If we have a $P\times N$ matrix $\mathbf{A}$ whose elements $A_i$ are samples (in this case, P samples) from a multivariate gaussian distribution in an $N$ dimensional space, we can define the Wishart random symmetric matrix through


Given this matrix, what is a good way of obtaining an order of magnitude estimation for the largest eigenvalue of $\mathbf{W}$ in the case where the number of samples $P$ is less than the degrees of freedom $N$ (i.e. $P<N$)?

In the end I'm looking to obtain an estimation for the order of magnitude of the eigenvalues $O(\rho(\lambda))(M,N)$ as a function of the number os samples M and the dimensionality of the space N.


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