# expectation of the function of Wishart matrix eigenvalues

For Given a $$N×M$$ random complex gaussian matrix $$X$$ where $$M=XX^H$$, let $$\lambda_1>\lambda_2>....>\lambda_N$$ be the ordered eigenvalues of $$M$$ my objective is to get an estimation of \begin{align} f = {(\sum\limits_{i = 1}^{Min(N,M)} {\frac{1}{{\sqrt {{\lambda _i}} }}} )^2} \end{align}

Let me take $$N< M$$. For $$N\gg 1$$ the fluctuations in the sum $$\sum_i \lambda_i^{-1/2}$$ are smaller than the expectation value by a factor $$1/N$$, so we may estimate $$\mathbb{E}[f]\approx\left(\mathbb{E}\left[\sum_i\lambda_i^{-1/2}\right]\right)^2=N^2\left(\int \rho_{\rm MP}(\lambda)\lambda^{-1/2}d\lambda\right)^2,$$ with $$\rho_{\rm MP}(\lambda)$$ the Marchenko-Pastur distribution.
For $$N=M$$ the expectation value of $$f$$ is divergent.