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We analytically compute the large-deviation probability of a diagonal matrixelement of two cases of random matrices, namely $\beta=[\vec H^\dagger\vecH]^{-1}_{11}$ and $\gamma=[\vec I_N+\rho\vec H^\dagger\vec H]^{-1}_{11}$, where$\vec H$ is a $M\times N$ complex Gaussian matrix with independent entries and$M\geq N$. These diagonal entries are related to the "signal to interferenceand noise ratio" (SINR) in multi-antenna communications. They depend not onlyon the eigenvalues but also on the corresponding eigenfunction weights, whichwe are able to evaluate on average constrained on the value of the SINR. Wealso show that beyond a lower and upper critical value of $\beta$, $\gamma$,the maximum and minimum eigenvalues, respectively, detach from the bulk.Responsible for this detachment is the fact that the corresponding eigenvalueweight becomes macroscopic (i.e. O(1)), and hence exerts a strong repulsion tothe eigenvalue.

Tails of Random Matrix Diagonal Elements: The Case of the Wishart Inverse