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\QTR{small}{Matematick\U{e1} anal\U{fd}za III online - Komplexn\U{e9} \U{10d}\U{ed}sla a funkcie komplexnej premennej - Komplexn\U{e9} \U{10d}\U{ed}sla a algebraick\U{e9} oper\U{e1}cie s nimi\dotfill \thepage }}
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\newtheorem{theorem}{Theorem}
\newtheorem{acknowledgement}[theorem]{Acknowledgement}
\newtheorem{algorithm}[theorem]{Algorithm}
\newtheorem{axiom}[theorem]{Axiom}
\newtheorem{case}[theorem]{Case}
\newtheorem{claim}[theorem]{Claim}
\newtheorem{conclusion}[theorem]{Conclusion}
\newtheorem{condition}[theorem]{Condition}
\newtheorem{conjecture}[theorem]{Conjecture}
\newtheorem{corollary}[theorem]{Corollary}
\newtheorem{criterion}[theorem]{Criterion}
\newtheorem{definition}[theorem]{Definition}
\newtheorem{example}[theorem]{Example}
\newtheorem{exercise}[theorem]{Exercise}
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{notation}[theorem]{Notation}
\newtheorem{problem}[theorem]{Problem}
\newtheorem{proposition}[theorem]{Proposition}
\newtheorem{remark}[theorem]{Remark}
\newtheorem{solution}[theorem]{Solution}
\newtheorem{summary}[theorem]{Summary}
\newenvironment{proof}[1][Proof]{\textbf{#1.} }{\ \rule{0.5em}{0.5em}}
\input{tcilatex}
\begin{document}
\author{A. U. Thor}
\title{Lab Report}
\date{The Date }
\maketitle
\begin{abstract}
A Laboratory report created with Scientific Notebook
\end{abstract}
\section{Matice}
\begin{center}
\begin{tabular}{|c|c|c|c|c|c|}
\hline
\textbf{%
%TCIMACRO{\hyperref{Obsah}{}{}{aindex.tex}}%
%BeginExpansion
\msihyperref{Obsah}{}{}{aindex.tex}%
%EndExpansion
} & \textbf{%
%TCIMACRO{\hyperref{Obsah kapitoly}{}{}{A3.tex}}%
%BeginExpansion
\msihyperref{Obsah kapitoly}{}{}{A3.tex}%
%EndExpansion
} & \textbf{%
%TCIMACRO{\hyperref{Nasleduj\'{u}ca str\'{a}nka}{}{}{A32.tex}}%
%BeginExpansion
\msihyperref{Nasleduj\'{u}ca str\'{a}nka}{}{}{A32.tex}%
%EndExpansion
} & \textbf{%
%TCIMACRO{\hyperref{Ot\'{a}zky}{}{}{OA3.tex}}%
%BeginExpansion
\msihyperref{Ot\'{a}zky}{}{}{OA3.tex}%
%EndExpansion
} & \textbf{%
%TCIMACRO{\hyperref{\textbf{Cvi\v{c}enia}}{}{}{CA3.tex}}%
%BeginExpansion
\msihyperref{\textbf{Cvi\v{c}enia}}{}{}{CA3.tex}%
%EndExpansion
} & \textbf{%
%TCIMACRO{\hyperref{Index}{}{}{INDA.tex}}%
%BeginExpansion
\msihyperref{Index}{}{}{INDA.tex}%
%EndExpansion
} \\ \hline
\end{tabular}
\end{center}
\subsection{Hodnos\v{t} matice}
\begin{theorem}
Nech~matica~$\mathbf{B}$~vznikne~z~matice~$\mathbf{A}$%
~jednou~ERO~resp.~ESO.~Potom~plat\'{\i}
1. ak s\'{u} riadky matice $\mathbf{A}$ line\'{a}rne nez\'{a}visl\'{e}, tak s%
\'{u} line\'{a}rne nez\'{a}visl\'{e} aj riadky matice $\mathbf{B}$.
