 # 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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