# Expected eigenvalues of a Wishart Matrix

I consider a $n\times n$ Wishart Matrix with expected value $p \cdot I_n$, i.e. a matrix of the form $$W = XX'$$ with $X$ a $n\times p$ matrix with independent standard normal entries.

It is easy to find in the literature the joint distribution of its eigenvalues $\lambda_1 > \dots > \lambda_n$, e.g. in Muirhead, Aspects of Multivariate Statistical Theory; if I get it right it is proportional to $$\exp\left( -{1\over 2} \sum \lambda_i \right) \left(\prod_i \lambda_i \right)^{(p-n-1)/2} \prod_{i<j} (\lambda_i-\lambda_j).$$ (The integration constant is known but ugly...)

I would like to know its expected value $E( (\lambda_1, \dots, \lambda_n) )$.

Is it known ? If not, could someone point out an easy way to generate vectors following this law for large values of $n$ (and $p$) ?

Post Scriptum I tried to use Gibbs Sampling but it is terribly inefficient: each eigenvalue being constrained by the neighboring values, many iterations are necessary to move away from the starting point.

• I'm confused by your question. The expectation reduces to $(E(\lambda_1),\cdots,E(\lambda_n))$. Are you interested in the expectation of a single eigenvalue then? – Alex R. May 4 '18 at 17:38
• @AlexR. I am interested in the expected value of all the eigenvalues, yes. – Elvis May 4 '18 at 19:33
• I guess I just fell off the turnip truck, but for given values of n and p, how about randomly generating instances of $X$, and for each instance of $X$, computing the eigenvalues of $XX'$? Is that too inefficient? – Mark L. Stone May 7 '18 at 13:24
• @MarkL.Stone I am interested in large values of $n$, for which computing the eigenvalues is very long. – Elvis May 7 '18 at 13:27
• How large are values of n and p? – Mark L. Stone May 7 '18 at 13:30