2. ak s\'{u} riadky matice $\mathbf{A}$ line\'{a}rne z\'{a}visl\'{e}, tak s%
\'{u} line\'{a}rne z\'{a}visl\'{e} aj riadky matice $\mathbf{B}$.
3. maxim\'{a}lny po\v{c}et line\'{a}rne nez\'{a}visl\'{y}ch riadkov v
maticiach $\mathbf{A},\,\mathbf{B}$ je rovnak\'{y}.
\end{theorem}
\begin{tabular}{|l|}
\hline
\textbf{%
%TCIMACRO{\hyperref{D\^{o}kaz}{}{}{DA311.tex}}%
%BeginExpansion
\msihyperref{D\^{o}kaz}{}{}{DA311.tex}%
%EndExpansion
} \\ \hline
\end{tabular}%
\textbf{\label{1} }
\begin{theorem}
Nech~matica~$\mathbf{B}$~vznikne~z~matice~$\mathbf{A}$%
~jednou~ERO~resp.~ESO.~Potom~plat\'{\i}:
1. St\'{l}pce matice $\mathbf{A}$ s\'{u} line\'{a}rne nez\'{a}visl\'{e} pr%
\'{a}ve vtedy, ke\v{d} s\'{u} line\'{a}rne nez\'{a}visl\'{e} st\'{l}pce
matice $\mathbf{B}$.
2. Maxim\'{a}lny po\v{c}et line\'{a}rne nez\'{a}visl\'{y}ch st\'{l}pcov v
maticiach $\mathbf{A}$, $\mathbf{B}$ je rovnak\'{y}.
\end{theorem}
\begin{tabular}[b]{|c|}
\hline
\textbf{D\^{o}kaz} \\ \hline
\end{tabular}
je podobn\'{y} ako v predch\'{a}dzaj\'{u}cej vete.\newline
\begin{theorem}
Nenulov\'{e} riadky stup\v{n}ovitej matice s\'{u} lin\'{a}rne nez\'{a}visl%
\'{e}.
\end{theorem}
\begin{tabular}{|c|}
\hline
\textbf{\textbf{%
%TCIMACRO{\hyperref{\textbf{\textbf{D\^{o}kaz}}}{}{}{DA312.tex}}%
%BeginExpansion
\msihyperref{\textbf{\textbf{D\^{o}kaz}}}{}{}{DA312.tex}%
%EndExpansion
}} \\ \hline
\end{tabular}%
\textbf{\label{2}}
\begin{theorem}
V ka\v{z}dej matici sa maxim\'{a}lny po\v{c}et line\'{a}rne nez\'{a}visl\'{y}%
ch riadkov rovn\'{a} maxim\'{a}lnemu po\v{c}tu line\'{a}rne nez\'{a}visl\'{y}%
ch st\'{l}pcov.
\end{theorem}
\begin{tabular}{|c|}
\hline
%TCIMACRO{\hyperref{\textbf{D\^{o}kaz}}{}{}{DA313.tex} }%
%BeginExpansion
\msihyperref{\textbf{D\^{o}kaz}}{}{}{DA313.tex}
%EndExpansion
\\ \hline
\end{tabular}%
\label{3}
\begin{definition}
\textsl{Hodnos\v{t}ou matice} $\mathbf{A}$ naz\'{y}vame maxim\'{a}lny po\v{c}%
et line\'{a}rne nez\'{a}visl\'{y}ch riadkov matice $\mathbf{A}$ a ozna\v{c}%
ujeme ju $h(\mathbf{A})$.
