Chapter 6.4, Problem 11P

In this problem, we indicate how to show that ${\text{x}}_1\left(t\right)$ and ${\text{x}}_2\left(t\right)$, as given by Eqs.(4), are linearly independent. Let ${\lambda }_1=\mu +iv$ and ${\bar{\lambda }}_1=\mu -iv$ be a pair of conjugate eigenvalues of the coefficient matrix $\text{A}$ of Eq.(1); let ${\text{v}}_1=\text{a}+i\text{b}$ and $\overline{{\text{v}}_1}=\text{a-}i\text{b}$ be the corresponding eigenvectors. Recall from Corollary 6.3.2 that if the eigenvalues of a real matrix are real and distinct, then the eigenvectors are linearly independent. The proof of Corollary 6.3.2 holds equally well if some of the eigenvalues are complex. Thus if $v\ne 0$ so that ${\lambda }_1\ne \overline{{\lambda }_1}$, then ${\text{v}}_{\text{1}}$ and $\overline{{\text{v}}_1}$ are linearly independent.

(a) First we show that $\text{a}$ and $\text{b}$ are linearly independent. Consider the equation $c_1{\text{a+c}}_2\text{b=0}$. Express $\text{a}$ and $\text{b}$ in terms of ${\text{v}}_{\text{1}}$ and $\overline{{\text{v}}_1}$, and then show that

$\left(c_1-i{c}_2\right){\text{v}}_1+\left(c_1+i{c}_2\right)\overline{{\text{v}}_1}=0$.

(b) Show that $c_1-i{c}_2=0$ and $c_1+i{c}_2=0$ and then that $c_1=0$ and $c_2=0$. Consequently, $\text{a}$ and $\text{b}$ are linearly independent.

(c) To show that ${\text{x}}_1\left(t\right)$ and ${\text{x}}_2\left(t\right)$ are linearly independent, consider the equation $c_1{\text{x}}_1\left(t_0\right)+c_2{\text{x}}_2\left(t_0\right)=0$, where $t_0$ is an arbitrary point. Rewrite this equation in terms of $\text{a}$ and $\text{b}$, and the proceed as in part (b), to show that $c_1=0$ and $c_2=0$. Hence ${\text{x}}_1\left(t\right)$ and ${\text{x}}_2\left(t\right)$ are linearly independent at the point $t_0$. Therefore they are linearly independent at every point and on every interval.