I found that $\mathbf H^t\mathbf H$ known as an Wishart matrix, when each row of $\mathbf H$ is an realization of i.i.d. Gaussian random vector of zero mean and identity covariance matrix ($\mathbf H$ is square). Then what is the expectation of $(c\mathbf I+\mathbf H^t\mathbf H)^{-1}$? (Expectation of scaled identity plus square Wishart matrix)
For an invert Wishart matrix of tall size, I found that its expectation is well-known as a scaled identity matrix. However, $\mathbf H$ is square and an identity matrix is added in this case.