\end{definition}
\begin{theorem}
Nech~$\mathbf{A},~\mathbf{B}$~s\'{u}~matice~typu~$m\times n$,~potom
1. $h(\mathbf{A})\leq \min \{m,~n\},$
2. ak~$\mathbf{A}\sim \mathbf{B,}$~tak~$h(\mathbf{A})=h(\mathbf{B}),$
3. $h(\mathbf{A}^{T})=h(\mathbf{A}).$
\end{theorem}
\begin{tabular}{|l|}
\hline
\textbf{%
%TCIMACRO{\hyperref{D\^{o}kaz}{}{}{DA314.tex}}%
%BeginExpansion
\msihyperref{D\^{o}kaz}{}{}{DA314.tex}%
%EndExpansion
} \\ \hline
\end{tabular}%
\label{4}
\begin{example}
Zistite,~\v{c}i~s\'{u}~line\'{a}rne~z\'{a}visl\'{e}~alebo~nez\'{a}visl\'{e}%
~vektory
a) $(1,2,0),(2,1.2),(4,0,1).$
b) $(2,0,3,4),(-3,1,2,1),(1,1,8,9).$
\end{example}
\begin{solution}
Vektory~zap\'{\i}\v{s}eme~do~matice~ako~riadky~a~ur\v{c}\'{\i}me~jej~hodnos%
\v{t}.\newline
a)%
\begin{equation*}
\left(
\begin{array}{ccc}
1 & 2 & 0 \\
2 & 1 & 2 \\
4 & 0 & 1%
\end{array}%
\right) \overset{\mathbf{r}_{3}:=\mathbf{r}_{3}-2\mathbf{r}_{2}}{\overset{%
\mathbf{r}_{2}:=\mathbf{r}_{2}-2\mathbf{r}_{1}}{\sim }}\left(
\begin{array}{ccc}
1 & 2 & 0 \\
0 & -3 & 2 \\
0 & -2 & -3%
\end{array}%
\right) \overset{\mathbf{r}_{2}:=\mathbf{r}_{2}-\mathbf{r}_{3}}{\overset{%
\mathbf{r}_{3}:=\mathbf{r}_{3}-2\mathbf{r}_{2}}{\sim }}\left(
\begin{array}{ccc}
1 & 2 & 0 \\
0 & -1 & 5 \\
0 & 0 & -13%
\end{array}%
\right)
\end{equation*}%
\newline
Hodnos\v{t}~matice~je~$3,$~preto~vektory~(v\v{s}etky~tri)~s\'{u}~line\'{a}%
rne~nez\'{a}visl\'{e}.
b)\newline
\begin{equation*}
\quad \left(
\begin{array}{cccc}
2 & 0 & 3 & 4 \\
-3 & 1 & 2 & 1 \\
1 & 1 & 8 & 9%
\end{array}%
\right) \overset{\mathbf{r}_{3}:=\mathbf{r}_{3}-2\mathbf{r}_{1}-\mathbf{r}%
_{2}}{\sim }\left(
\begin{array}{cccc}
2 & 0 & 3 & 4 \\
-3 & 1 & 2 & 1 \\
0 & 0 & 0 & 0%
\end{array}%
\right) =\mathbf{B}
\end{equation*}%
\newline
Matica $\mathbf{B}$ m\'{a} dva nenulov\'{e} riadky, jej hodnos\v{t} je
najviac dva, a preto dan\'{e} vektory (s\'{u} a\v{z} tri) s\'{u} line\'{a}%
rne~z\'{a}visl\'{e}. $\ \square $
\end{solution}
\begin{example}
Ur\v{c}te hodnos\v{t} matice
\begin{equation*}
\mathbf{A}=\left(
\begin{array}{rrrr}
3 & a & 10 & 1 \\
2 & -1 & a & 3 \\
5 & 10 & 30 & -5%
\end{array}%
\right)
\end{equation*}
v z\'{a}vislosti od parametra $a\in \mathbf{R}$.
