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 1


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 $P(x)$.

  • $\begingroup$ @ 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 $\endgroup$
    – hichem hb
    Jun 11, 2020 at 13:44
  • 1
    $\begingroup$ yes, the order is irrelevant. $\endgroup$ Jun 11, 2020 at 14:50
  • $\begingroup$ 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? $\endgroup$
    – hichem hb
    Aug 18, 2020 at 16:10
  • $\begingroup$ As a test, you could first try large K and use the Marcenko -Pastur distribution. $\endgroup$ Aug 18, 2020 at 18:50
  • $\begingroup$ using he Marcenko -Pastur distribution i can't find a finite form of this equation? $\endgroup$
    – hichem hb
    Aug 18, 2020 at 22:12

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