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Mathematik-Online problems:

Problem 70: Scalar Product, Complex/Unitary 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

a)
Analyse if

$\displaystyle \left<x, y\right>:=\sum_{k=1}^n x_k\,y_k, \qquad \forall x=(x_1,\,\ldots , x_n),\ y=(y_1,\,\ldots , y_n)\in\mathbb{C}^n, $

defines a scalar product in $ \mathbb{C}^n$.
b)
Find a symmetric complex matrix $ A$ that is not diagonalisable.
c)
Find an unitary matrix $ B$ with $ B^{-1}\neq B^{\rm {t}}$.

(Authors: Kimmerle/Höfert)

see also:

  automatisch erstellt am 12.  8. 2008