I know that if I generate a random matrix $\mathbf{A}\in\mathbb{C}^{n\times n}$ in MATLAB, then $\mathbf{B} = \mathbf{A}\mathbf{A}^{\dagger}\in\mathbb{C}^{n\times n}$ is a Hermitian (i.e., $\mathbf{B}^{\dagger} = \mathbf{B}$) positive semidefinite (PSD) matrix (i.e., $\mathbf{B} \succeq 0$).
In my case, $n = 8$ and I want the rows $\left[\mathbf{B}\right]_{3,:}$ to $\left[\mathbf{B}\right]_{8,:}$ to be equal to the null vector, i.e., $\left[\mathbf{B}\right]_{n,:} = \begin{bmatrix} 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\end{bmatrix}$ for $n=3,\dots,8$.
If I create the Hermitian PSD matrix $\mathbf{B}$ from a random matrix $\mathbf{A}$ as mentioned above and then replace rows 3 to 8 by the null vector, I will lose both the Hermitian and PSD properties.
My question is the following: is there a way to generate an $8\times 8$ Hermitian PSD matrix with rows 3 to 8 being equal to the null vector?
Thanks.