\end{example}
\begin{solution}
\newline
\begin{equation*}
\mathbf{A}\overset{\mathbf{s}_{1}:=:\mathbf{s}_{4}}{\overset{\mathbf{s}%
_{3}:=:\mathbf{s}_{4}}{\sim }}\left(
\begin{array}{cccc}
1 & 3 & 10 & a \\
3 & 2 & a & -1 \\
-5 & 5 & 30 & 10%
\end{array}%
\right) \overset{\mathbf{r}_{2}:=\mathbf{r}_{2}-3\mathbf{r}_{1}}{\overset{%
\mathbf{r}_{3}:=\frac{1}{5}\mathbf{r}_{3}+\mathbf{r}_{1}}{\sim }}\left(
\begin{array}{cccc}
1 & 3 & 10 & a \\
0 & -7 & a-30 & -1-3a \\
0 & 4 & 16 & 2+a%
\end{array}%
\right) \overset{\mathbf{r}_{2}:=:\mathbf{r}_{3}}{\overset{\mathbf{r}_{3}:=4%
\mathbf{r}_{3}+7\mathbf{r}_{2}}{\sim }}\left(
\begin{array}{cccc}
1 & 3 & 10 & a \\
0 & 4 & 16 & 2+a \\
0 & 0 & 4a-8 & 10-5a%
\end{array}%
\right) .
\end{equation*}%
\newline
Maticu $\mathbf{A}$ sme upravili na stup\v{n}ovit\'{u}, ktor\'{a} m\'{a} pre
$a=2$ dva, pre $a\neq 2$ tri nenulov\'{e} riadky. Preto ak ~$a=2$,~tak~$h(%
\mathbf{A})=2,$\newline
ak~$a\neq 2,\,$tak~$h(\mathbf{A})=3$. $\square $
\end{solution}
\begin{definition}
Matica typu $n\times n$ sa naz\'{y}va \textsl{\v{s}tvorcov\'{a} matica stup%
\v{n}a }$n$. \v{S}tvorcov\'{a}~matica~sa~naz\'{y}va
\emph{diagon\'{a}lna},~ak~v\v{s}etky~prvky~mimo~hlavnej~diagon\'{a}%
ly~sa~rovnaj\'{u}~nule,
\emph{jednotkov\'{a}},~ak~je~diagon\'{a}lna~a~v\v{s}%
etky~prvky~na~hlavnej~diagon\'{a}le~s\'{u}~jednotky(ozna\v{c}ujeme~ju~$%
\mathbf{I}$~alebo~$\mathbf{I}_{n}$),
\emph{horn\'{a}~trojuholn\'{\i}kov\'{a}},~ak~v\v{s}%
etky~prvky~pod~hlavnou~diagon\'{a}lou~s\'{u}~nuly,
\emph{doln\'{a} trojuholn\'{\i}kov\'{a}}, ak v\v{s}etky prvky nad hlavnou
diagon\'{a}lou s\'{u} nuly.
\end{definition}
\begin{example}
Matica
\begin{equation*}
\left(
\begin{array}{ccc}
2, & 0, & 0 \\
0, & 0, & 0 \\
0, & 0, & 3%
\end{array}%
\right)
\end{equation*}%
je diagon\'{a}lna matica, matica%
\begin{equation*}
\mathbf{I}_{3}=\left(
\begin{array}{ccc}
1, & 0, & 0 \\
0, & 1, & 0 \\
0, & 0, & 1%
\end{array}%
\right)
\end{equation*}%
je jednotkov\'{a} matica,
\begin{equation*}
\left(
\begin{array}{cccc}
2, & -1, & 0, & 3 \\
0, & 0, & 1, & 5 \\
0, & 0, & 3, & 0 \\
0, & 0, & 0, & 1%
\end{array}%
\right)
\end{equation*}%
je horn\'{a} trojuholn\'{\i}kov\'{a} matica, matica%
\begin{equation*}
\left(
\begin{array}{cccc}
2, & 0, & 0, & 0 \\
-1, & 0, & 0, & 0 \\
0, & 1, & 3, & 0 \\
3, & 5, & 0, & 1%
\end{array}%
\right)
\end{equation*}
~je~doln\'{a}~trojuholn\'{\i}kov\'{a}~matica. $\square $
\end{example}
\begin{definition}
\v{S}tvorcov\'{a}~matica~$\mathbf{A}$~stup\v{n}a~$n$~sa~naz\'{y}va\newline
\textit{regul\'{a}rna},~ak~$h(\mathbf{A})=n$,\newline
\textit{singul\'{a}rna}, ak $h(\mathbf{A})