0
$\begingroup$

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}

$\endgroup$
2
$\begingroup$

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 $P(x)$.

$\endgroup$
6
  • $\begingroup$ @ 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 '20 at 13:44
  • 1
    $\begingroup$ yes, the order is irrelevant. $\endgroup$ – Carlo Beenakker Jun 11 '20 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 '20 at 16:10
  • $\begingroup$ As a test, you could first try large K and use the Marcenko -Pastur distribution. $\endgroup$ – Carlo Beenakker Aug 18 '20 at 18:50
  • $\begingroup$ using he Marcenko -Pastur distribution i can't find a finite form of this equation? $\endgroup$ – hichem hb Aug 18 '20 at 22:12

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.