Does this Dirac Delta Exist in multivariate form?

Question

Is there a multi-variate equivalent to the Dirac-Delta? Like $\delta (X - A)$ that is infinity whenever $X=A$ and zero everywhere else and X, A positive-definite n by n squares so that I could integrate this over the Haar measure say as in \begin{eqnarray} \int_{X>0} \delta (X-A)=I_n \end{eqnarray} ? Thanks!

Answer 1

Yes, you can use the identity \begin{eqnarray} \delta(X-A) = \int_{-\infty}^\infty \frac{e^{ik(X-A)}}{2 \pi}dk \end{eqnarray}. Then you can compute the Matrix exponential.