Let $\textbf{W}=\textbf{H}^{\dagger}\textbf{H}$ with $\{\cdot\}^{\dagger}$ being conjugate transpose operator. In some materials I have read so far, there is a common statement that the $k$-th diagonal element of the inverse matrix $\textbf{W}^{-1}$ can be presented as \begin{align} [\textbf{W}^{-1}]_{k,k} = \frac{1} {\textbf{h}_k^{\dagger}\textbf{h}_k -\textbf{h}_k\textbf{H}_k(\textbf{H}_k^{\dagger}\textbf{H}_k)^{-1}\textbf{H}_k^{\dagger}\textbf{h}_k} \end{align} where $\textbf{h}_k$ is the $k$-th column of $\textbf{H}$, and $\textbf{H}_k$ is the submatrix obtained by striking $\textbf{h}_k$ out of $\textbf{H}$.
I understand that the above expression must be derived from a certain analysis relating to Schur complement, but I cannot give a proof myself.
Would you please show me a proof for the above-mentioned expression?