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Mathematik-Online problems:

Problem 56: Complex Conjugated Eigenvalues and Eigenvectors

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
Let $ A$ be a matrix in $ \mathbb{C}^{n\times n}$.
a)
Show: If all entries of $ A$ are real and $ \lambda\in\mathbb{C}$ is an eigenvalue of $ A$, than $ \overline{\lambda}$ is eigenvalue of $ A$, too. Which relation exists between the eigenvectors corresponding to $ \lambda$ and $ \overline{\lambda}$?
b)
Assume $ A^4=E$. Which complex numbers are possible eigenvalues of $ A$?
c)
Assume $ A^4+A^2+E = 0$. Is it possible for $ A$ to have real eigenvalues?

(Authors: Kimmerle/Roggenkamp/Höfert)

see also:

[Solutions]
  automatisch erstellt am 14. 10. 2004