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Mathematik-Online problems:

Problem 81: Eigenvalues and Eigenvectors of Cyclic Matrices

A B C D E F G H I J K L M N O P Q R S T U V W X Y Z
Show that the vectors

$\displaystyle v_1=(1, 1, 1)^{\rm {t}}, \quad v_2=(1, \omega, \omega^2)^{\rm {t... ...t}}, \qquad \quad {\mbox{for}} \quad \omega={e}^{\frac{2\pi{\mathrm{i}}}{3}}, $

are eigenvectors of the cyclic matrix

$\displaystyle C=\left(\begin{array}{ccc} 0 & a & 1 \\ 1 & 0 & a \\ a & 1 & 0\end{array}\right) \ , $

and find the according eigenvalues. For which $ a$ all eigenvalues are real and for which $ a$ the eigenvalues are purely imaginary?
(Authors: Höllig/Höfert)

Solution:

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  automatisch erstellt am 12.  3. 2018