Chapter 3.P1, Problem 4P

Given estimates ${\hat{K}}_{ij}$ of the entries of $K$ and estimates ${\hat{\lambda }}_1$ and ${\hat{\lambda }}_2$ of the eigenvalues of $K$, show how to obtain an estimate ${\hat{K}}_{01}$ of $K_{01}$ using the relations in Problem 1(b).

Assume that all the rate constants in Eq. (1) are positive.

(a) Show that the eigenvalues of the matrix $K$ are real, distinct, and negative.

Hint: Show that the discriminant of the characteristic polynomial of $K$ is positive.

(b) If ${\lambda }_1$ and ${\lambda }_2$ are the eigenvalues of $K$, show that ${\lambda }_1+{\lambda }_2=-\left(k_{01}+k_{12}+k_{21}\right)$ and ${\lambda }_1{\lambda }_2=k_{12}{k}_{01}$.