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\begin{document}
\setcounter{page}{24} \noindent\textit{Algebra liniowa -- dr
Micha\l \ G\'ora\hfill Zestaw 11. Formy kwadratowe}\vspace{-11pt}%
\newline\noindent\rule{\textwidth}{0.5pt}

\begin{description}
\item[Zadanie 1.] Poni\.{z}sze formy kwadratowe zapisz w postaci macierzowej,
tj. $h\left(  x\right)  =x^{T}Ax$:

\begin{itemize}
\item[a)] $h\left(  x_{1},x_{2}\right)  =3x_{1}^{2}+4x_{1}x_{2}+3x_{2}^{2}$,

\item[b)] $h\left(  x_{1},x_{2},x_{3}\right)  =10x_{1}^{2}+4x_{1}x_{2}%
+x_{1}x_{3}+2x_{2}x_{3}+5x_{2}^{2}+2x_{3}^{2}$,

\item[c)] $h\left(  x_{1},x_{2},x_{3},x_{4},x_{5},x_{6}\right)  =5x_{1}%
^{2}+8x_{1}x_{2}+2x_{3}^{2}+5x_{2}^{2}+6x_{3}x_{4}+2x_{4}^{2}+x_{5}%
^{2}+12x_{5}x_{6}+x_{6}^{2}.$
\end{itemize}

\item[Zadanie 2.] Zbadaj okre\'{s}lono\'{s}\'{c} form kwadratowych:

\begin{itemize}
\item[a)] $\varphi(x_{1},x_{2})=x_{1}^{2}+3x_{1}x_{2}-x_{2}^{2}$,

\item[b)] $\varphi(x_{1},x_{2},x_{3})=x_{1}^{2}+x_{2}^{2}+2x_{1}x_{2}%
+4x_{1}x_{3}+2x_{2}x_{3}$,

\item[c)] $\varphi(x_{1},x_{2},x_{3},x_{4})=-x_{1}^{2}-x_{2}^{2}-5x_{3}%
^{2}+4x_{1}x_{3}$.
\end{itemize}

\item[Zadanie 3.] Wyznacz wszystkie warto\'{s}ci parametru $\alpha
\in\mathbb{R}$, dla kt\'{o}rych odwzorowanie

\begin{itemize}
\item[a)] $\varphi_{1}(x_{1},x_{2})=x_{1}^{2}+\alpha x_{1}x_{2}+x_{2}%
^{2}+\alpha^{2}-4$

\item[b)] $\varphi_{2}(x_{1},x_{2},x_{3})=x_{1}^{2}+2x_{1}x_{2}+\alpha
x_{1}x_{3}+x_{3}^{2}$

\item[c)] $\varphi_{3}(x_{1},x_{2},x_{3})=4x_{1}^{2}+4x_{1}x_{2}+2x_{2}%
^{2}+2\alpha x_{1}x_{3}+x_{3}^{2}$

jest form\k{a} kwadratow\k{a}, kt\'{o}ra

\begin{itemize}
\item jest dodatnio okre\'{s}lona,

\item nie jest ujemnie okre\'{s}lona,

\item jest dodatnio p\'{o}\l okre\'{s}lona.
\end{itemize}
\end{itemize}

\item[Zadanie 4.] Rozwa\.{z}my dwie formy kwadratowe%
\begin{align*}
F(x_{1},x_{2},x_{3})  &  =5x_{1}^{2}+x_{2}^{2}+2\alpha x_{1}x_{3}+2x_{1}%
x_{2},\\
G(x_{1},x_{2},x_{3})  &  =x_{1}^{2}-x_{2}^{2}-2x_{1}x_{2}-x_{3}^{2}\text{.}%
\end{align*}
Dla jakich warto\'{s}ci parametru $\alpha\in\mathbb{R}$ nier\'{o}wno\'{s}\'{c}
$F(x_{1},x_{2},x_{3})>G(x_{1},x_{2},x_{3})$ jest prawdziwa dla ka\.{z}dego
$(x_{1},x_{2},x_{3})\in\mathbb{R}^{3}\backslash\left\{  \left(  0,0,0\right)
\right\}  $?

