Consider a diagonal $N\times N$ matrix $\boldsymbol{H}=\text{diag}(\omega_1,\omega_2, \dots,\omega_N)$ with ordering $\omega_1<\omega_2<\dots<\omega_N$, where the eigenvector $\boldsymbol{t}_1$ corresponding to the smallest eigenvalue $\omega_1$ is known. I want to construct an arrow-head matrix (AHM) $\boldsymbol{M}$, with real entries, as given below from $\boldsymbol{H}$ using the known eigenpair $\{\omega_1,\boldsymbol{t}_1\}$
$ \boldsymbol{M} = \begin{pmatrix} \Omega_1 & c_1 & c_2 & \dots & c_N\\ c_1 & \Omega_2 & 0 & \dots & 0\\ c_2 & 0 & \ddots & 0 & \vdots\\ \vdots & \vdots & 0 & \ddots & 0\\ c_N & 0 & \dots & 0 & \Omega_N\\ \end{pmatrix}, \hspace{0.2cm} c_i\neq0,\forall i$.
with elements $c_i$, which are non-zero, but might be positive or negativ. There exists a work of Peng et al.,Linear Algebra and its Applications 416 (2006), 336, who treat such problems for AHM with $c_i>0,\forall i$, but is there a generalization for real $c_i\neq0,\forall i$?