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linear algebra – Change of Variables in a Gaussian integral in matrix figure Answer

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linear algebra – Change of Variables in a Gaussian integral in matrix figure

I’ve an issue through which I’ve to compute the next integral: $$mathop{idotsintlimits_{mathbb{R}^okay}}_{sum_{i=1}^{okay}y_i=x} e^{-N^2r(sum_{i=1}^{okay}y_i^2-frac{1}{okay}x^2)} dy_1dots dy_{okay},$$
the place this notation signifies that I wish to combine over $mathbb{R}^okay$ restricted to the monotonous the place $sum_{i=1}^{okay}y_i=x$ (a convolution of gaussians) and $N$ and $r$ are optimistic actual constants. I’ve tried two completely different strategies for computing this integral, however they’re yielding completely different outcomes. I’d value it very mighty if somebody may have a look and inform me what I’m doing grievance.

Method 1

In system 1 I simply wrote it as
$$mathop{idotsintlimits_{mathbb{R}^okay}}_{sum_{i=1}^{okay}y_i=x} e^{-N^2r(sum_{i=1}^{okay}y_i^2-frac{1}{okay}x^2)} dy_1dots dy_{okay}=int_{-infty}^{infty}dotsint_{-infty}^{infty} e^{-N^2r((x-y_1)^2+sum_{i=1}^{k-2}(y_i-y_{i+1})^2+y_{k-1}^2-frac{1}{okay}x^2)} dy_1dots dy_{k-1}=sqrt{frac{1} {pi r^{k-1}okay}}frac{pi^{okay}}{N^{k-1}}$$

I deduced this components by induction, first integrating in $y_{k-1}$, then $y_{k-2}$ and so forth.

Method 2

In system 2 I attempted writting the duty in a matrix figure $$mathop{idotsintlimits_{mathbb{R}^okay}}_{sum_{i=1}^{okay}y_i=x} e^{-N^2r(sum_{i=1}^{okay}y_i^2-frac{1}{okay}x^2)} dy_1dots dy_{okay}=mathop{idotsintlimits_{mathbb{R}^okay}}_{sum_{i=1}^{okay}y_i=x} e^{-N^2r(vec{y},Qvec{y})} dy_1dots dy_{okay}$$
the place commence{equation}
Q:=left(commence{array}{cccccccc}
(1-frac{1}{okay})& -frac{1}{okay} & -frac{1}{okay} & cdots & -frac{1}{okay}
-frac{1}{okay} & (1-frac{1}{okay}) & -frac{1}{okay} & cdots & -frac{1}{okay}
vdots & ddots & & &vdots
-frac{1}{okay} & dots & &-frac{1}{okay} &(1-frac{1}{okay})
aim{array}privilege)
aim{equation}
.

This matrix $Q$ has eigenvalues $lambda_0=0$, $lambda_l=1$ and corresponding normalized eigenvetors commence{equation}
vec{lambda}_l=frac{1}{sqrt{okay}}left(commence{array}{c}
1
e^{frac{2pi i}{okay}1l}
vdots
e^{frac{2pi i}{okay}(k-1)l}
aim{array}privilege)
aim{equation}
for $0le lle k-1$.

As I grasp it, the restriction within the integral signifies that I should not combine within the $lambda_0$ route, since on this route I will need to have all elements equal, and the one place the place the elements are equal and the certain is happy is $(frac{x}{okay},dots,frac{x}{okay})$. So my integration ought to occour within the orthogonal complement of this vector, which is a hyperplane of dimension $k-1$. Everything appears to bridle thus far, so I diagonalized the matrix $Q=ULambda U^{-1}$ and so

$$(vec{y},Qvec{y})=(vec{xi},Lambdavec{xi})=sum_{i=1}^{k-1}xi_i^2.$$

The change of variables $vec{xi}=U^{-1}vec{y}$ has a Jacobian $frac{1}{sqrt{okay^{k-1}}}$, since $U^{-1}$ is the DFT matrix instances $frac{1}{sqrt{okay^{k-1}}}$ and the DFT matrix is understood to breathe unitary. So

$$mathop{idotsintlimits_{mathbb{R}^okay}}_{sum_{i=1}^{okay}y_i=x} e^{-N^2r(vec{y},Qvec{y})} dy_1dots dy_{okay}=idotsintlimits_{mathbb{R}^okay} e^{-N^2rsum_{i=1}^{k-1}xi_i^2} frac{1}{sqrt{okay^{k-1}}}dxi_1dots dxi_{k-1}= sqrt{frac{pi^{k-1}}{okay^{k-1}r^{k-1}}}frac{1}{N^{k-1}}$$.

These two outcomes are completely different and I can’t design out why.

Thank you all in forward in your ameliorate!

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