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Mathematics-Online problems:

Interactive Problem 76: Jordan Form of a Matrix

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
The real matrix

$\displaystyle A=\left(\begin{array}{rrr} -1 & -1 & 0 \\ 2 & -3 & -1 \\ 0 & 2 & -5 \end{array} \right) $

has an integral eigenvalue $ \lambda$ with algebraic multiplicity $ 3$. Find $ \lambda$:

$ \lambda =$ .

Find the Jordan canonical form $ J$ of $ A$. Start with the greatest Jordan block:

$ J= \left(\rule{0pt}{8ex}\right.$
0
0
0 0
$ \left.\rule{0pt}{8ex}\right)$ .

Find a symmetric matrix $ T$, consisting of the smallest possible integral and nonnegative entries, so that $ T^{-1}AT=J$ holds:

$ T= \left(\rule{0pt}{8ex}\right.$
$ \left.\rule{0pt}{8ex}\right)$ .

Which of the following states holds for every $ 3\times 3$-matrix $ A$ with entries in the reals? $ J_{A}$ denotes the Jordan canonical form of $ A$.

n/a
There is a symmetric matrix $ T$ with $ T^{-1}AT=J_{A}$
There is a real matrix $ T$ with $ T^{-1}AT=J_{A}$
There is a unitary matrix $ T$ with $ T^{-1}AT=J_{A}$
There is a quadratic matrix $ T$ with $ T^{-1}AT=J_{A}$

   

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