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Mathematics-Online problems:

Interactive Problem 73: 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}{rrr} 19 & -9 \\ 36 & -17 \end{array} \right). $

$ A$ has a twofold eigenvalue $ \lambda$. Find $ \lambda$:

$ \lambda=$ .

Give the Jordan canonical form $ J$ of $ A$:

$ J= \left(\rule{0pt}{5ex}\right.$
0
$ \left.\rule{0pt}{5ex}\right)$ .

Find an eigenvector $ v$ of $ A$, corresponding to the eigenvalue $ \lambda$. Find a vector$ W$ with $ (A-\lambda E)w=99v$ and define $ T=(v,w)$ (a $ 2\times 2$-matrix consisting of the column vectors $ v$ and $ w$). Give $ T^{-1}AT$:

$ T^{-1}AT= \left(\rule{0pt}{5ex}\right.$
$ \left.\rule{0pt}{5ex}\right)$ .

Find a real matrix $ B$, so that $ B^{99}=A$ holds:

$ B= \frac{1}{11} \left(\rule{0pt}{5ex}\right.$
$ \left.\rule{0pt}{5ex}\right)$ .

   
(Authors: Hertweck/Höfert)
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  automatically generated: 8/11/2017