# 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}

## 1 Answer

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 – hichem hb Jun 11 '20 at 13:44
• yes, the order is irrelevant. – Carlo Beenakker Jun 11 '20 at 14:50
• sir I have tried to estimate the expectation using the distribution of the unordered eigenvalue. however, when I compare this result to the result obtained numerically (using Matlab) I find that there are not similar? – hichem hb Aug 18 '20 at 16:10
• As a test, you could first try large K and use the Marcenko -Pastur distribution. – Carlo Beenakker Aug 18 '20 at 18:50
• using he Marcenko -Pastur distribution i can't find a finite form of this equation? – hichem hb Aug 18 '20 at 22:12