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Mathematics-Online problems:

Interactive Problem 72: Diagonal Form and Conjugated 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
Given the matrices

$\displaystyle A=\left(\begin{array}{rrr} 1 & 0 & 8 \\ 3 & 1 & 2 \\ 2 & 0 & 1 \... ... & t & 0 \\ t & 1 & 0 \\ 0 & t & 1 \end{array} \right)\quad (t\in\mathbb{R}). $

a) Inscribe the eigenvalues of $ A$ in ascending order into the diagonal of the following matrix $ D$:

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

Complete the following matrix so that $ S^{-1}AS=D$ holds:

$ S= \left(\rule{0pt}{8ex}\right.$
$ 1$ $ 1$ $ 2$
$ \left.\rule{0pt}{8ex}\right)$ .

b) Find the positive value $ t_{0}$ so that $ A$ and $ B_{t_{0}}$ are conjugated (i.e. there exists a regular matrix $ U$ with $ U^{-1}AU=B_{t_{0}}$):

$ t_{0}=$ .

As from now let be $ B=B_{t_{0}}$. Complete the following matrix $ T$ so that $ TB_{t}T^{-1}$ is a diagonal matrix for all $ t\in\mathbb{R}$ and $ TBT^{-1}=D$ is true:

$ T= \left(\rule{0pt}{8ex}\right.$
$ 1$
$ 1$
$ 1$
$ \left.\rule{0pt}{8ex}\right)$ .

c) Find a matrix $ M$ with integral entries and eigenvalues $ 1$, $ 2$ and $ 4$ so that $ M^{-1}AM=B$ holds.
Solution:

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

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