symmetry matrix.
$l = I + V^{T}\Sigma^{-1}NV$ ($I = R \times R $ matrix ) but I am not so sure that I understand $l$ matrix is symmetry matrix. here, $N$ is also symmetry matrix.
$\Sigma \times N = D$ is symmetry matrix multiply symmetry matrix => symmetry and $(V^{T}DV)^{T}=V^{T}D^{T}V=V^{T}DV $(because D is semmetry) => symmetry )
First of all, 2 sites (there are many others) that I have obtained with the following keywords:
"multivariate normal" "completing the square" "simple"
http://granite.ices.utexas.edu/coursewiki/images/f/f4/Multivar_normal.pdf
https://learnbayes.org/index.php
I understand that it is in the context of the reduction of a gaussian distribution (generalization to $n$ dimensions of the "completing the square" technique) that you deal with this formula (where I have dropped the "exp")
$$y^{T}V^{T}\Sigma^{-1}S_{X}Vy- \frac{1}{2}y^{T}ly=C - \frac{1}{2}(y-a)^{T}l(y-a) \ \ \text{with} \ \ a=l^{-1}V^{T}\Sigma^{-1}S_{X}V$$
where $C$ is a constant (you see that I have added a missing $Vy$ and a missing $V$).
I understand that $\Sigma$ is a variance matrix and $S_X$ another one. But I need to know the dimensions of the different matrices, etc... (for example, I assume $\ell$ is a number, but maybe its a matrix) for avoiding to give an inapropriate answer to a misunderstood question.
I am sorry but it is too inefficient (on both sides) if you give insufficient context information or incomplete formulas. Moreover, you should try to show what you have attempted and where you situate the point of misunderstanding.
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