Specifically, the operator $\circ$ denotes the hadamard product, the matrix $\mathbf{T}$ is a low rank toeplitz PSD matrix , the matrix $\mathbf{A}$ is a matrix in vandermonde structure and its entries $A(m,n)=e^{-j(m-1) \pi sin(\theta_n)},\theta_n=[-\pi/2,\pi/2]$, the matrix $\mathbf{C}$ is a random matrix whose entries are only $0$ or $1$. And we have already ensured that the matrix $\mathbf{T} - \mathbf{A} \mathbf{A}^H$ is a PSD matrix.
Is there any possibility that $\mathbf{T} - ( \mathbf{C} \circ \mathbf{A} )( \mathbf{C} \circ \mathbf{A} )^H$ is also PSD? If it is not PSD, how can we modify the matrix $\mathbf{C}$ to make it a PSD matrix?