Multivariate complex Gaussian integral proof by induction on matrix dimension.

I'm looking for a hint on how to carry out an inductive proof of the following equality:

$$\int_{\mathbb{R}^n}dx\,e^{-\frac12 x^TAx}=\frac{(2\pi)^{n/2}}{\sqrt{\det A}} .$$

where $A$ is a symmetric $n \times n$ complex matrix with positive definite real part. I know this can be proven by diagonalizing $A$, since every symmetric matrix can be diagonalized, but I am interested in a proof by induction.

I can't seem to figure out how to carry out the inductive step, since the dimension of the matrix changes and I don't see how I can relate the $n+1$ dimensional case to the $n$ case. I suppose there must be some standard technique when doing induction on matrix dimension, but I don't know how to do start.

• You can prove that every symmetric matrix can be diagonalized by using induction. This can be implemented in your proof. – Sungjin Kim Nov 6 '16 at 18:06
• I suppose so, although I wouldn't know how to start that either. And I don't know if that is the point of this exercise. – Spine Feast Nov 6 '16 at 18:10