\item[Zadanie 5.*] Niech $X$ b\k{e}dzie przestrzeni\k{a} wektorow\k{a}
wyposa\.{z}on\k{a} w iloczyn skalarny $\left\langle \cdot,\cdot\right\rangle
$. Niech $f_{1},\ldots,f_{n}\in X$. Uzasadnij r\'{o}wnowa\.{z}no\'{s}\'{c}%
\[
f_{1},\ldots,f_{n}-\text{ liniowo niezale\.{z}ne }\Leftrightarrow\det
G\neq0\text{,}%
\]
gdzie $G=\left[  g_{ij}\right]  _{i,j=1}^{n}\overset{df}{=}\left[
\left\langle f_{i},f_{j}\right\rangle \right]  _{i,j=1}^{n}$.
(\footnote{Macierz $\left[  \left\langle f_{i},f_{j}\right\rangle \right]
_{i,j=1}^{n}$ to macierz Grama.})

\item[Zadanie 6.] Niech $A=A^{T}\in\mathbb{R}^{n\times n}$ oraz niech
$x\rightarrow x^{T}Ax$ b\k{e}dzie form\k{a} kwadratow\k{a} dodatnio
okre\'{s}lon\k{a}. Poka\.{z}, \.{z}e $a_{ii}>0,$ $i=1,\ldots,n$. Czy
istniej\k{a} podobne warunki dla form ujemnie okre\'{s}lonych (p\'{o}\l okre\'{s}lonych)?

\item[Zadanie 7.] Niech $A=A^{T}\in\mathbb{R}^{n\times n}\ $b\k{e}dzie
macierz\k{a} o warto\'{s}ciach w\l asnych $\lambda_{1}\left(  A\right)
\leq\ldots\leq\lambda_{n}\left(  A\right)  $. Wyznacz wszystkie warto\'{s}ci
parametru\ $\alpha\in\mathbb{R}$, dla kt\'{o}rych forma kwadratowa
\[
x\rightarrow x^{T}\left(  A+\alpha I\right)  ^{n}x
\]
jest dodatnio okre\'{s}lona dla ka\.{z}dego $n\in\mathbb{N}$.
\end{description}

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

\begin{description}
\item[Zadanie 1:] a) $\left[
\begin{array}
[c]{c}%
x_{1}\\
x_{2}%
\end{array}
\right]  ^{T}\left[
\begin{array}
[c]{cc}%
3 & 2\\
2 & 3
\end{array}
\right]  \left[
\begin{array}
[c]{c}%
x_{1}\\
x_{2}%
\end{array}
\right]  ;$ b) $\left[
\begin{array}
[c]{c}%
x_{1}\\
x_{2}\\
x_{3}%
\end{array}
\right]  ^{T}\left[
\begin{array}
[c]{ccc}%
10 & 2 & \frac{1}{2}\\
2 & 5 & 1\\
\frac{1}{2} & 1 & 2
\end{array}
\right]  \left[
\begin{array}
[c]{c}%
x_{1}\\
x_{2}\\
x_{3}%
\end{array}
\right]  ;$

c) $\left[
\begin{array}
[c]{c}%
x_{1}\\
x_{2}\\
x_{3}\\
x_{4}\\
x_{5}\\
x_{6}%
\end{array}
\right]  ^{T}\left[
\begin{array}
[c]{cccccc}%
5 & 4 & 0 & 0 & 0 & 0\\
4 & 5 & 0 & 0 & 0 & 0\\
0 & 0 & 2 & 3 & 0 & 0\\
0 & 0 & 3 & 2 & 0 & 0\\
0 & 0 & 0 & 0 & 1 & 6\\
0 & 0 & 0 & 0 & 6 & 1
\end{array}
\right]  \left[
\begin{array}
[c]{c}%
x_{1}\\
x_{2}\\
x_{3}\\
x_{4}\\
x_{5}\\
x_{6}%
\end{array}
\right]  ;$

\item[Zadanie 2:] a) $\varphi(x_{1},x_{2})=\left(  x_{1}+\frac{3}{2}%
x_{2}\right)  ^{2}-\frac{13}{4}x_{2}^{2}$ -- nieokre\'{s}lona;

b) $\varphi(x_{1},x_{2},x_{3})=\left(  x_{1}+x_{2}+2x_{3}\right)
^{2}-4\left(  x_{3}+\frac{x_{2}}{4}\right)  ^{2}+\frac{x_{2}^{2}}{4}$ -- nieokre\'{s}lona;

c) $\varphi(x_{1},x_{2},x_{3},x_{4})=-\left(  x_{1}-2x_{3}\right)  ^{2}%
-x_{2}^{2}-x_{3}^{2}$ --\ ujemnie okre\'{s}lona;

\item[Zadanie 3:]
\[%
\begin{tabular}
[c]{|c|c|c|c|}\hline
Forma & a) & b) & c)\\\hline
jest okre\'{s}lona dodatnio & $\varnothing$ & $\varnothing$ & $\alpha
\in\left(  -\sqrt{2},\sqrt{2}\right)  $\\\hline
nie jest okre\'{s}lona ujemnie & $\alpha=\pm2$ & $\alpha\in\mathbb{R}$ &
$\alpha\in\mathbb{R}$\\\hline
jest dodatnio p\'{o}\l okre\'{s}lona & $\alpha=\pm2$ & $\varnothing$ &
$\alpha=\pm\sqrt{2}$\\\hline
\end{tabular}
\ \ \ \
\]


