Consider the following $(n+1)\times(n+1)$ real matrix $$A=\begin{pmatrix}a&p^t\\p&D\end{pmatrix},$$ where $D$ is an $n\times n$ diagonal matrix with strictly positive entries, $a>0$, and also elements of the vector $p$ are non-negative (some could be zero).

Let $A$ be positive semidefinite: $x^*Ax\geq0$ $\forall x\in\mathbb{C}^{n+1}$.

Is it possible to determine the eigenavlues of $A$ in terms of $a,D,$ and $p$? If not, how much can be said about the eigenvalues of $A$ except that they are non-negative and majorizes $\{a,d_i\}$?


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