Integral of a complex exponential

[mhill]

Homework Statement

let be $$A_{i,j}$$ a Hermitian Matrix with only real values then

$$\int_{V} dV e^{iA_{j,k}x^{j}x^{k}}= \delta (DetA) (2\pi)^{n}$$ (1)

Homework Equations

$$\int_{V} dV e^{iA_{j,k}x^{j}x^{k}} = \delta (DetA) (2\pi)^{n}$$

The Attempt at a Solution

the idea is that the integral (1) is divergent when the Matrix A is not invertible Det=A and the Dirac delta is not defined at x=0 ,for example if detA=0 then at least one of the eigenvalues is 0 so the exponential takes the value 1 and the integral is divergent

 

[Avodyne]

Try working in the basis in which A is diagonal.