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\begin{document}
\setcounter{page}{19} \noindent\textit{Algebra liniowa -- dr
Micha\l \ G\'ora\hfill Zestaw 8. Warto\'{s}ci i wektory w\l asne}\vspace{-11pt}%
\newline\noindent\rule{\textwidth}{0.5pt}


\begin{description}
\item[Zadanie 1.] Wyznacz warto\'{s}ci i wektory w\l asne macierzy:%
\[%
\begin{tabular}
[c]{lll}%
a) $\left[
\begin{array}
[c]{cc}%
1 & 5\\
0 & 3
\end{array}
\right]  $;$\medskip$ & b) $\left[
\begin{array}
[c]{ccc}%
a & 0 & 0\\
0 & b & 0\\
0 & 0 & c
\end{array}
\right]  $; & c) $\left[
\begin{array}
[c]{rrr}%
1 & 2 & 0\\
0 & 2 & 0\\
-2 & -2 & -1
\end{array}
\right]  $;\\
d) $\left[
\begin{array}
[c]{rrr}%
1 & -1 & -1\\
1 & 1 & 0\\
3 & 0 & 1
\end{array}
\right]  $; & e) $\left[
\begin{array}
[c]{rrr}%
2 & -2 & 0\\
-2 & 1 & -2\\
0 & -2 & 0
\end{array}
\right]  $; & f) $\left[
\begin{array}
[c]{rrr}%
2 & -1 & -1\\
3 & -2 & -3\\
-1 & 1 & 2
\end{array}
\right]  .$%
\end{tabular}
\
\]


\item[Zadanie 2.] Niech $\lambda$ b\k{e}dzie warto\'{s}ci\k{a} w\l asn\k{a}
macierzy $A$. Wyznacz warto\'{s}ci w\l asne macierzy:

\begin{itemize}
\item[a)] $A^{-1}$ (je\.{z}eli istnieje),

\item[b)] $A+\alpha I$, dla $\alpha\in\mathbb{R}$,

\item[c)] $\left(  A+I\right)  ^{n}$, dla $n\in\mathbb{N}$,

\item[d)] $\left(  A+I\right)  ^{-n}$, dla $n\in\mathbb{N}$, gdzie
$B^{-n}\overset{df}{=}\left(  B^{-1}\right)  ^{n}$ (je\.{z}eli $B^{-1}$ istnieje).
\end{itemize}

\item[Zadanie 3.] Wielomian charakterystyczny macierzy $A$ ma posta\'c:
\[
\varphi_{A}(\lambda)=-\lambda^{9}+\frac{2}{9}\lambda^{8}+\lambda^{5}+\frac
{2}{9}\lambda^{4}-\lambda+\frac{2}{9}.%
\]
Wyznacz:

\begin{itemize}
\item[a)] wymiar macierzy $A$,

\item[b)] $\det(A)$,

\item[c)] $\operatorname{rank}(A)$,

\item[d)] $\operatorname*{tr}(A)$.
\end{itemize}

\item[Zadanie 4.] Dla macierz $A=\left[
\begin{array}
[c]{rrr}%
2 & 3 & 0\\
-1 & 2 & 1\\
2 & 3 & -2
\end{array}
\right]  $ oblicz:%
\[%
\begin{array}
[c]{ll}%
\text{a) }\lambda_{1}\lambda_{2}\lambda_{3}\text{,} & \text{b) }\lambda
_{1}^{2}+\lambda_{2}^{2}+\lambda_{3}^{2}\text{,}%
\end{array}
\]
gdzie $\lambda_{1},$ $\lambda_{2},$ $\lambda_{3}$ to warto\'{s}ci w\l asne
macierzy $A$.

\item[Zadanie 5.] Niech $A,B\in\mathbb{R}^{n\times n}:\det\left(  AB\right)
\neq0$. Udowodnij, \.{z}e je\.{z}eli $\lambda$ jest warto\'{s}ci\k{a}
w\l asn\k{a} macierzy $AB$, to $\lambda$ jest r\'{o}wnie\.{z}
warto\'{s}ci\k{a} w\l asn\k{a} macierzy $BA$. Czy za\l o\.{z}enie $\det\left(
AB\right)  \neq0$ mo\.{z}na os\l abi\'{c}?

\item[Zadanie 6.] Niech $w\left(  x\right)  =a_{k}x^{k}+\ldots+a_{1}x+a_{0},$
$a_{i}\in\mathbb{C}$ $\left(  i=0,\ldots,k\right)  $ oraz niech $\lambda
_{1},\ldots,\lambda_{n}\in\mathbb{C}$ b\k{e}d\k{a} warto\'{s}ciami w\l asnymi
macierzy $A\in\mathbb{C}^{n\times n}$. Wyznacz warto\'{s}ci w\l asne macierzy
\linebreak$w\left(  A\right)  =a_{k}A^{k}+\ldots+a_{1}A+a_{0}I$.

