Expectation of sum of last $k$ eigenvalues of $\mathbf{A}\mathbf{A}^T$

Question

Supposing the eigenvalues of $\mathbf{A}\mathbf{A}^T$ are $\sigma_1,\sigma_2,\cdots,\sigma_{k+1},\cdots\sigma_{n+k}$($\sigma_1\leqslant\sigma_2\leqslant\cdots\leqslant\sigma_{k+1}\cdots\leqslant\sigma_{n+k}$) and the elements of $\mathbf{A}$ are zero mean and unit variance, is there any way to find $\mathbb{E}(\sigma_{k+1}+\cdots+\sigma_{n+k})$?