Chapter 3.P1, Problem 2P

Estimating Eigenvalues and Eigenvectors of $K$ from Transient Concentration Data. Denote by ${\text{x}}^{\text{*}}\left(t_k\right)=x_1^{*}\left(t_k\right)i+x_2^{*}\left(t_k\right)j,k=1,2,3,\dots$ measurements of the concentrations in each of the compartments. We assume that the eigenvalues of $K$ satisfy ${\lambda }_2<{\lambda }_1<0$. Denote the eigenvectors of ${\lambda }_1$ and ${\lambda }_2$ by

${\text{V}}_1=\left(\begin{array}{c}{v}_{11}\\ v_{21}\end{array}\right)$ and ${\text{V}}_2=\left(\begin{array}{c}{v}_{12}\\ v_{22}\end{array}\right)$,

respectively. The solution of Eq. (1) can be expressed as

$\text{X}\left(t\right)=\alpha e^{{\lambda }_{1}t}{\text{v}}_1+\beta e^{{\lambda }_{2}t}{\text{v}}_2$, (i)

where $\alpha$ and $\beta$, assumed to be nonzero, depend on initial conditions. From Eq. (i), we note that

$\text{x}\left(t\right)=e^{{\lambda }_{1}t}[\alpha {\text{v}}_1+\beta e^{\left({\lambda }_2-{\lambda }_1\right)t}{\text{v}}_2]~\alpha e^{{\lambda }_{1}t}{\text{v}}_1$ if $e^{\left({\lambda }_2-{\lambda }_1\right)t}~0$.

(a) For values of $t$ such that $e^{\left({\lambda }_2-{\lambda }_1\right)t}~0$, explain why the graphs of $\ln x_1\left(t\right)$ and $\ln x_2\left(t\right)$ should be approximately straight lines with slopes equal to ${\lambda }_1$ and intercepts equal to $\ln \alpha v_{11}$ and $\ln \alpha v_{21}$, respectively. Thus estimates of ${\lambda }_1$, $\alpha v_{11}$, and $\alpha v_{21}$ may be obtained by fitting straight lines to the data $\ln x_1^{*}\left(t_n\right)$ and $\ln x_2^{*}\left(t_n\right)$ corresponding to values of $t_n$, where graphs of the logarithms of the data are approximately linear, as shown in Figure 3.P.2.

(b) Given that both components of the data $x^{*}\left(t_n\right)$ are accurately represented by a sum of exponential functions of the form (i), explain how to find estimates of ${\lambda }_2$, $\beta v_{12}$, and $\beta v_{22}$ using the residual data ${\text{x}}_r^{*}\left(t_n\right)={\text{x}}^{*}\left(t_n\right)-{\hat{v}}^{\left(\alpha \right)}{e}^{{\lambda }_1{t}_n}$, where estimates of ${\lambda }_1$ and $\alpha {\text{v}}_1$ are denoted by ${\hat{\lambda }}_1$ and ${\hat{v}}^{\left(\alpha \right)}$, respectively.