# Condition for positive semidefiniteness for this special block matrix

Consider a finite matrix with the following structure

$$\left( \begin{array}{ccccc} 1 & \vec{u} & \vec{u} & \vec{u} &...\\ \vec{u}^T & X & P & P & ...\\ \vec{u}^T & P^T & Y & P & ... \\ \vec{u}^T & P^T & P^T & Z & ... \\ ... & ... & ... & ... & ... \end{array} \right)$$

where, e.g., $\vec{u}=(a, b)$, with $a, b$ real constants, $P$ is a $2 \times 2$ real matrix (eventually such that $P=P^T$), $X = \bigl( \begin{smallmatrix} 1 & x \\ x & 1 \end{smallmatrix} \bigr)$, with $x$ real unknown, and similarly for $Y$, $Z$, ...

Knowing $\vec{u}$ and $P$, I would like to know if there exist some $x,y,z,...$, such that the matrix is positive semidefinite.

Do you know if there is a way of doing this without running a SDP on the full matrix, but for example only on some minors? Or, is it possible to decompose it in a block-diagonal structure, and run the SDP on the blocks?