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For $x^{\prime }=Ax$ with constant matrix $A$, an eigenvector $v$ with eigenvalue $\lambda$ gives $x(t)=e^{\lambda t}v$. If $A$ has an eigenbasis, every solution is the linear combination ${\sum }_j{c}_j{e}^{{\lambda }_{j}t}{v}_j$, with constants fixed by the initial condition. For a real matrix, complex conjugate eigenpairs combine into real oscillatory solutions.

For $A=\left(\begin{array}{cc}-2& 1\\ 1& -2\end{array}\right)$, the eigenvectors $(1,1)$ and $(1,-1)$ have eigenvalues $-1,-3$. Data $x(0)=(1,0)$ giveSubstitution verifies both equations and the data. A defective matrix needs generalized eigenvectors and polynomial-exponential Jordan chains, as in the defective repeated eigenvalue in a cyclic conversion model; eigenvectors alone then do not span all solutions.^[ref]

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.