\item[Zadanie 7.] Dla podanych macierzy wyznacz ich macierze Jordana:%
\[%
\begin{tabular}
[c]{llll}%
a) $\left(
\begin{array}
[c]{cc}%
1 & 2\\
2 & 1
\end{array}
\right)  ,$ & b) $\left(
\begin{array}
[c]{cc}%
2 & 1\\
-1 & 0
\end{array}
\right)  ,$ & c) $\left(
\begin{array}
[c]{ccc}%
2 & 0 & 0\\
0 & -1 & 2\\
1 & -2 & -1
\end{array}
\right)  ,$ & d) $\left(
\begin{array}
[c]{ccc}%
1 & 2 & 1\\
0 & 2 & 0\\
1 & 0 & 1
\end{array}
\right)  .$%
\end{tabular}
\ \ \ \
\]

\pagebreak
\item[Zadanie 8.] Niech $A$ b\k{e}dzie macierz\k{a} z poprzedniego zadania
(trzy przypadki: a), b), d)). Wyznacz:

\begin{itemize}
\item[a)] $A^{2010};$

\item[b)] $I+A+A^{2}+\ldots+A^{99}$.
\end{itemize}
\end{description}

\noindent\rule{0.5\textwidth}{0.5pt}\\
\hspace{-5pt}
\noindent {\large Odpowiedzi:}

\begin{description}
\item[Zadanie 1:] $\left\{  1,\left[
\begin{array}
[c]{c}%
1\\
0
\end{array}
\right]  \right\}  ,\left\{  3,\left[
\begin{array}
[c]{c}%
5\\
2
\end{array}
\right]  \right\}  ;$

b) $\left\{  a,\left[
\begin{array}
[c]{c}%
1\\
0\\
0
\end{array}
\right]  \right\}  ,$ $\left\{  b,\left[
\begin{array}
[c]{c}%
0\\
1\\
0
\end{array}
\right]  \right\}  ,$ $\left\{  c,\left[
\begin{array}
[c]{c}%
0\\
0\\
1
\end{array}
\right]  \right\}  ;$

c) $\left\{  -1,\left[
\begin{array}
[c]{c}%
0\\
0\\
1
\end{array}
\right]  \right\}  ,$ $\left\{  1,\left[
\begin{array}
[c]{c}%
-1\\
0\\
1
\end{array}
\right]  \right\}  ,$ $\left\{  2,\left[
\begin{array}
[c]{c}%
-2\\
-1\\
2
\end{array}
\right]  \right\}  ;$

d) $\left\{  1,\left[
\begin{array}
[c]{c}%
0\\
-1\\
1
\end{array}
\right]  \right\}  ,$ $\left\{  1\pm2i,\left[
\begin{array}
[c]{c}%
\pm2i\\
1\\
3
\end{array}
\right]  \right\}  ;$

e) $\left\{  1,\left[
\begin{array}
[c]{c}%
-2\\
-1\\
2
\end{array}
\right]  \right\}  ,$ $\left\{  -2,\left[
\begin{array}
[c]{c}%
1\\
2\\
2
\end{array}
\right]  \right\}  ,$ $\left\{  4,\left[
\begin{array}
[c]{c}%
2\\
-2\\
1
\end{array}
\right]  \right\}  ;$

f) $\left\{  0,\left[
\begin{array}
[c]{c}%
-1\\
-3\\
1
\end{array}
\right]  \right\}  ,$ $\left\{  1,\left[
\begin{array}
[c]{c}%
1\\
0\\
1
\end{array}
\right]  ,\left[
\begin{array}
[c]{c}%
1\\
1\\
0
\end{array}
\right]  \right\}  $;

\item[Zadanie 2:] a) $\frac{1}{\lambda};$ b) $\lambda+\alpha;$ c) $\left(
\lambda+1\right)  ^{n};$ d) $\frac{1}{\left(  \lambda+1\right)  ^{n}};$

\item[Zadanie 3:] a) $9;$ b) $\frac{2}{9};$ c) $9$; d)$ -\frac{2}{9};$

\item[Zadanie 4:] a) $-14;$ b) $12$;

\item[Zadanie 5:] Wskaz\'{o}wka: uzasadnij, \.{z}e macierze $AB$ i $BA$ s\k{a} podobne;

\item[Zadanie 6:] $w\left(  \lambda_{1}\right)  ,\ldots,w\left(  \lambda
_{n}\right)  $;

\item[Zadanie 7:] a) $J=\left(
\begin{array}
[c]{cc}%
-1 & 0\\
0 & 3
\end{array}
\right)  ;$ b) $J=\left(
\begin{array}
[c]{cc}%
1 & 1\\
0 & 1
\end{array}
\right)  ;$ c) $J=\left(
\begin{array}
[c]{ccc}%
2 & 0 & 0\\
0 & -1-2i & 0\\
0 & 0 & -1+2i
\end{array}
\right)  ;$

d) $J=\left(
\begin{array}
[c]{ccc}%
2 & 1 & 0\\
0 & 2 & 0\\
0 & 0 & 0
\end{array}
\right)  $.

\item[Zadanie 8:] Wskaz\'{o}wka: a) $A=PJP^{-1}\Rightarrow A^{2010}%
=PJ^{2010}P^{-1}$;

b) $A=PJP^{-1}\Rightarrow I+A+A^{2}+\ldots+A^{99}=P\left(  I+J+J^{2}%
+\ldots+J^{99}\right)  P^{-1}$.
\end{description}


\end{document}