Science:Math Exam Resources/Courses/MATH307/December 2010/Question 05 (c)/Solution 1

For real matrices, just as complex eigenvalues, also complex eigenvectors come in conjugate pairs. Hence

$\displaystyle v_{2}={\overline {v_{3}}}$

Hence v₂ + v₃ is a real vector and therefore we ensure that the solution vector [c, b, a]^T in part (b) is real by choosing real coefficients s and t and setting s = t:

$\left\{{\begin{bmatrix}c\\b\\a\end{bmatrix}}{\text{ such that }}{\begin{bmatrix}c\\b\\a\end{bmatrix}}=s(v_{2}+v_{3}){\text{ for some }}s\in \mathbb {R} \right\}$