# expectation of the trace of the square root of wishart matrix

Let $$X(N,N)$$ be Wishart matrix with rank(X)=K in order to estimate the expectation of the trace of the square root of X i.e $$X^{1/2}$$ I want to know if is possible to use the unordered Wishart distribution function to estimate this value?

\begin{align} E[trace(\sqrt X )]=? \end{align}

Since the trace is invariant under unitary transformations, you can work in a basis where $$X$$ is diagonal, with nonzero elements $$x_n$$, $$n=1,2,\ldots K$$ on the diagonal; denote by $$P(x)$$ their marginal distribution; then $$\mathbb{E}[{\rm tr}\, \sqrt X]=\mathbb{E}\left[\sum_{n=1}^K\sqrt{x_n}\right]=K\int P(x)\sqrt{x}\,dx.$$ For $$K\gg 1$$ you can use the Marcenko-Pastur distribution for $$P(x)$$.
• @ Carlo Beenakker sir can I used the distribution of the unordered eigenvalue \begin{align} f(\lambda ) = \frac{1}{K}{\sum {\frac{{i!}}{{(i + N - K)!}}{{[L_i^{N - K}(\lambda )]}^2}\lambda } ^{N - K}}{e^{ - \lambda }}\end{align} wher $L$ represent the lagrangien polynom Jun 11, 2020 at 13:44