In the study of measure transformation in Gaussian space, there is a fundamental issue that I want to write down here, which is Fredholm determinant. Let me try to use half an hour to explain the intuition of this determinant in matrix form with finite dimension.

So suppose a matrix , our goal is to find out what definition is that makes sense for trace class operator(i.e.). We know that its trace , and assume the eigenvalues of are , we have

, so

.

And because

Iet’s take a look at these terms in the summation,

when , it’s 1,

when , it’s ,

when , it’s ,

which is the trace of operator on a linear space with basis where is is the wedge product form.

Similarly, we have the expression given in widipedia about “Fredholm determinant” that for a general trace-class operator

and this new operator is a linear operator on space formed by the basis

Finally, I want to mention that when , easy to check that indeed.

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