Regular readers of this blog will know that I love my integrals. For those that share the same enjoyment, I recommend this channel that often covers integrals from the MIT integration bee. For example, see this case of writing an integral in terms of itself. Very cool.

In this post, I want to think about the m-dimensional Gaussian. It is a common integral in QFT. Here’s how we compute it when of the form,

Two comments: is a real symmetric matrix, . X is a column vector,

Since is real and symmetric, we make use of a result from spectral theorem. In particular, consider the following spectral theorem proposition:

This implies has eigenvalues . It can also be diagonalised into a matrix by an orthogonal matrix , such that

Here, it should be highlighted is an matrix. From these considerations,

At this point, we perform variable substitution. Notice,

From the 1-dimensional case, we know: . So,

Now, recall that:

From this, we can simplify our last result