I need to prove a statement in my research. The statement seems to be fundamental linear algebra, and numerical studies in MATLAB supported this statement, but I wasn't able to prove it after a few days of effort... Any insight would be greatly appreciated!

The problem is as follows:

Let $A$ be a $p$-by-$p$ diagonal matrix with distinct positive diagonal elements $a_1, \dots, a_p$. Let $Z := A^{1/2} 1_p$, where $1_p$ is a $p$-vector of ones. Let $$P := I - Z \left( Z^T Z \right)^{-1}Z^T$$ where $I$ is a $p$-by-$p$ identity matrix. Let $\xi_1,\dots,\xi_{p-1}$ be the $p-1$ nonzero eigenvalues of $AP$. Show that $1+t\xi_i$, $i=1,\dots,p-1$ are $p-1$ nonzero eigenvalues of $P+tAP$, where $t \in \mathbb R$ and $t \neq 0$.