\item[Zadanie 4:] $\alpha\in\left(  -\sqrt{2},\sqrt{2}\right)  $;

\item[Zadanie 5*:] Zauwa\.{z}my, \.{z}e $\forall\beta=\left(  \beta_{1}%
,\ldots,\beta_{n}\right)  \in\mathbb{R}^{n}$ zachodzi:%
\[
\beta^{T}G\beta=%
%TCIMACRO{\tsum \limits_{k=1}^{n}}%
%BeginExpansion
{\textstyle\sum\limits_{k=1}^{n}}
%EndExpansion%
%TCIMACRO{\tsum \limits_{i=1}^{n}}%
%BeginExpansion
{\textstyle\sum\limits_{i=1}^{n}}
%EndExpansion
\beta_{k}\beta_{i}\left\langle f_{k},f_{i}\right\rangle =\left\langle
%TCIMACRO{\tsum \limits_{k=1}^{n}}%
%BeginExpansion
{\textstyle\sum\limits_{k=1}^{n}}
%EndExpansion
\beta_{k}f_{k},%
%TCIMACRO{\tsum \limits_{i=1}^{n}}%
%BeginExpansion
{\textstyle\sum\limits_{i=1}^{n}}
%EndExpansion
\beta_{i}f_{i}\right\rangle =\left\Vert
%TCIMACRO{\tsum \limits_{k=1}^{n}}%
%BeginExpansion
{\textstyle\sum\limits_{k=1}^{n}}
%EndExpansion
\beta_{k}f_{k}\right\Vert ^{2}\text{.}%
\]


\begin{description}
\item[$\Rightarrow$] $\beta^{T}G\beta\geq0$ oraz $\beta^{T}G\beta
=0\Leftrightarrow%
%TCIMACRO{\tsum \limits_{k=1}^{n}}%
%BeginExpansion
{\textstyle\sum\limits_{k=1}^{n}}
%EndExpansion
\beta_{k}f_{k}=0\overset{\text{lin.}}{\underset{\text{niezal.}}%
{\Longleftrightarrow}}\beta=0$. Macierz $G$ jest wi\k{e}c dodatnio
okre\'{s}lona. Zatem $\det G>0$.

\item[$\Leftarrow$] Je\.{z}eli $f_{1},\ldots,f_{n}$ s\k{a} liniowo
zale\.{z}ne, to $\exists\beta\in\mathbb{R}^{n}\backslash\left\{
\mathbf{0}\right\}  :%
%TCIMACRO{\tsum \limits_{k=1}^{n}}%
%BeginExpansion
{\textstyle\sum\limits_{k=1}^{n}}
%EndExpansion
\beta_{k}f_{k}=0.$ \newline W\'{o}wczas:%
\[
\forall i=1,\ldots,n:0=\left\langle f_{i},%
%TCIMACRO{\tsum \limits_{k=1}^{n}}%
%BeginExpansion
{\textstyle\sum\limits_{k=1}^{n}}
%EndExpansion
\beta_{k}f_{k}\right\rangle =%
%TCIMACRO{\tsum \limits_{k=1}^{n}}%
%BeginExpansion
{\textstyle\sum\limits_{k=1}^{n}}
%EndExpansion
\left\langle f_{i},f_{k}\right\rangle \beta_{k}\Rightarrow G\beta
=0\text{.}%
\]
Oznacza to, \.ze jednorodne r\'ownanie liniowe $G\beta=0$ o macierzy kwadratowej $G$ posiada rozwi\k{a}zanie niezerowe; tym samym $\det G=0$.
\end{description}

\item[Zadanie 6:] Wskaz\'{o}wka: $x\thicksim e_{i}$ $\left(  i=1,\ldots
,n\right)  $, gdzie $\left\{  e_{i}\right\}  _{i=1}^{n}$-- baza kanoniczna
$\mathbb{R}^{n}$.

Dla ujemnie okre\'{s}lonych: $a_{ii}<0$ $\left(  i=1,\ldots,n\right)  ;$ dla
p\'{o}\l okre\'{s}lonych dodatnio: $a_{ii}\geqslant0$ $\left(  i=1,\ldots
,n\right)  ;$ dla p\'{o}\l okre\'{s}lonych ujemnie: $a_{ii}\leqslant0$
$\left(  i=1,\ldots,n\right)  ;$

\item[Zadanie 7:] $\alpha>-\lambda_{1}\left(  A\right)  .$
\end{description}


\end{document}