# Mutual Compatibility/Incompatibility of Quasi-Hermitian Quantum Observables
## 5 Example  
One of the expected outcomes of our present methodological study can be seen in its possible relevance in the theory of quantum gravity. In this context, indeed, many conventional quantization methods cease to be applicable because they rely upon the availability of a classical \"background\" represented, typically, by space or space-time. In the case of a gravitational field, in contrast, one also has to quantize the space-time itself, in principle at least [21]. As a consequence, a consistent quantum theory of gravity must be, in [PERSON]'s words, \"background-independent\" [22]. In this sense, a minimal number of the candidates for observables \(A_{j}\) is two, and the demonstration of their mathematical compatibility is of deep and fundamental relevance indeed. In a narrower setting of quantum cosmology, in addition, one has to quantize classical scenarios in which one often encounters several different forms of singularities. Thus, typically, the latter candidates \(A_{j}\) have to admit the existence of the singularities represented, in the Hilbert-space operator context, by so-called [PERSON]'s [23] exceptional points (EPs). In a realistic setting, these operators even have to be non-stationary and non-Hermitian, and, in parallel, the physical Hilbert-space inner-product metric itself also has to be non-stationary and non-Hermitian. In spite of all of these technical challenges, the practical applicability of the general theory has still been demonstrated and illustrated, in the existing literature, by a number of simple physical examples. A few particularly suitable examples of such a type were described in Ref. [24]. From the purely phenomenological point of view, one of the important merits of the latter classes of models is that all of them admit the non-Hermitian exceptional-point degeneracy [23]. This means that all of them could be understood as mimicking the Big-Bang-like singularity in quantum cosmology. The additional methodological merit is that all of these families of solvable models are represented by matrices \(A_{j}\) of arbitrary dimensions \(N\). As we already mentioned above, nevertheless, we do not need the large-matrix models (because the conditions of their mutual compatibility become over-restrictive at \(N\geq 4\)) or the minimal models with \(N=2\) (for them, the parametric freedom survives). Thus, we decided to pick up, for our present illustration purposes, just the one-parametric sets of the \(N=3\) matrices such that  
\[{A_{1}}^{\dagger}=\left[ \begin{array}{ccc}-2&\sqrt{2-2\,s^{2}}&0\\ -\sqrt{2-2\,s^{2}}&0&\sqrt{2-2\,s^{2}}\\ 0&-\sqrt{2-2\,s^{2}}&2\end{array} \right] \tag{20}\]  
and  
\[{A_{2}}^{\dagger}=\left[ \begin{array}{ccc}-2\,ia&\sqrt{2}&0\\ \sqrt{2}&0&\sqrt{2}\\ 0&\sqrt{2}&2\,ia\end{array} \right]. \tag{21}\]  
They seem totally different (notice that the former one is asymmetric but real, while the latter one is complex-symmetric). Moreover, as long as we can quickly determine their diagonal partners of Equation (8), say,  
\[\mathbf{a_{1}}=\left[ \begin{array}{ccc}0&0&0\\ 0&-2\,s&0\\ 0&0&2\,s\end{array} \right] \tag{22}\]and  
\[\mathbf{a_{2}}=\left[ \begin{array}{ccc}0&0&0\\ 0&-2\,\sqrt{-a^{2}+1}&0\\ 0&0&2\,\sqrt{-a^{2}+1}\end{array} \right] \tag{23}\]  
we may also find closed one-parametric forms of the arbitrarily normalized related matrices  
\[\Omega_{1}^{\ddagger}=\left[ \begin{array}{ccc}-2+2\,s^{2}&2\,s+1+s^{2}&-2\,s +1+s^{2}\\ -2\,\sqrt{2-2\,s^{2}}&\sqrt{2-2\,s^{2}}(s+1)&-\sqrt{2-2\,s^{2}}(-1+s)\\ -2+2\,s^{2}&1-s^{2}&1-s^{2}\end{array} \right] \tag{24}\]  
plus, similarly, matrix \(\Omega_{2}^{\ddagger}\) (which appeared, incidentally, just a bit too large for its display in print). In the next step of illustration, we will fix parameters \(s=1/2\) and \(a=1/2\), yielding a simpler formula for  
\[\Omega_{1}^{\ddagger}=\left[ \begin{array}{ccc}-3/2&9/4&1/4\\ -\sqrt{6}&3/4\,\sqrt{6}&1/4\,\sqrt{6}\\ -3/2&3/4&3/4\end{array} \right] \tag{25}\]  
as well as a more comfortably printable formula for  
\[\Omega_{2}^{\ddagger}=\left[