 Solution

If $K^2\ne W$, the three eigenvalues are distinct, so a basis of eigenvectors givesThe transformed coordinates then have the displayed exponential form. If $K^2<W$, this is a complex eigenbasis; real concentrations are recovered by conjugate-paired coefficients. A real basis instead gives a decaying two-dimensional rotation block with decay rate $-K$ and angular frequency $\sqrt{W-K^2}$.

The proposed diagonalization is not valid for every positive choice of rates. For example $ϵ=\delta =1$, $\eta =4$ gives $K=-3$, $W=9$, with repeated eigenvalue $-3$ and only one corresponding eigenvector. More generally the defective repeated eigenvalue in a cyclic conversion model occurs when $K^2=W$. A Jordan basis then givesThe extra factor $t$ cannot be represented by two independent pure exponential coordinates in a diagonal basis.