= Solution
If $K^2\ne W$, the three <eigenvalues> are distinct, so a basis of <eigenvectors> gives
$$
\boxed{M'=\operatorname{diag}(-\rho_1,-\rho_2,0),\qquad
\rho_1=-K-\sqrt{K^2-W},\quad\rho_2=-K+\sqrt{K^2-W}.}
$$
The 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 \b[not valid for every positive choice of rates]. For example $\epsilon=\delta=1$, $\eta=4$ 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 gives
$$
M'=\begin{pmatrix}K&1&0\\0&K&0\\0&0&0\end{pmatrix},\qquad
x'(t)=\begin{pmatrix}(c_1+c_2t)e^{Kt}\\c_2e^{Kt}\\c_3\end{pmatrix}.
$$
The extra factor $t$ cannot be represented by two independent pure exponential coordinates in a diagonal basis.
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