As shown the picture. $V_m$ is a rank-deficient initial matrix. Do the unitary transform first and expand the kernel by a set of orthogonal vectors. These operations make the right side of the matrix 0. How to obtain this unitary transformation?
$$U^{\dagger}V_m U = \begin{bmatrix} V'_{m11} & 0_{(m-k) \times k} \\ V^{'}_{m21} & 0_{k \times k} \end{bmatrix} $$
![enter image description here][2]