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Mathematics-Online problems:

Interactive Problem 74: Jordan Form and Matrix Powers

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
Given the real matrix

$\displaystyle A=\left(\begin{array}{rrrr} 1 & 1 & 0 & 1 \\ 2 & 2 & 0 & 0 \\ 0 & 0 & 2 & 2 \\ 0 & 0 & 1 & 1 \end{array} \right). $

a) Find the Jordan canonical form $ J$ of $ A$. Start with the smallest eigenvalue:

$ J= \left(\rule{0pt}{10ex}\right.$
0 0
0 0
0 0
0 0 0
$ \left.\rule{0pt}{10ex}\right)$ .

Complete the following matrix $ T$, so that $ T^{-1}AT=J$ holds:

$ T= \left(\rule{0pt}{10ex}\right.$
0 0
$ 3$ $ 3$
$ \left.\rule{0pt}{10ex}\right)$ .

b) Find the inverse of the transformation matrix $ T$:

$ T^{-1}= \frac{1}{54} \left(\rule{0pt}{10ex}\right.$
$ \left.\rule{0pt}{10ex}\right)$ .

Find (using $ A=TJT^{-1}$) the following matrix:

$ \big(\frac{1}{3}\big)^{999}A^{1001}= \left(\rule{0pt}{10ex}\right.$
$ \left.\rule{0pt}{10ex}\right)$ .

   
(Authors: Hertweck/Höfert)

see also:

  automatically generated: 8/11/2017